Abstract group 1728.47343: $D_4:S_3^3$

• Label: 1728.47343
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $17$
• Trans deg.: $48$
• Rank: $5$

Group information

Description:	$D_4:S_3^3$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^3.C_2^5.C_2^5$, of order \(27648\)\(\medspace = 2^{10} \cdot 3^{3} \)
Composition factors:	$C_2$ x 6, $C_3$ x 3
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	3	4	6	12
Elements	1	287	26	224	550	640	1728
Conjugacy classes	1	19	7	20	45	43	135
Divisions	1	19	7	16	45	35	123
Autjugacy classes	1	10	5	8	20	16	60

Dimension	1	2	4	8	16
Irr. complex chars.	32	56	36	10	1	135
Irr. rational chars.	32	48	28	12	3	123

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$48$
Rank:	$5$
Inequivalent generating 5-tuples:	$159989760000$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	16	16	16
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{6}=d^{6}=e^{12}=[a,b]=[a,c]=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $17$ $\langle(1,2)(3,5)(4,6)(7,8)(10,11), (1,3)(2,5)(4,7)(6,8)(12,13)(14,17)(15,16), (1,3) \!\cdots\! \rangle$
Direct product:	$S_3$ $\, \times\, $ $(D_4:S_3^2)$
Semidirect product:	$D_4$ $\,\rtimes\,$ $S_3^3$	$(D_6:D_6)$ $\,\rtimes\,$ $D_6$	$(D_6.D_6)$ $\,\rtimes\,$ $D_6$	$(D_6.D_6)$ $\,\rtimes\,$ $D_6$	all 148
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_6$ . $D_6^2$ (3)	$C_{12}$ . $D_6^2$ (2)	$C_2$ . $(S_3\times D_6^2)$	$C_4$ . $(C_2\times S_3^3)$	all 52

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2}^{5} $
Schur multiplier:	$C_{2}^{9}$
Commutator length:	$1$

Subgroups

There are 32068 subgroups in 4178 conjugacy classes, 654 normal (74 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $S_3\times D_6^2$
Commutator:	$G' \simeq$ $C_3^2\times C_6$	$G/G' \simeq$ $C_2^5$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $S_3\times D_6^2$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times C_3^3$	$G/\operatorname{Fit} \simeq$ $C_2^3$
Radical:	$R \simeq$ $D_4:S_3^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^5$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4:C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 1728: $D_4:S_3^3$

Order 864: $C_6.D_6^2$ x 4, $C_6.D_6^2$ x 4, $C_6.D_6^2$ x 2, $C_6.D_6^2$ x 2, $C_2^2:S_3^3$ x 2, $C_6.D_6^2$ x 2, $C_6.D_6^2$ x 2, $C_6.D_6^2$ x 2, $C_4.S_3^3$ x 2, $D_{12}:S_3^2$ x 2, $C_6.D_6^2$, $C_6.D_6^2$, $C_6.D_6^2$, $C_4\times S_3^3$, $D_{12}:S_3^2$, $D_{12}:S_3^2$, $C_4.S_3^3$

Order 576: $C_4.D_6^2$ x 2, $C_4.D_6^2$

Order 432: $C_2.S_3^3$ x 17, $C_6^2.D_6$ x 8, $C_2.S_3^3$ x 8, $C_6^2:D_6$ x 6, $C_6^2.D_6$ x 4, $C_6^2.D_6$ x 4, $C_6^2.D_6$ x 4, $C_6^2.D_6$ x 4, $C_6^2:D_6$ x 4, $C_6^2.D_6$ x 4, $C_6^2.D_6$ x 4, $C_2.S_3^3$ x 4, $C_2.S_3^3$ x 4, $C_2.S_3^3$ x 4, $C_2.S_3^3$ x 4, $D_6:S_3^2$ x 4, $D_6:S_3^2$ x 4, $C_{12}:S_3^2$ x 3, $C_{12}\times S_3^2$ x 3, $C_2.S_3^3$ x 3, $C_3:D_6^2$ x 2, $C_6^2.D_6$ x 2, $C_{12}.C_6^2$ x 2, $C_6^2:D_6$ x 2, $C_6^2.D_6$ x 2, $C_{12}.S_3^2$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_{12}.S_3^2$ x 2, $C_{12}.S_3^2$ x 2, $C_{12}.S_3^2$ x 2, $C_6^2:D_6$ x 2, $C_6^2.D_6$ x 2, $C_{12}.S_3^2$ x 2, $C_{12}.S_3^2$ x 2, $C_{12}.S_3^2$ x 2, $C_{12}.S_3^2$ x 2, $C_2.S_3^3$ x 2, $C_2.S_3^3$ x 2, $C_2.S_3^3$ x 2, $D_6:S_3^2$ x 2, $D_6:S_3^2$ x 2, $C_2\times S_3^3$, $C_6^2:D_6$, $C_6^2:D_6$, $C_{12}:C_6^2$, $C_{12}:S_3^2$, $C_{12}.S_3^2$, $C_{12}:S_3^2$, $C_{12}:S_3^2$, $C_{12}:S_3^2$, $C_{12}.S_3^2$, $C_2.S_3^3$

Order 288: $D_4:S_3^2$ x 24, $D_{12}:D_6$ x 12, $D_{12}:D_6$ x 12, $D_4\times S_3^2$ x 7, $C_2.D_6^2$ x 7, $D_{12}:D_6$ x 6, $C_6^2.C_2^3$ x 4, $C_2.D_6^2$ x 4, $D_{12}:D_6$ x 4, $C_6^2.C_2^3$ x 2, $D_6^2:C_2$ x 2, $C_2.D_6^2$ x 2, $D_{12}:D_6$ x 2, $Q_8\times S_3^2$ x 2, $C_2.D_6^2$ x 2, $C_6^2.C_2^3$ x 2, $C_2.D_6^2$, $C_6^2:C_2^3$

Order 216: $C_6.S_3^2$ x 18, $C_6:S_3^2$ x 17, $C_6.S_3^2$ x 15, $C_6.S_3^2$ x 9, $C_6.S_3^2$ x 9, $C_3^2:D_{12}$ x 8, $C_6^2:C_6$ x 6, $C_6^2:C_6$ x 6, $C_3^2:D_{12}$ x 6, $C_3^2:D_{12}$ x 6, $C_3^3:Q_8$ x 6, $S_3^3$ x 4, $C_6^2.C_6$ x 4, $C_6.C_6^2$ x 4, $C_3^2:D_{12}$ x 4, $C_3^3:Q_8$ x 4, $C_3^3:D_4$ x 4, $C_6\times S_3^2$ x 3, $C_3:S_3\times C_{12}$ x 3, $C_6.C_6^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6^2:S_3$ x 2, $C_6^2:C_6$ x 2, $S_3\times C_6^2$ x 2, $C_6^2:S_3$ x 2, $C_3^3:Q_8$ x 2, $C_3^3:Q_8$ x 2, $C_3^3:Q_8$ x 2, $C_3^3:D_4$ x 2, $C_6:S_3^2$, $D_4\times C_3^3$, $C_3^2:D_{12}$, $C_4\times C_3^2:S_3$, $C_3^2:D_{12}$, $C_3^2\times D_{12}$

Order 192: $D_{12}:C_2^3$ x 3

Order 144: $D_6.D_6$ x 64, $C_4\times S_3^2$ x 55, $D_6:D_6$ x 34, $C_2^2.S_3^2$ x 33, $C_{12}.D_6$ x 32, $C_{12}.D_6$ x 32, $D_6:D_6$ x 28, $C_{12}:D_6$ x 18, $C_{12}.D_6$ x 16, $C_{12}:D_6$ x 14, $C_{12}\times D_6$ x 14, $D_6.D_6$ x 14, $S_3\times D_{12}$ x 14, $D_{12}:C_6$ x 12, $D_6.D_6$ x 12, $C_{12}.D_6$ x 12, $D_{12}:S_3$ x 12, $D_{12}:S_3$ x 12, $C_{12}.D_6$ x 12, $D_6^2$ x 9, $C_{12}:D_6$ x 7, $C_{12}:D_6$ x 7, $C_{12}.D_6$ x 6, $C_{12}.D_6$ x 6, $C_6^2:C_2^2$ x 4, $C_6^2:C_4$ x 4, $D_{12}:C_6$ x 4, $C_{12}.D_6$ x 4, $C_6:D_{12}$ x 4, $C_6^2:C_2^2$ x 2, $C_4.C_6^2$ x 2, $C_6^2:C_2^2$ x 2, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_2^2.S_3^2$ x 2, $C_4:C_6^2$, $C_6:D_{12}$

Order 108: $C_3:S_3^2$ x 20, $C_3^2:D_6$ x 9, $C_3^2\times D_6$ x 9, $C_3^2:D_6$ x 7, $C_3^2:C_{12}$ x 7, $C_3^2:C_{12}$ x 7, $C_3\times S_3^2$ x 6, $C_3\times C_6^2$ x 2, $C_3:S_3^2$ x 2, $C_3^2\times C_{12}$, $C_3^3:C_4$

Order 96: $D_4:D_6$ x 50, $D_4\times D_6$ x 10, $D_4:D_6$ x 9, $D_{12}:C_2^2$ x 9, $C_{12}:C_2^3$ x 9, $C_6.C_2^4$ x 3, $Q_8:D_6$ x 3, $Q_8\times D_6$ x 3

Order 72: $S_3\times D_6$ x 69, $C_6.D_6$ x 63, $S_3\times C_{12}$ x 54, $C_6.D_6$ x 54, $C_6\wr C_2$ x 52, $C_3:D_{12}$ x 52, $C_6:C_{12}$ x 42, $C_6:D_6$ x 33, $C_6^2:C_2$ x 28, $C_{12}:S_3$ x 28, $C_3^2:Q_8$ x 24, $C_3^2:Q_8$ x 24, $C_6\times D_6$ x 18, $D_4\times C_3^2$ x 14, $C_3:D_{12}$ x 14, $C_3\times D_{12}$ x 14, $D_6:S_3$ x 14, $C_6\times C_{12}$ x 7, $C_6.D_6$ x 7, $C_3^2:Q_8$ x 6, $C_2\times C_6^2$ x 2, $Q_8\times C_3^2$ x 2

Order 64: $D_4:C_2^3$

Order 54: $C_3^2:C_6$ x 6, $S_3\times C_3^2$ x 6, $C_3^2:S_3$ x 4, $C_3^2\times C_6$ x 3

Order 48: $C_4\times D_6$ x 109, $S_3\times D_4$ x 87, $D_4:S_3$ x 78, $D_{12}:C_2$ x 78, $D_4:C_6$ x 26, $D_{12}:C_2$ x 26, $S_3\times Q_8$ x 26, $C_6:D_4$ x 20, $C_2^2\times D_6$ x 11, $C_6\times D_4$ x 10, $C_2\times D_{12}$ x 10, $C_2^2\times C_{12}$ x 9, $C_6.C_2^3$ x 9, $C_6:Q_8$ x 9, $C_6\times Q_8$ x 3

Order 36: $C_6:S_3$ x 78, $S_3^2$ x 72, $C_6\times S_3$ x 66, $C_3:C_{12}$ x 42, $C_6^2$ x 23, $C_3\times C_{12}$ x 14, $C_3^2:C_4$ x 14

Order 32: $D_4:C_2^2$ x 24, $C_2^2\times D_4$ x 3, $C_2^3\times C_4$ x 3, $C_2^2\times Q_8$

Order 27: $C_3^3$

Order 24: $C_4\times S_3$ x 145, $C_2\times D_6$ x 111, $C_3:D_4$ x 94, $C_2\times C_{12}$ x 55, $C_6:C_4$ x 55, $D_{12}$ x 47, $C_3\times D_4$ x 47, $C_3:Q_8$ x 42, $C_3\times Q_8$ x 14, $C_2^2\times C_6$ x 11

Order 18: $C_3\times S_3$ x 36, $C_3:S_3$ x 34, $C_3\times C_6$ x 27

Order 16: $D_4:C_2$ x 64, $C_2^2\times C_4$ x 40, $C_2\times D_4$ x 36, $C_2\times Q_8$ x 12, $C_2^4$ x 3

Order 12: $D_6$ x 183, $C_2\times C_6$ x 56, $C_{12}$ x 35, $C_3:C_4$ x 35

Order 9: $C_3^2$ x 7

Order 8: $C_2\times C_4$ x 72, $D_4$ x 48, $C_2^3$ x 31, $Q_8$ x 16

Order 6: $S_3$ x 52, $C_6$ x 45

Order 4: $C_2^2$ x 55, $C_4$ x 16

Order 3: $C_3$ x 7

Order 2: $C_2$ x 19

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1728: $D_4:S_3^3$

Order 864: $C_6.D_6^2$, $C_6.D_6^2$, $C_6.D_6^2$, $C_6.D_6^2$, $C_6.D_6^2$, $C_2^2:S_3^3$, $C_6.D_6^2$, $C_6.D_6^2$, $C_6.D_6^2$, $C_6.D_6^2$, $C_6.D_6^2$, $C_4\times S_3^3$, $D_{12}:S_3^2$, $C_4.S_3^3$, $D_{12}:S_3^2$, $D_{12}:S_3^2$, $C_4.S_3^3$

Order 576: $C_4.D_6^2$, $C_4.D_6^2$

Order 432: $C_2.S_3^3$ x 7, $C_6^2:D_6$ x 2, $C_{12}:S_3^2$ x 2, $C_6^2.D_6$ x 2, $C_{12}\times S_3^2$ x 2, $C_2.S_3^3$ x 2, $C_2.S_3^3$ x 2, $C_3:D_6^2$, $C_2\times S_3^3$, $C_6^2:D_6$, $C_6^2.D_6$, $C_6^2:D_6$, $C_{12}.C_6^2$, $C_{12}:C_6^2$, $C_6^2:D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_{12}:S_3^2$, $C_{12}.S_3^2$, $C_{12}.S_3^2$, $C_6^2:D_6$, $C_6^2:D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_{12}:S_3^2$, $C_{12}:S_3^2$, $C_{12}:S_3^2$, $C_{12}.S_3^2$, $C_{12}.S_3^2$, $C_{12}.S_3^2$, $C_6^2:D_6$, $C_6^2:D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_{12}.S_3^2$, $C_{12}.S_3^2$, $C_{12}.S_3^2$, $C_{12}.S_3^2$, $C_{12}.S_3^2$, $C_2.S_3^3$, $C_2.S_3^3$, $C_2.S_3^3$, $C_2.S_3^3$, $C_2.S_3^3$, $C_2.S_3^3$, $C_2.S_3^3$, $D_6:S_3^2$, $D_6:S_3^2$, $D_6:S_3^2$, $D_6:S_3^2$, $C_2.S_3^3$

Order 288: $D_4:S_3^2$ x 12, $D_{12}:D_6$ x 4, $D_{12}:D_6$ x 4, $D_4\times S_3^2$ x 3, $C_2.D_6^2$ x 3, $C_6^2.C_2^3$ x 2, $D_{12}:D_6$ x 2, $D_{12}:D_6$ x 2, $C_6^2.C_2^3$, $D_6^2:C_2$, $C_2.D_6^2$, $C_2.D_6^2$, $D_{12}:D_6$, $Q_8\times S_3^2$, $C_2.D_6^2$, $C_2.D_6^2$, $C_6^2.C_2^3$, $C_6^2:C_2^3$

Order 216: $C_6:S_3^2$ x 6, $C_6.S_3^2$ x 6, $C_6.S_3^2$ x 5, $C_6.S_3^2$ x 4, $C_6.S_3^2$ x 4, $C_6\times S_3^2$ x 2, $C_6^2:C_6$ x 2, $C_3:S_3\times C_{12}$ x 2, $C_6^2:C_6$ x 2, $C_6.C_6^2$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:D_{12}$ x 2, $C_6.S_3^2$ x 2, $C_6.S_3^2$ x 2, $C_3^3:Q_8$ x 2, $C_3^2:D_{12}$ x 2, $C_6^2:S_3$, $C_6^2:C_6$, $S_3\times C_6^2$, $C_6:S_3^2$, $S_3^3$, $D_4\times C_3^3$, $C_6^2:S_3$, $C_3^2:D_{12}$, $C_4\times C_3^2:S_3$, $C_6^2.C_6$, $C_3^2:D_{12}$, $C_3^3:Q_8$, $C_6.C_6^2$, $C_3^2\times D_{12}$, $C_3^3:Q_8$, $C_3^3:Q_8$, $C_3^2:D_{12}$, $C_3^3:Q_8$, $C_3^3:D_4$, $C_3^3:D_4$

Order 192: $D_{12}:C_2^3$ x 2

Order 144: $C_4\times S_3^2$ x 23, $D_6.D_6$ x 16, $C_{12}.D_6$ x 12, $C_2^2.S_3^2$ x 12, $C_{12}.D_6$ x 12, $C_{12}:D_6$ x 11, $D_6:D_6$ x 11, $D_6:D_6$ x 8, $C_{12}.D_6$ x 8, $C_{12}:D_6$ x 6, $S_3\times D_{12}$ x 6, $C_{12}\times D_6$ x 5, $D_6.D_6$ x 5, $D_6^2$ x 4, $D_{12}:C_6$ x 4, $D_{12}:S_3$ x 4, $D_{12}:S_3$ x 4, $C_{12}.D_6$ x 4, $C_{12}:D_6$ x 3, $D_6.D_6$ x 3, $C_{12}:D_6$ x 3, $C_{12}.D_6$ x 3, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $D_{12}:C_6$ x 2, $C_{12}.D_6$ x 2, $C_6^2:C_2^2$, $C_4.C_6^2$, $C_4:C_6^2$, $C_6^2:C_2^2$, $C_{12}.D_6$, $C_{12}.D_6$, $C_6:D_{12}$, $C_6^2:C_2^2$, $C_6^2:C_4$, $C_{12}.D_6$, $C_2^2.S_3^2$, $C_6:D_{12}$

Order 108: $C_3:S_3^2$ x 5, $C_3^2:D_6$ x 4, $C_3^2\times D_6$ x 4, $C_3^2:D_6$ x 3, $C_3^2:C_{12}$ x 3, $C_3^2:C_{12}$ x 3, $C_3\times S_3^2$ x 2, $C_3\times C_6^2$, $C_3:S_3^2$, $C_3^2\times C_{12}$, $C_3^3:C_4$

Order 96: $D_4:D_6$ x 19, $D_4\times D_6$ x 5, $D_4:D_6$ x 4, $D_{12}:C_2^2$ x 4, $C_{12}:C_2^3$ x 4, $C_6.C_2^4$ x 2, $Q_8:D_6$ x 2, $Q_8\times D_6$ x 2

Order 72: $S_3\times C_{12}$ x 22, $C_6.D_6$ x 22, $C_6.D_6$ x 22, $S_3\times D_6$ x 19, $C_{12}:S_3$ x 14, $C_6\wr C_2$ x 14, $C_3:D_{12}$ x 14, $C_6:D_6$ x 12, $C_6:C_{12}$ x 12, $C_6^2:C_2$ x 10, $C_3^2:Q_8$ x 9, $D_4\times C_3^2$ x 8, $C_3:D_{12}$ x 8, $C_6\times D_6$ x 7, $C_3^2:Q_8$ x 7, $C_3\times D_{12}$ x 6, $D_6:S_3$ x 4, $C_6\times C_{12}$ x 3, $C_6.D_6$ x 3, $C_3^2:Q_8$ x 2, $C_2\times C_6^2$, $Q_8\times C_3^2$

Order 64: $D_4:C_2^3$

Order 54: $C_3^2:C_6$ x 3, $S_3\times C_3^2$ x 3, $C_3^2\times C_6$ x 2, $C_3^2:S_3$ x 2

Order 48: $C_4\times D_6$ x 41, $S_3\times D_4$ x 33, $D_4:S_3$ x 23, $D_{12}:C_2$ x 23, $S_3\times Q_8$ x 11, $D_4:C_6$ x 10, $D_{12}:C_2$ x 10, $C_6:D_4$ x 7, $C_2^2\times D_6$ x 5, $C_6\times D_4$ x 5, $C_2\times D_{12}$ x 5, $C_2^2\times C_{12}$ x 4, $C_6.C_2^3$ x 4, $C_6:Q_8$ x 4, $C_6\times Q_8$ x 2

Order 36: $C_6:S_3$ x 24, $C_6\times S_3$ x 21, $C_3:C_{12}$ x 16, $S_3^2$ x 15, $C_6^2$ x 9, $C_3\times C_{12}$ x 8, $C_3^2:C_4$ x 8

Order 32: $D_4:C_2^2$ x 12, $C_2^2\times D_4$ x 2, $C_2^3\times C_4$ x 2, $C_2^2\times Q_8$

Order 27: $C_3^3$

Order 24: $C_4\times S_3$ x 56, $C_2\times D_6$ x 34, $C_3:D_4$ x 28, $C_2\times C_{12}$ x 21, $C_6:C_4$ x 21, $D_{12}$ x 18, $C_3\times D_4$ x 18, $C_3:Q_8$ x 13, $C_3\times Q_8$ x 6, $C_2^2\times C_6$ x 5

Order 18: $C_3\times S_3$ x 14, $C_3\times C_6$ x 13, $C_3:S_3$ x 13

Order 16: $D_4:C_2$ x 20, $C_2^2\times C_4$ x 19, $C_2\times D_4$ x 17, $C_2\times Q_8$ x 7, $C_2^4$ x 2

Order 12: $D_6$ x 49, $C_2\times C_6$ x 20, $C_{12}$ x 16, $C_3:C_4$ x 16

Order 9: $C_3^2$ x 5

Order 8: $C_2\times C_4$ x 30, $D_4$ x 18, $C_2^3$ x 13, $Q_8$ x 6

Order 6: $C_6$ x 20, $S_3$ x 20

Order 4: $C_2^2$ x 20, $C_4$ x 8

Order 3: $C_3$ x 5

Order 2: $C_2$ x 10

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1728: $D_4:S_3^3$ ($C_1$)

Order 864: $C_6.D_6^2$ ($C_2$) x 4, $C_6.D_6^2$ ($C_2$) x 4, $C_6.D_6^2$ ($C_2$) x 2, $C_6.D_6^2$ ($C_2$) x 2, $C_2^2:S_3^3$ ($C_2$) x 2, $C_6.D_6^2$ ($C_2$) x 2, $C_6.D_6^2$ ($C_2$) x 2, $C_6.D_6^2$ ($C_2$) x 2, $C_4.S_3^3$ ($C_2$) x 2, $D_{12}:S_3^2$ ($C_2$) x 2, $C_6.D_6^2$ ($C_2$), $C_6.D_6^2$ ($C_2$), $C_6.D_6^2$ ($C_2$), $C_4\times S_3^3$ ($C_2$), $D_{12}:S_3^2$ ($C_2$), $D_{12}:S_3^2$ ($C_2$), $C_4.S_3^3$ ($C_2$)

Order 432: $C_2.S_3^3$ ($C_2^2$) x 17, $C_6^2.D_6$ ($C_2^2$) x 8, $C_2.S_3^3$ ($C_2^2$) x 8, $C_6^2:D_6$ ($C_2^2$) x 6, $C_6^2.D_6$ ($C_2^2$) x 4, $C_6^2.D_6$ ($C_2^2$) x 4, $C_6^2.D_6$ ($C_2^2$) x 4, $C_6^2.D_6$ ($C_2^2$) x 4, $C_6^2:D_6$ ($C_2^2$) x 4, $C_6^2.D_6$ ($C_2^2$) x 4, $C_6^2.D_6$ ($C_2^2$) x 4, $C_2.S_3^3$ ($C_2^2$) x 4, $C_2.S_3^3$ ($C_2^2$) x 4, $C_2.S_3^3$ ($C_2^2$) x 4, $C_2.S_3^3$ ($C_2^2$) x 4, $D_6:S_3^2$ ($C_2^2$) x 4, $D_6:S_3^2$ ($C_2^2$) x 4, $C_{12}:S_3^2$ ($C_2^2$) x 3, $C_{12}\times S_3^2$ ($C_2^2$) x 3, $C_2.S_3^3$ ($C_2^2$) x 3, $C_3:D_6^2$ ($C_2^2$) x 2, $C_6^2.D_6$ ($C_2^2$) x 2, $C_{12}.C_6^2$ ($C_2^2$) x 2, $C_6^2:D_6$ ($C_2^2$) x 2, $C_6^2.D_6$ ($C_2^2$) x 2, $C_{12}.S_3^2$ ($C_2^2$) x 2, $C_6^2:D_6$ ($C_2^2$) x 2, $C_6^2:D_6$ ($C_2^2$) x 2, $C_{12}.S_3^2$ ($C_2^2$) x 2, $C_{12}.S_3^2$ ($C_2^2$) x 2, $C_{12}.S_3^2$ ($C_2^2$) x 2, $C_6^2:D_6$ ($C_2^2$) x 2, $C_6^2.D_6$ ($C_2^2$) x 2, $C_{12}.S_3^2$ ($C_2^2$) x 2, $C_{12}.S_3^2$ ($C_2^2$) x 2, $C_{12}.S_3^2$ ($C_2^2$) x 2, $C_{12}.S_3^2$ ($C_2^2$) x 2, $C_2.S_3^3$ ($C_2^2$) x 2, $C_2.S_3^3$ ($C_2^2$) x 2, $C_2.S_3^3$ ($C_2^2$) x 2, $D_6:S_3^2$ ($C_2^2$) x 2, $D_6:S_3^2$ ($C_2^2$) x 2, $C_2\times S_3^3$ ($C_2^2$), $C_6^2:D_6$ ($C_2^2$), $C_6^2:D_6$ ($C_2^2$), $C_{12}:C_6^2$ ($C_2^2$), $C_{12}:S_3^2$ ($C_2^2$), $C_{12}.S_3^2$ ($C_2^2$), $C_{12}:S_3^2$ ($C_2^2$), $C_{12}:S_3^2$ ($C_2^2$), $C_{12}:S_3^2$ ($C_2^2$), $C_{12}.S_3^2$ ($C_2^2$), $C_2.S_3^3$ ($C_2^2$)

Order 288: $D_{12}:D_6$ ($S_3$) x 2, $D_4:S_3^2$ ($S_3$)

Order 216: $C_6.S_3^2$ ($C_2^3$) x 18, $C_6.S_3^2$ ($C_2^3$) x 15, $C_6:S_3^2$ ($C_2^3$) x 9, $C_6.S_3^2$ ($C_2^3$) x 9, $C_6.S_3^2$ ($C_2^3$) x 9, $C_3^2:D_{12}$ ($C_2^3$) x 8, $C_6^2:C_6$ ($C_2^3$) x 6, $C_6^2:C_6$ ($C_2^3$) x 6, $C_3^2:D_{12}$ ($C_2^3$) x 6, $C_3^2:D_{12}$ ($C_2^3$) x 6, $C_3^3:Q_8$ ($C_2^3$) x 6, $C_6^2.C_6$ ($C_2^3$) x 4, $C_6.C_6^2$ ($C_2^3$) x 4, $C_3^2:D_{12}$ ($C_2^3$) x 4, $C_3^3:Q_8$ ($C_2^3$) x 4, $C_3^3:D_4$ ($C_2^3$) x 4, $C_6\times S_3^2$ ($C_2^3$) x 3, $C_3:S_3\times C_{12}$ ($C_2^3$) x 3, $C_6.C_6^2$ ($C_2^3$) x 3, $C_6.S_3^2$ ($C_2^3$) x 3, $C_6.S_3^2$ ($C_2^3$) x 3, $C_6^2:S_3$ ($C_2^3$) x 2, $C_6^2:C_6$ ($C_2^3$) x 2, $S_3\times C_6^2$ ($C_2^3$) x 2, $C_6^2:S_3$ ($C_2^3$) x 2, $C_3^3:Q_8$ ($C_2^3$) x 2, $C_3^3:Q_8$ ($C_2^3$) x 2, $C_3^3:Q_8$ ($C_2^3$) x 2, $C_3^3:D_4$ ($C_2^3$) x 2, $C_6:S_3^2$ ($C_2^3$), $D_4\times C_3^3$ ($C_2^3$), $C_3^2:D_{12}$ ($C_2^3$), $C_4\times C_3^2:S_3$ ($C_2^3$), $C_3^2:D_{12}$ ($C_2^3$), $C_3^2\times D_{12}$ ($C_2^3$)

Order 144: $D_6.D_6$ ($D_6$) x 8, $C_{12}.D_6$ ($D_6$) x 4, $D_6:D_6$ ($D_6$) x 4, $D_6.D_6$ ($D_6$) x 4, $D_6.D_6$ ($D_6$) x 4, $C_4\times S_3^2$ ($D_6$) x 3, $C_{12}.D_6$ ($D_6$) x 2, $C_{12}:D_6$ ($D_6$) x 2, $D_6:D_6$ ($D_6$) x 2, $C_2^2.S_3^2$ ($D_6$) x 2, $C_{12}.D_6$ ($D_6$) x 2, $D_{12}:S_3$ ($D_6$) x 2, $D_{12}:S_3$ ($D_6$) x 2, $C_{12}.D_6$ ($D_6$) x 2, $C_{12}:D_6$ ($D_6$), $C_{12}.D_6$ ($D_6$)

Order 108: $C_3^2:C_{12}$ ($C_2^4$) x 7, $C_3^2:C_{12}$ ($C_2^4$) x 7, $C_3^2:D_6$ ($C_2^4$) x 5, $C_3^2\times D_6$ ($C_2^4$) x 5, $C_3:S_3^2$ ($D_4:C_2$) x 4, $C_3^2:D_6$ ($C_2^4$) x 3, $C_3\times C_6^2$ ($C_2^4$) x 2, $C_3^2\times C_{12}$ ($C_2^4$), $C_3^3:C_4$ ($C_2^4$)

Order 72: $C_6.D_6$ ($C_2\times D_6$) x 18, $C_6\wr C_2$ ($C_2\times D_6$) x 12, $C_6.D_6$ ($C_2\times D_6$) x 9, $C_6:C_{12}$ ($C_2\times D_6$) x 8, $C_3:D_{12}$ ($C_2\times D_6$) x 8, $C_6^2:C_2$ ($C_2\times D_6$) x 6, $S_3\times C_{12}$ ($C_2\times D_6$) x 6, $C_3^2:Q_8$ ($C_2\times D_6$) x 6, $C_6\times D_6$ ($C_2\times D_6$) x 4, $C_6.D_6$ ($C_2\times D_6$) x 4, $C_3^2:Q_8$ ($C_2\times D_6$) x 4, $D_6:S_3$ ($C_2\times D_6$) x 4, $S_3\times D_6$ ($C_2\times D_6$) x 3, $D_4\times C_3^2$ ($C_2\times D_6$) x 3, $C_{12}:S_3$ ($C_2\times D_6$) x 3, $C_6:D_6$ ($C_2\times D_6$) x 2, $C_3^2:Q_8$ ($C_2\times D_6$) x 2, $C_3\times D_{12}$ ($C_2\times D_6$) x 2, $C_3:D_{12}$ ($C_2\times D_6$)

Order 54: $C_3^2:S_3$ ($D_4:C_2^2$) x 2, $C_3^2:C_6$ ($D_4:C_2^2$) x 2, $S_3\times C_3^2$ ($D_4:C_2^2$) x 2, $C_3^2\times C_6$ ($C_2^5$)

Order 48: $D_4:S_3$ ($S_3^2$) x 2, $S_3\times D_4$ ($S_3^2$)

Order 36: $C_3:C_{12}$ ($C_2^2\times D_6$) x 14, $C_6\times S_3$ ($C_2^2\times D_6$) x 10, $C_3^2:C_4$ ($C_2^2\times D_6$) x 7, $C_6^2$ ($C_2^2\times D_6$) x 6, $C_6:S_3$ ($C_2^2\times D_6$) x 5, $C_3\times C_{12}$ ($C_2^2\times D_6$) x 3

Order 27: $C_3^3$ ($D_4:C_2^3$)

Order 24: $C_3:D_4$ ($S_3\times D_6$) x 6, $C_6:C_4$ ($S_3\times D_6$) x 4, $C_4\times S_3$ ($S_3\times D_6$) x 3, $C_3\times D_4$ ($S_3\times D_6$) x 3, $C_3:Q_8$ ($S_3\times D_6$) x 2, $C_2\times D_6$ ($S_3\times D_6$) x 2, $D_{12}$ ($S_3\times D_6$)

Order 18: $C_3\times S_3$ ($D_4:D_6$) x 4, $C_3\times C_6$ ($C_2^3\times D_6$) x 3, $C_3:S_3$ ($D_4:D_6$) x 2

Order 12: $C_3:C_4$ ($D_6^2$) x 7, $C_2\times C_6$ ($D_6^2$) x 6, $D_6$ ($D_6^2$) x 5, $C_{12}$ ($D_6^2$) x 3

Order 9: $C_3^2$ ($D_{12}:C_2^3$) x 3

Order 8: $D_4$ ($S_3^3$)

Order 6: $C_6$ ($C_2\times D_6^2$) x 3, $S_3$ ($D_4:S_3^2$) x 2

Order 4: $C_2^2$ ($C_2\times S_3^3$) x 2, $C_4$ ($C_2\times S_3^3$)

Order 3: $C_3$ ($C_4.D_6^2$) x 2, $C_3$ ($C_4.D_6^2$)

Order 2: $C_2$ ($S_3\times D_6^2$)

Order 1: $C_1$ ($D_4:S_3^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1728: $D_4:S_3^3$ ($C_1$)

Order 864: $C_6.D_6^2$ ($C_2$), $C_6.D_6^2$ ($C_2$), $C_6.D_6^2$ ($C_2$), $C_6.D_6^2$ ($C_2$), $C_6.D_6^2$ ($C_2$), $C_2^2:S_3^3$ ($C_2$), $C_6.D_6^2$ ($C_2$), $C_6.D_6^2$ ($C_2$), $C_6.D_6^2$ ($C_2$), $C_6.D_6^2$ ($C_2$), $C_6.D_6^2$ ($C_2$), $C_4\times S_3^3$ ($C_2$), $D_{12}:S_3^2$ ($C_2$), $C_4.S_3^3$ ($C_2$), $D_{12}:S_3^2$ ($C_2$), $D_{12}:S_3^2$ ($C_2$), $C_4.S_3^3$ ($C_2$)

Order 432: $C_2.S_3^3$ ($C_2^2$) x 7, $C_6^2:D_6$ ($C_2^2$) x 2, $C_{12}:S_3^2$ ($C_2^2$) x 2, $C_6^2.D_6$ ($C_2^2$) x 2, $C_{12}\times S_3^2$ ($C_2^2$) x 2, $C_2.S_3^3$ ($C_2^2$) x 2, $C_2.S_3^3$ ($C_2^2$) x 2, $C_3:D_6^2$ ($C_2^2$), $C_2\times S_3^3$ ($C_2^2$), $C_6^2:D_6$ ($C_2^2$), $C_6^2.D_6$ ($C_2^2$), $C_6^2:D_6$ ($C_2^2$), $C_{12}.C_6^2$ ($C_2^2$), $C_{12}:C_6^2$ ($C_2^2$), $C_6^2:D_6$ ($C_2^2$), $C_6^2.D_6$ ($C_2^2$), $C_6^2.D_6$ ($C_2^2$), $C_{12}:S_3^2$ ($C_2^2$), $C_{12}.S_3^2$ ($C_2^2$), $C_{12}.S_3^2$ ($C_2^2$), $C_6^2:D_6$ ($C_2^2$), $C_6^2:D_6$ ($C_2^2$), $C_6^2.D_6$ ($C_2^2$), $C_6^2.D_6$ ($C_2^2$), $C_6^2.D_6$ ($C_2^2$), $C_{12}:S_3^2$ ($C_2^2$), $C_{12}:S_3^2$ ($C_2^2$), $C_{12}:S_3^2$ ($C_2^2$), $C_{12}.S_3^2$ ($C_2^2$), $C_{12}.S_3^2$ ($C_2^2$), $C_{12}.S_3^2$ ($C_2^2$), $C_6^2:D_6$ ($C_2^2$), $C_6^2:D_6$ ($C_2^2$), $C_6^2.D_6$ ($C_2^2$), $C_6^2.D_6$ ($C_2^2$), $C_6^2.D_6$ ($C_2^2$), $C_{12}.S_3^2$ ($C_2^2$), $C_{12}.S_3^2$ ($C_2^2$), $C_{12}.S_3^2$ ($C_2^2$), $C_{12}.S_3^2$ ($C_2^2$), $C_{12}.S_3^2$ ($C_2^2$), $C_2.S_3^3$ ($C_2^2$), $C_2.S_3^3$ ($C_2^2$), $C_2.S_3^3$ ($C_2^2$), $C_2.S_3^3$ ($C_2^2$), $C_2.S_3^3$ ($C_2^2$), $C_2.S_3^3$ ($C_2^2$), $C_2.S_3^3$ ($C_2^2$), $D_6:S_3^2$ ($C_2^2$), $D_6:S_3^2$ ($C_2^2$), $D_6:S_3^2$ ($C_2^2$), $D_6:S_3^2$ ($C_2^2$), $C_2.S_3^3$ ($C_2^2$)

Order 288: $D_4:S_3^2$ ($S_3$), $D_{12}:D_6$ ($S_3$)

Order 216: $C_6.S_3^2$ ($C_2^3$) x 6, $C_6:S_3^2$ ($C_2^3$) x 5, $C_6.S_3^2$ ($C_2^3$) x 5, $C_6.S_3^2$ ($C_2^3$) x 4, $C_6.S_3^2$ ($C_2^3$) x 4, $C_6\times S_3^2$ ($C_2^3$) x 2, $C_6^2:C_6$ ($C_2^3$) x 2, $C_3:S_3\times C_{12}$ ($C_2^3$) x 2, $C_6^2:C_6$ ($C_2^3$) x 2, $C_6.C_6^2$ ($C_2^3$) x 2, $C_3^2:D_{12}$ ($C_2^3$) x 2, $C_3^2:D_{12}$ ($C_2^3$) x 2, $C_6.S_3^2$ ($C_2^3$) x 2, $C_6.S_3^2$ ($C_2^3$) x 2, $C_3^3:Q_8$ ($C_2^3$) x 2, $C_3^2:D_{12}$ ($C_2^3$) x 2, $C_6^2:S_3$ ($C_2^3$), $C_6^2:C_6$ ($C_2^3$), $S_3\times C_6^2$ ($C_2^3$), $C_6:S_3^2$ ($C_2^3$), $D_4\times C_3^3$ ($C_2^3$), $C_6^2:S_3$ ($C_2^3$), $C_3^2:D_{12}$ ($C_2^3$), $C_4\times C_3^2:S_3$ ($C_2^3$), $C_6^2.C_6$ ($C_2^3$), $C_3^2:D_{12}$ ($C_2^3$), $C_3^3:Q_8$ ($C_2^3$), $C_6.C_6^2$ ($C_2^3$), $C_3^2\times D_{12}$ ($C_2^3$), $C_3^3:Q_8$ ($C_2^3$), $C_3^3:Q_8$ ($C_2^3$), $C_3^2:D_{12}$ ($C_2^3$), $C_3^3:Q_8$ ($C_2^3$), $C_3^3:D_4$ ($C_2^3$), $C_3^3:D_4$ ($C_2^3$)

Order 144: $C_{12}.D_6$ ($D_6$) x 2, $D_6.D_6$ ($D_6$) x 2, $C_4\times S_3^2$ ($D_6$) x 2, $C_{12}.D_6$ ($D_6$), $C_{12}:D_6$ ($D_6$), $C_{12}:D_6$ ($D_6$), $D_6:D_6$ ($D_6$), $D_6:D_6$ ($D_6$), $C_2^2.S_3^2$ ($D_6$), $D_6.D_6$ ($D_6$), $D_6.D_6$ ($D_6$), $C_{12}.D_6$ ($D_6$), $C_{12}.D_6$ ($D_6$), $D_{12}:S_3$ ($D_6$), $D_{12}:S_3$ ($D_6$), $C_{12}.D_6$ ($D_6$)

Order 108: $C_3^2:D_6$ ($C_2^4$) x 3, $C_3^2\times D_6$ ($C_2^4$) x 3, $C_3^2:C_{12}$ ($C_2^4$) x 3, $C_3^2:C_{12}$ ($C_2^4$) x 3, $C_3^2:D_6$ ($C_2^4$) x 2, $C_3\times C_6^2$ ($C_2^4$), $C_3:S_3^2$ ($D_4:C_2$), $C_3^2\times C_{12}$ ($C_2^4$), $C_3^3:C_4$ ($C_2^4$)

Order 72: $C_6.D_6$ ($C_2\times D_6$) x 6, $C_6.D_6$ ($C_2\times D_6$) x 4, $C_6\wr C_2$ ($C_2\times D_6$) x 3, $S_3\times C_{12}$ ($C_2\times D_6$) x 3, $S_3\times D_6$ ($C_2\times D_6$) x 2, $D_4\times C_3^2$ ($C_2\times D_6$) x 2, $C_6^2:C_2$ ($C_2\times D_6$) x 2, $C_{12}:S_3$ ($C_2\times D_6$) x 2, $C_6:C_{12}$ ($C_2\times D_6$) x 2, $C_3^2:Q_8$ ($C_2\times D_6$) x 2, $C_3^2:Q_8$ ($C_2\times D_6$) x 2, $C_3:D_{12}$ ($C_2\times D_6$) x 2, $C_6:D_6$ ($C_2\times D_6$), $C_6\times D_6$ ($C_2\times D_6$), $C_6.D_6$ ($C_2\times D_6$), $C_3:D_{12}$ ($C_2\times D_6$), $C_3^2:Q_8$ ($C_2\times D_6$), $C_3\times D_{12}$ ($C_2\times D_6$), $D_6:S_3$ ($C_2\times D_6$)

Order 54: $C_3^2\times C_6$ ($C_2^5$), $C_3^2:S_3$ ($D_4:C_2^2$), $C_3^2:C_6$ ($D_4:C_2^2$), $S_3\times C_3^2$ ($D_4:C_2^2$)

Order 48: $D_4:S_3$ ($S_3^2$), $S_3\times D_4$ ($S_3^2$)

Order 36: $C_3:C_{12}$ ($C_2^2\times D_6$) x 5, $C_6\times S_3$ ($C_2^2\times D_6$) x 4, $C_3^2:C_4$ ($C_2^2\times D_6$) x 3, $C_6:S_3$ ($C_2^2\times D_6$) x 3, $C_3\times C_{12}$ ($C_2^2\times D_6$) x 2, $C_6^2$ ($C_2^2\times D_6$) x 2

Order 27: $C_3^3$ ($D_4:C_2^3$)

Order 24: $C_3:D_4$ ($S_3\times D_6$) x 2, $C_4\times S_3$ ($S_3\times D_6$) x 2, $C_3\times D_4$ ($S_3\times D_6$) x 2, $C_6:C_4$ ($S_3\times D_6$), $D_{12}$ ($S_3\times D_6$), $C_3:Q_8$ ($S_3\times D_6$), $C_2\times D_6$ ($S_3\times D_6$)

Order 18: $C_3\times C_6$ ($C_2^3\times D_6$) x 2, $C_3:S_3$ ($D_4:D_6$), $C_3\times S_3$ ($D_4:D_6$)

Order 12: $D_6$ ($D_6^2$) x 3, $C_3:C_4$ ($D_6^2$) x 3, $C_2\times C_6$ ($D_6^2$) x 2, $C_{12}$ ($D_6^2$) x 2

Order 9: $C_3^2$ ($D_{12}:C_2^3$) x 2

Order 8: $D_4$ ($S_3^3$)

Order 6: $C_6$ ($C_2\times D_6^2$) x 2, $S_3$ ($D_4:S_3^2$)

Order 4: $C_2^2$ ($C_2\times S_3^3$), $C_4$ ($C_2\times S_3^3$)

Order 3: $C_3$ ($C_4.D_6^2$), $C_3$ ($C_4.D_6^2$)

Order 2: $C_2$ ($S_3\times D_6^2$)

Order 1: $C_1$ ($D_4:S_3^3$)

Series

Derived series

$D_4:S_3^3$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_1$

Chief series

$D_4:S_3^3$

$\rhd$

$C_6.D_6^2$

$\rhd$

$C_{12}.C_6^2$

$\rhd$

$D_4\times C_3^3$

$\rhd$

$D_4\times C_3^2$

$\rhd$

$C_3\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$D_4:S_3^3$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_3^3$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$D_4$

Character theory

Complex character table

See the $135 \times 135$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $123 \times 123$ rational character table.