Abstract group 1728.47319: $C_{12}:D_6^2$

• Label: 1728.47319
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \)
• Perm deg.: $15$
• Trans deg.: $96$
• Rank: $5$

Group information

Description:	$C_{12}:D_6^2$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^4:(S_3\times D_{12})$, of order \(442368\)\(\medspace = 2^{14} \cdot 3^{3} \)
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	447	26	64	822	368	1728
Conjugacy classes	1	31	7	8	69	46	162
Divisions	1	31	7	8	69	28	144
Autjugacy classes	1	7	5	3	17	9	42

Dimension	1	2	4	8	16
Irr. complex chars.	32	72	48	10	0	162
Irr. rational chars.	32	56	40	14	2	144

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$96$
Rank:	$5$
Inequivalent generating 5-tuples:	$9999360000$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{6}=d^{6}=e^{12}=[a,b]=[a,c]=[a,e]=[b,d]=[b,e]=[c,d]=[c,e]=1, d^{a}=d^{5}, c^{b}=c^{5}, e^{d}=e^{11} \rangle$
Permutation group:	Degree $15$ $\langle(4,5)(6,7)(8,9)(10,11)(12,14)(13,15), (6,7)(8,9)(10,11)(12,15)(13,14), (2,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 14 & 27 \\ 27 & 8 \end{array}\right), \left(\begin{array}{rr} 23 & 0 \\ 0 & 23 \end{array}\right), \left(\begin{array}{rr} 34 & 33 \\ 33 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 54 \\ 24 & 43 \end{array}\right), \left(\begin{array}{rr} 1 & 33 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 23 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 52 & 63 \\ 39 & 13 \end{array}\right), \left(\begin{array}{rr} 1 & 22 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 43 & 0 \\ 0 & 43 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/66\Z)$
Direct product:	$C_2$ $\, \times\, $ $S_3$ ${}^2$ $\, \times\, $ $D_{12}$
Semidirect product:	$D_6^2$ $\,\rtimes\,$ $D_6$	$D_6$ $\,\rtimes\,$ $D_6^2$	$C_{12}$ $\,\rtimes\,$ $D_6^2$ (2)	$(D_6.D_6)$ $\,\rtimes\,$ $D_6$	all 116
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_6^2$ . $D_6$	$D_6$ . $D_6^2$	$C_2$ . $(S_3\times D_6^2)$	$(C_2\times C_6)$ . $D_6^2$ (2)	all 22

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{66}\Z)$.

Homology

Abelianization:	$C_{2}^{5} $
Schur multiplier:	$C_{2}^{10}$
Commutator length:	$1$

Subgroups

There are 57780 subgroups in 6210 conjugacy classes, 780 normal (34 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2\times S_3^3$
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:	$\operatorname{Fit} \simeq$ $C_3\times C_6\times C_{12}$	$G/\operatorname{Fit} \simeq$ $C_2^3$
Radical:	$R \simeq$ $C_{12}:D_6^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4\times C_2^3$
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: $C_{12}:D_6^2$

Order 864: $D_{12}\times S_3^2$ x 16, $D_6^2:S_3$ x 4, $S_3\times D_6^2$ x 2, $C_6.D_6^2$ x 2, $D_6^2:C_6$ x 2, $C_6.D_6^2$ x 2, $C_6.D_6^2$, $C_6.D_6^2$, $C_6.D_6^2$

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

Order 432: $C_2\times S_3^3$ x 32, $D_6:S_3^2$ x 32, $C_{12}:S_3^2$ x 16, $C_{12}:S_3^2$ x 16, $D_6:S_3^2$ x 16, $C_{12}:S_3^2$ x 8, $C_{12}:S_3^2$ x 8, $C_{12}\times S_3^2$ x 8, $C_3:D_6^2$ x 6, $C_3\times D_6^2$ x 5, $C_6^2.D_6$ x 4, $C_6^2.D_6$ x 4, $C_3:D_6^2$ 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, $C_6^2.D_6$ x 2, $C_6^2.D_6$, $C_6^2.D_6$, $C_{12}:C_6^2$, $C_6^2.D_6$

Order 288: $D_6\times D_{12}$ x 53, $D_4\times S_3^2$ x 16, $C_2\times D_6^2$ x 6, $D_6^2:C_2$ x 4, $C_6^2:D_4$ x 4, $C_6^2:C_2^3$ x 2, $C_6^2:D_4$ x 2, $C_6^2.C_2^3$ x 2, $D_6^2:C_2$ x 2, $C_6^2:D_4$ x 2, $D_{12}:D_6$, $C_2.D_6^2$, $C_6^2:C_2^3$

Order 216: $S_3^3$ x 64, $C_6:S_3^2$ x 48, $C_6\times S_3^2$ x 44, $C_6:S_3^2$ x 16, $C_3^2:D_{12}$ x 16, $C_3^2:D_{12}$ x 16, $C_3^2:D_{12}$ x 8, $C_6.C_6^2$ x 8, $C_3^2:D_{12}$ x 8, $C_3^2:D_{12}$ x 8, $C_6.S_3^2$ x 8, $C_6^2:C_6$ x 5, $S_3\times C_6^2$ x 4, $C_3^2:D_{12}$ x 4, $C_3:S_3\times C_{12}$ x 4, $C_3^2\times D_{12}$ x 4, $C_6.S_3^2$ x 4, $C_6^2:S_3$ x 2, $C_6.C_6^2$ x 2, $C_3\times C_6\times C_{12}$, $C_6^2.C_6$

Order 192: $C_{12}:C_2^4$ x 2, $C_2^3\times D_{12}$

Order 144: $S_3\times D_{12}$ x 168, $D_6^2$ x 125, $C_6:D_{12}$ x 58, $D_6:D_6$ x 32, $C_6:D_{12}$ x 29, $C_6\times D_{12}$ x 29, $C_{12}\times D_6$ x 29, $C_{12}:D_6$ x 16, $D_6:D_6$ x 16, $C_6^2:C_2^2$ x 10, $C_{12}:D_6$ x 8, $C_{12}:D_6$ x 8, $C_4\times S_3^2$ x 8, $C_6^2:C_2^2$ x 6, $C_6^2:C_2^2$ x 4, $C_2\times C_6\times C_{12}$ x 2, $C_6^2:C_2^2$ x 2, $C_6^2:C_4$ x 2, $D_6:D_6$ x 2, $D_6.D_6$ x 2, $C_4:C_6^2$, $C_{12}:D_6$, $C_2^2.S_3^2$

Order 108: $C_3:S_3^2$ x 48, $C_3\times S_3^2$ x 48, $C_3^2:D_6$ x 22, $C_3^2\times D_6$ x 20, $C_3:S_3^2$ x 16, $C_3^2:D_6$ x 8, $C_3^2:C_{12}$ x 4, $C_3^2\times C_{12}$ x 2, $C_3^2:C_{12}$ x 2, $C_3\times C_6^2$

Order 96: $D_4\times D_6$ x 49, $C_2^2\times D_{12}$ x 32, $C_2^3\times D_6$ x 6, $C_2^3:D_6$ x 4, $C_{12}:C_2^3$ x 2, $C_{12}:C_2^3$ x 2, $C_2^3\times C_{12}$

Order 72: $S_3\times D_6$ x 428, $C_6\times D_6$ x 115, $C_3:D_{12}$ x 104, $C_6:D_6$ x 67, $C_3:D_{12}$ x 52, $C_3\times D_{12}$ x 52, $S_3\times C_{12}$ x 52, $C_6\times C_{12}$ x 17, $C_6:C_{12}$ x 17, $C_6\wr C_2$ x 16, $C_6^2:C_2$ x 8, $D_6:S_3$ x 8, $C_6.D_6$ x 8, $C_2\times C_6^2$ x 4, $D_4\times C_3^2$ x 4, $C_{12}:S_3$ x 4, $C_6.D_6$ x 4, $C_6.D_6$

Order 64: $D_4\times C_2^3$

Order 54: $C_3^2:C_6$ x 12, $S_3\times C_3^2$ x 12, $C_3^2:S_3$ x 4, $C_3^2\times C_6$ x 3

Order 48: $C_2\times D_{12}$ x 162, $C_2^2\times D_6$ x 136, $S_3\times D_4$ x 136, $C_6:D_4$ x 50, $C_6\times D_4$ x 25, $C_4\times D_6$ x 25, $C_2^2\times C_{12}$ x 18, $C_2^3\times C_6$ x 5, $C_6.C_2^3$ x 2

Order 36: $S_3^2$ x 240, $C_6\times S_3$ x 222, $C_6:S_3$ x 126, $C_6^2$ x 27, $C_3\times C_{12}$ x 18, $C_3:C_{12}$ x 18, $C_3^2:C_4$ x 2

Order 32: $C_2^2\times D_4$ x 28, $C_2^5$ x 2, $C_2^3\times C_4$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 553, $D_{12}$ x 136, $C_3:D_4$ x 72, $C_2^2\times C_6$ x 61, $C_2\times C_{12}$ x 54, $C_4\times S_3$ x 36, $C_3\times D_4$ x 36, $C_6:C_4$ x 13

Order 18: $C_3\times S_3$ x 72, $C_3:S_3$ x 40, $C_3\times C_6$ x 33

Order 16: $C_2\times D_4$ x 112, $C_2^4$ x 45, $C_2^2\times C_4$ x 14

Order 12: $D_6$ x 478, $C_2\times C_6$ x 137, $C_{12}$ x 28, $C_3:C_4$ x 10

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 183, $D_4$ x 64, $C_2\times C_4$ x 28

Order 6: $S_3$ x 76, $C_6$ x 69

Order 4: $C_2^2$ x 163, $C_4$ x 8

Order 3: $C_3$ x 7

Order 2: $C_2$ x 31

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1728: $C_{12}:D_6^2$

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

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

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

Order 288: $D_6\times D_{12}$ x 10, $C_2\times D_6^2$ x 2, $C_6^2:C_2^3$, $C_6^2:D_4$, $C_6^2.C_2^3$, $D_6^2:C_2$, $D_6^2:C_2$, $C_6^2:D_4$, $D_4\times S_3^2$, $D_{12}:D_6$, $C_2.D_6^2$, $C_6^2:C_2^3$, $C_6^2:D_4$

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

Order 192: $C_{12}:C_2^4$, $C_2^3\times D_{12}$

Order 144: $D_6^2$ x 13, $S_3\times D_{12}$ x 9, $C_6:D_{12}$ x 8, $C_{12}\times D_6$ x 8, $C_6:D_{12}$ x 8, $C_6\times D_{12}$ x 7, $C_6^2:C_2^2$ x 3, $C_6^2:C_2^2$ x 2, $C_4:C_6^2$, $C_2\times C_6\times C_{12}$, $C_6^2:C_2^2$, $C_{12}:D_6$, $C_{12}:D_6$, $C_6^2:C_2^2$, $C_6^2:C_4$, $C_{12}:D_6$, $D_6:D_6$, $D_6:D_6$, $D_6:D_6$, $C_2^2.S_3^2$, $D_6.D_6$, $C_{12}:D_6$, $C_4\times S_3^2$

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

Order 96: $C_2^2\times D_{12}$ x 9, $D_4\times D_6$ x 8, $C_2^3\times D_6$ x 2, $C_{12}:C_2^3$, $C_2^3\times C_{12}$, $C_2^3:D_6$, $C_{12}:C_2^3$

Order 72: $S_3\times D_6$ x 20, $C_6\times D_6$ x 16, $C_6:D_6$ x 11, $C_6\times C_{12}$ x 7, $C_3:D_{12}$ x 7, $S_3\times C_{12}$ x 7, $C_3:D_{12}$ x 7, $C_6:C_{12}$ x 6, $C_3\times D_{12}$ x 6, $C_2\times C_6^2$ x 2, $D_4\times C_3^2$, $C_6^2:C_2$, $C_6.D_6$, $C_{12}:S_3$, $C_6\wr C_2$, $D_6:S_3$, $C_6.D_6$, $C_6.D_6$

Order 64: $D_4\times C_2^3$

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

Order 48: $C_2\times D_{12}$ x 23, $C_2^2\times D_6$ x 16, $C_2^2\times C_{12}$ x 7, $S_3\times D_4$ x 7, $C_6:D_4$ x 6, $C_4\times D_6$ x 6, $C_6\times D_4$ x 5, $C_2^3\times C_6$ x 2, $C_6.C_2^3$

Order 36: $C_6\times S_3$ x 23, $C_6:S_3$ x 16, $C_6^2$ x 9, $S_3^2$ x 9, $C_3\times C_{12}$ x 6, $C_3:C_{12}$ x 5, $C_3^2:C_4$

Order 32: $C_2^2\times D_4$ x 7, $C_2^5$, $C_2^3\times C_4$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 38, $C_2\times C_{12}$ x 15, $D_{12}$ x 15, $C_2^2\times C_6$ x 11, $C_3:D_4$ x 5, $C_4\times S_3$ x 5, $C_6:C_4$ x 4, $C_3\times D_4$ x 4

Order 18: $C_3\times C_6$ x 12, $C_3\times S_3$ x 10, $C_3:S_3$ x 7

Order 16: $C_2\times D_4$ x 13, $C_2^4$ x 7, $C_2^2\times C_4$ x 5

Order 12: $D_6$ x 36, $C_2\times C_6$ x 21, $C_{12}$ x 9, $C_3:C_4$ x 3

Order 9: $C_3^2$ x 5

Order 8: $C_2^3$ x 16, $D_4$ x 7, $C_2\times C_4$ x 7

Order 6: $C_6$ x 17, $S_3$ x 12

Order 4: $C_2^2$ x 16, $C_4$ x 3

Order 3: $C_3$ x 5

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1728: $C_{12}:D_6^2$ ($C_1$)

Order 864: $D_{12}\times S_3^2$ ($C_2$) x 16, $D_6^2:S_3$ ($C_2$) x 4, $S_3\times D_6^2$ ($C_2$) x 2, $C_6.D_6^2$ ($C_2$) x 2, $D_6^2:C_6$ ($C_2$) x 2, $C_6.D_6^2$ ($C_2$) x 2, $C_6.D_6^2$ ($C_2$), $C_6.D_6^2$ ($C_2$), $C_6.D_6^2$ ($C_2$)

Order 432: $D_6:S_3^2$ ($C_2^2$) x 32, $C_2\times S_3^3$ ($C_2^2$) x 16, $C_{12}:S_3^2$ ($C_2^2$) x 16, $C_{12}:S_3^2$ ($C_2^2$) x 16, $D_6:S_3^2$ ($C_2^2$) x 16, $C_{12}:S_3^2$ ($C_2^2$) x 8, $C_{12}:S_3^2$ ($C_2^2$) x 8, $C_{12}\times S_3^2$ ($C_2^2$) x 8, $C_3:D_6^2$ ($C_2^2$) x 6, $C_3\times D_6^2$ ($C_2^2$) x 5, $C_6^2.D_6$ ($C_2^2$) x 4, $C_6^2.D_6$ ($C_2^2$) x 4, $C_3:D_6^2$ ($C_2^2$) 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, $C_6^2.D_6$ ($C_2^2$) x 2, $C_6^2.D_6$ ($C_2^2$), $C_6^2.D_6$ ($C_2^2$), $C_{12}:C_6^2$ ($C_2^2$), $C_6^2.D_6$ ($C_2^2$)

Order 288: $D_6\times D_{12}$ ($S_3$) x 2, $C_2.D_6^2$ ($S_3$)

Order 216: $C_6:S_3^2$ ($C_2^3$) x 24, $C_6\times S_3^2$ ($C_2^3$) x 20, $C_3^2:D_{12}$ ($C_2^3$) x 16, $C_3^2:D_{12}$ ($C_2^3$) x 16, $C_6:S_3^2$ ($C_2^3$) x 8, $C_6\times S_3^2$ ($D_4$) x 8, $C_3^2:D_{12}$ ($C_2^3$) x 8, $C_6.C_6^2$ ($C_2^3$) x 8, $C_3^2:D_{12}$ ($C_2^3$) x 8, $C_3^2:D_{12}$ ($C_2^3$) x 8, $C_6.S_3^2$ ($C_2^3$) x 8, $C_6^2:C_6$ ($C_2^3$) x 5, $S_3\times C_6^2$ ($C_2^3$) x 4, $C_3^2:D_{12}$ ($C_2^3$) x 4, $C_3:S_3\times C_{12}$ ($C_2^3$) x 4, $C_3^2\times D_{12}$ ($C_2^3$) x 4, $C_6.S_3^2$ ($C_2^3$) x 4, $C_6^2:S_3$ ($C_2^3$) x 2, $C_6.C_6^2$ ($C_2^3$) x 2, $C_3\times C_6\times C_{12}$ ($C_2^3$), $C_6^2.C_6$ ($C_2^3$)

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

Order 108: $C_3\times S_3^2$ ($C_2\times D_4$) x 16, $C_3^2:D_6$ ($C_2^4$) x 10, $C_3^2\times D_6$ ($C_2^4$) x 8, $C_3^2\times D_6$ ($C_2\times D_4$) x 8, $C_3^2:D_6$ ($C_2^4$) x 4, $C_3^2:D_6$ ($C_2\times D_4$) x 4, $C_3^2:C_{12}$ ($C_2^4$) x 4, $C_3^2\times C_{12}$ ($C_2^4$) x 2, $C_3^2:C_{12}$ ($C_2^4$) x 2, $C_3\times C_6^2$ ($C_2^4$)

Order 72: $S_3\times D_6$ ($C_2\times D_6$) x 20, $S_3\times C_{12}$ ($C_2\times D_6$) x 16, $C_3:D_{12}$ ($C_2\times D_6$) x 16, $C_6\times D_6$ ($C_2\times D_6$) x 8, $S_3\times D_6$ ($D_{12}$) x 8, $C_3:D_{12}$ ($C_2\times D_6$) x 8, $C_3\times D_{12}$ ($C_2\times D_6$) x 8, $C_6.D_6$ ($C_2\times D_6$) x 8, $C_6:D_6$ ($C_2\times D_6$) x 5, $C_{12}:S_3$ ($C_2\times D_6$) x 4, $C_6:C_{12}$ ($C_2\times D_6$) x 4, $C_6.D_6$ ($C_2\times D_6$) x 4, $C_6\times C_{12}$ ($C_2\times D_6$) x 3, $C_6.D_6$ ($C_2\times D_6$)

Order 54: $S_3\times C_3^2$ ($C_2^2\times D_4$) x 8, $C_3^2:C_6$ ($C_2^2\times D_4$) x 4, $C_3^2\times C_6$ ($C_2^2\times D_4$) x 2, $C_3^2\times C_6$ ($C_2^5$)

Order 48: $C_4\times D_6$ ($S_3^2$) x 2, $C_2\times D_{12}$ ($S_3^2$)

Order 36: $C_6\times S_3$ ($C_2^2\times D_6$) x 16, $S_3^2$ ($C_2\times D_{12}$) x 16, $C_6:S_3$ ($C_2^2\times D_6$) x 10, $C_3:C_{12}$ ($C_2^2\times D_6$) x 8, $C_6\times S_3$ ($S_3\times D_4$) x 8, $C_6\times S_3$ ($C_2\times D_{12}$) x 8, $C_3\times C_{12}$ ($C_2^2\times D_6$) x 6, $C_6:S_3$ ($C_2\times D_{12}$) x 4, $C_6^2$ ($C_2^2\times D_6$) x 3, $C_3^2:C_4$ ($C_2^2\times D_6$) x 2

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

Order 24: $C_4\times S_3$ ($S_3\times D_6$) x 8, $D_{12}$ ($S_3\times D_6$) x 4, $C_2\times D_6$ ($S_3\times D_6$) x 4, $C_2\times C_{12}$ ($S_3\times D_6$) x 3, $C_6:C_4$ ($S_3\times D_6$) x 2

Order 18: $C_3\times S_3$ ($D_4\times D_6$) x 8, $C_3\times S_3$ ($C_2^2\times D_{12}$) x 8, $C_3\times C_6$ ($D_4\times D_6$) x 4, $C_3:S_3$ ($C_2^2\times D_{12}$) x 4, $C_3\times C_6$ ($C_2^3\times D_6$) x 3, $C_3\times C_6$ ($C_2^2\times D_{12}$) x 2

Order 12: $D_6$ ($D_6^2$) x 8, $D_6$ ($S_3\times D_{12}$) x 8, $C_{12}$ ($D_6^2$) x 6, $C_3:C_4$ ($D_6^2$) x 4, $C_2\times C_6$ ($D_6^2$) x 3

Order 9: $C_3^2$ ($C_{12}:C_2^4$) x 2, $C_3^2$ ($C_2^3\times D_{12}$)

Order 8: $C_2\times C_4$ ($S_3^3$)

Order 6: $S_3$ ($D_6\times D_{12}$) x 8, $C_6$ ($D_6\times D_{12}$) x 4, $C_6$ ($C_2\times D_6^2$) x 3, $C_6$ ($D_4\times S_3^2$) x 2

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

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

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

Order 1: $C_1$ ($C_{12}:D_6^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1728: $C_{12}:D_6^2$ ($C_1$)

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

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

Order 288: $D_6\times D_{12}$ ($S_3$), $C_2.D_6^2$ ($S_3$)

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

Order 144: $D_6^2$ ($D_6$) x 2, $C_{12}\times D_6$ ($D_6$) x 2, $C_6:D_{12}$ ($D_6$), $C_{12}:D_6$ ($D_6$), $C_6\times D_{12}$ ($D_6$), $C_6:D_{12}$ ($D_6$), $C_2^2.S_3^2$ ($D_6$), $D_6.D_6$ ($D_6$), $S_3\times D_{12}$ ($D_6$), $C_4\times S_3^2$ ($D_6$)

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

Order 72: $C_6\times D_6$ ($C_2\times D_6$) x 3, $C_6:D_6$ ($C_2\times D_6$) x 2, $S_3\times D_6$ ($C_2\times D_6$) x 2, $C_6\times C_{12}$ ($C_2\times D_6$) x 2, $C_6:C_{12}$ ($C_2\times D_6$) x 2, $S_3\times C_{12}$ ($C_2\times D_6$) x 2, $S_3\times D_6$ ($D_{12}$), $C_6.D_6$ ($C_2\times D_6$), $C_3:D_{12}$ ($C_2\times D_6$), $C_{12}:S_3$ ($C_2\times D_6$), $C_3\times D_{12}$ ($C_2\times D_6$), $C_3:D_{12}$ ($C_2\times D_6$), $C_6.D_6$ ($C_2\times D_6$), $C_6.D_6$ ($C_2\times D_6$)

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

Order 48: $C_2\times D_{12}$ ($S_3^2$), $C_4\times D_6$ ($S_3^2$)

Order 36: $C_6\times S_3$ ($C_2^2\times D_6$) x 3, $C_3\times C_{12}$ ($C_2^2\times D_6$) x 2, $C_3:C_{12}$ ($C_2^2\times D_6$) x 2, $C_6^2$ ($C_2^2\times D_6$) x 2, $C_6:S_3$ ($C_2^2\times D_6$) x 2, $C_3^2:C_4$ ($C_2^2\times D_6$), $C_6:S_3$ ($C_2\times D_{12}$), $C_6\times S_3$ ($S_3\times D_4$), $C_6\times S_3$ ($C_2\times D_{12}$), $S_3^2$ ($C_2\times D_{12}$)

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

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

Order 18: $C_3\times C_6$ ($C_2^3\times D_6$) x 2, $C_3\times C_6$ ($D_4\times D_6$), $C_3\times C_6$ ($C_2^2\times D_{12}$), $C_3:S_3$ ($C_2^2\times D_{12}$), $C_3\times S_3$ ($D_4\times D_6$), $C_3\times S_3$ ($C_2^2\times D_{12}$)

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

Order 9: $C_3^2$ ($C_{12}:C_2^4$), $C_3^2$ ($C_2^3\times D_{12}$)

Order 8: $C_2\times C_4$ ($S_3^3$)

Order 6: $C_6$ ($C_2\times D_6^2$) x 2, $C_6$ ($D_4\times S_3^2$), $C_6$ ($D_6\times D_{12}$), $S_3$ ($D_6\times D_{12}$)

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: $C_2$ ($S_3\times D_6^2$), $C_2$ ($D_{12}\times S_3^2$)

Order 1: $C_1$ ($C_{12}:D_6^2$)

Series

Derived series

$C_{12}:D_6^2$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_1$

Chief series

$C_{12}:D_6^2$

$\rhd$

$D_{12}\times S_3^2$

$\rhd$

$C_{12}:S_3^2$

$\rhd$

$C_3^2\times D_{12}$

$\rhd$

$C_3\times D_{12}$

$\rhd$

$C_3\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{12}:D_6^2$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_3^3$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $162 \times 162$ 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 $144 \times 144$ rational character table (warning: may be slow to load).