Abstract group 1728.24984: $C_{12}^2.D_6$

• Label: 1728.24984
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \cdot 3 \)
• Perm deg.: $21$
• Trans deg.: $576$
• Rank: $3$

Group information

Description:	$C_{12}^2.D_6$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$\PSU(3,2).C_6^2.C_2^6.C_2^2$, of order \(663552\)\(\medspace = 2^{13} \cdot 3^{4} \)
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	8	12	24
Elements	1	27	26	36	270	576	504	288	1728
Conjugacy classes	1	7	9	12	43	20	108	16	216
Divisions	1	7	9	7	43	5	53	3	128
Autjugacy classes	1	4	3	5	10	3	13	2	41

Dimension	1	2	4	8	16
Irr. complex chars.	32	104	80	0	0	216
Irr. rational chars.	8	42	49	24	5	128

Minimal presentations

Permutation degree:	$21$
Transitive degree:	$576$
Rank:	$3$
Inequivalent generating triples:	$1344$

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:	$\langle a, b, c, d \mid a^{8}=b^{3}=c^{6}=d^{12}=[a,d]=[b,c]=[b,d]=1, b^{a}=b^{2}, c^{a}=c^{5}, d^{c}=d^{11} \rangle$
Permutation group:	Degree $21$ $\langle(2,3)(5,6)(7,8,10,12,9,11,13,14)(15,16)(17,18), (7,9)(8,11)(10,13)(12,14) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 73 & 70 \\ 10 & 53 \end{array}\right), \left(\begin{array}{rr} 71 & 0 \\ 0 & 71 \end{array}\right), \left(\begin{array}{rr} 73 & 54 \\ 18 & 1 \end{array}\right), \left(\begin{array}{rr} 10 & 63 \\ 19 & 80 \end{array}\right), \left(\begin{array}{rr} 31 & 30 \\ 30 & 61 \end{array}\right), \left(\begin{array}{rr} 46 & 45 \\ 45 & 1 \end{array}\right), \left(\begin{array}{rr} 73 & 0 \\ 0 & 73 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right), \left(\begin{array}{rr} 49 & 30 \\ 10 & 59 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/90\Z)$
Direct product:	$D_{12}$ $\, \times\, $ $(C_3^2:C_8)$
Semidirect product:	$D_6$ $\,\rtimes\,$ $(C_{12}.D_6)$	$C_{12}$ $\,\rtimes\,$ $(C_{12}.D_6)$	$C_{12}$ $\,\rtimes\,$ $(C_{12}.D_6)$	$(C_{12}^2.S_3)$ $\,\rtimes\,$ $C_2$	all 21
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{12}^2$ . $D_6$ (2)	$(C_{12}\times D_6)$ . $D_6$	$(C_{12}.D_6)$ . $D_6$ (2)	$(C_{12}\times D_{12})$ . $S_3$	all 77

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

Homology

Abelianization:	$C_{2}^{2} \times C_{8} $
Schur multiplier:	$C_{2}^{2} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 4288 subgroups in 956 conjugacy classes, 288 normal (58 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $C_6:S_3^2$
Commutator:	$G' \simeq$ $C_3^2\times C_6$	$G/G' \simeq$ $C_2^2\times C_8$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_6:S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_{12}^2$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_{12}^2.D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

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

Order 864: $C_3^2:C_8\times D_6$ x 2, $(C_3^2\times D_6):C_8$ x 2, $C_{12}^2:C_6$, $C_{12}^2.S_3$, $C_{12}^2.C_6$

Order 576: $C_4^2.S_3^2$ x 4, $C_{12}^2.C_2^2$

Order 432: $C_{12}.S_3^2$ x 4, $C_{12}:C_6^2$ x 2, $C_2\times C_3^3:C_8$ x 2, $C_6^2.C_{12}$ x 2, $C_6^2.D_6$ x 2, $C_{12}:C_6^2$, $C_3\times C_{12}^2$, $C_6^2.D_6$

Order 288: $C_{12}.D_{12}$ x 9, $S_3\times C_6:C_8$ x 8, $C_{12}.D_{12}$ x 8, $C_{12}\times D_{12}$ x 4, $C_{12}:C_{24}$ x 4, $C_6^2:C_8$ x 2, $C_6^2:C_8$ x 2, $C_{12}^2:C_2$, $C_{12}^2.C_2$

Order 216: $C_3^2\times D_{12}$ x 4, $C_6.C_6^2$ x 4, $C_3^2:C_{24}$ x 3, $C_3\times C_6\times C_{12}$ x 3, $C_3^3:C_8$ x 2, $S_3\times C_6^2$ x 2, $C_6.C_6^2$ x 2

Order 192: $C_4^2.D_6$ x 4, $C_8\times D_{12}$

Order 144: $C_{12}.D_6$ x 24, $C_{12}.D_6$ x 16, $C_{12}^2$ x 9, $D_6:C_{12}$ x 8, $C_6:C_{24}$ x 8, $C_{12}\times D_6$ x 8, $C_{12}:C_{12}$ x 4, $C_6\times D_{12}$ x 4, $C_2\times C_6\times C_{12}$ x 2, $C_6^2:C_4$ x 2, $C_4:C_6^2$, $C_{12}:C_{12}$

Order 108: $C_3^2\times D_6$ x 8, $C_3^2\times C_{12}$ x 5, $C_3^2:C_{12}$ x 2, $C_3\times C_6^2$

Order 96: $C_{12}:C_8$ x 9, $C_{12}.D_4$ x 8, $C_{12}.C_2^3$ x 8, $C_{12}:C_8$ x 4, $D_4\times C_{12}$ x 4, $D_6:C_8$ x 2, $C_8\times D_6$ x 2, $C_4\times D_{12}$, $C_4\times C_{24}$

Order 72: $C_6\times C_{12}$ x 33, $C_3^2:C_8$ x 21, $C_3\times D_{12}$ x 16, $S_3\times C_{12}$ x 16, $C_3:C_{24}$ x 12, $C_6\times D_6$ x 8, $C_6:C_{12}$ x 8, $D_4\times C_3^2$ x 4, $C_2\times C_6^2$ x 2

Order 64: $C_8\times D_4$

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

Order 48: $C_6:C_8$ x 42, $C_4\times C_{12}$ x 9, $C_2^2\times C_{12}$ x 8, $C_2^2:C_{12}$ x 8, $C_6\times D_4$ x 4, $S_3\times C_8$ x 4, $C_4:C_{12}$ x 4, $C_4\times D_6$ x 2, $C_2\times C_{24}$ x 2, $D_6:C_4$ x 2, $C_2\times D_{12}$, $C_{12}:C_4$

Order 36: $C_3\times C_{12}$ x 47, $C_6\times S_3$ x 32, $C_6^2$ x 17, $C_3:C_{12}$ x 8

Order 32: $C_2^2:C_8$ x 2, $C_2^2\times C_8$ x 2, $C_4\times C_8$, $C_4\times D_4$, $C_4:C_8$

Order 27: $C_3^3$

Order 24: $C_2\times C_{12}$ x 51, $C_3:C_8$ x 30, $C_3\times D_4$ x 16, $C_2^2\times C_6$ x 8, $D_{12}$ x 4, $C_4\times S_3$ x 4, $C_{24}$ x 3, $C_6:C_4$ x 2, $C_2\times D_6$ x 2

Order 18: $C_3\times C_6$ x 31, $C_3\times S_3$ x 16

Order 16: $C_2\times C_8$ x 8, $C_2^2:C_4$ x 2, $C_2^2\times C_4$ x 2, $C_4:C_4$, $C_4^2$, $C_2\times D_4$

Order 12: $C_{12}$ x 53, $C_2\times C_6$ x 41, $D_6$ x 8, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 9

Order 8: $C_2\times C_4$ x 9, $C_8$ x 5, $D_4$ x 4, $C_2^3$ x 2

Order 6: $C_6$ x 43, $S_3$ x 4

Order 4: $C_2^2$ x 9, $C_4$ x 7

Order 3: $C_3$ x 9

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

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

Order 864: $C_{12}^2:C_6$, $C_3^2:C_8\times D_6$, $C_{12}^2.S_3$, $C_{12}^2.C_6$, $(C_3^2\times D_6):C_8$

Order 576: $C_{12}^2.C_2^2$, $C_4^2.S_3^2$

Order 432: $C_6^2.C_{12}$ x 2, $C_{12}:C_6^2$, $C_{12}:C_6^2$, $C_3\times C_{12}^2$, $C_2\times C_3^3:C_8$, $C_6^2.D_6$, $C_6^2.D_6$, $C_{12}.S_3^2$

Order 288: $C_{12}.D_{12}$ x 3, $C_{12}^2:C_2$, $C_6^2:C_8$, $C_{12}\times D_{12}$, $S_3\times C_6:C_8$, $C_6^2:C_8$, $C_{12}^2.C_2$, $C_{12}:C_{24}$, $C_{12}.D_{12}$

Order 216: $C_3\times C_6\times C_{12}$ x 3, $C_3^2:C_{24}$ x 2, $C_6.C_6^2$ x 2, $C_3^3:C_8$, $S_3\times C_6^2$, $C_6.C_6^2$, $C_3^2\times D_{12}$

Order 192: $C_4^2.D_6$, $C_8\times D_{12}$

Order 144: $C_{12}.D_6$ x 6, $C_{12}^2$ x 3, $C_6:C_{24}$ x 2, $D_6:C_{12}$, $C_{12}:C_{12}$, $C_{12}.D_6$, $C_4:C_6^2$, $C_2\times C_6\times C_{12}$, $C_6\times D_{12}$, $C_{12}\times D_6$, $C_{12}:C_{12}$, $C_6^2:C_4$

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

Order 96: $C_{12}:C_8$ x 3, $C_{12}:C_8$, $C_4\times D_{12}$, $C_4\times C_{24}$, $C_{12}.D_4$, $D_6:C_8$, $D_4\times C_{12}$, $C_{12}.C_2^3$, $C_8\times D_6$

Order 72: $C_6\times C_{12}$ x 12, $C_3^2:C_8$ x 5, $S_3\times C_{12}$ x 2, $C_3:C_{24}$ x 2, $C_2\times C_6^2$, $C_6\times D_6$, $D_4\times C_3^2$, $C_6:C_{12}$, $C_3\times D_{12}$

Order 64: $C_8\times D_4$

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

Order 48: $C_6:C_8$ x 6, $C_4\times C_{12}$ x 3, $C_2\times C_{24}$ x 2, $C_6\times D_4$, $C_2^2\times C_{12}$, $S_3\times C_8$, $C_2\times D_{12}$, $C_4\times D_6$, $C_4:C_{12}$, $C_2^2:C_{12}$, $D_6:C_4$, $C_{12}:C_4$

Order 36: $C_3\times C_{12}$ x 13, $C_6^2$ x 6, $C_6\times S_3$ x 3, $C_3:C_{12}$

Order 32: $C_2^2:C_8$, $C_2^2\times C_8$, $C_4\times C_8$, $C_4\times D_4$, $C_4:C_8$

Order 27: $C_3^3$

Order 24: $C_2\times C_{12}$ x 12, $C_3:C_8$ x 5, $C_4\times S_3$ x 2, $C_{24}$ x 2, $C_6:C_4$, $D_{12}$, $C_2^2\times C_6$, $C_2\times D_6$, $C_3\times D_4$

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

Order 16: $C_2\times C_8$ x 4, $C_4:C_4$, $C_2^2:C_4$, $C_4^2$, $C_2\times D_4$, $C_2^2\times C_4$

Order 12: $C_{12}$ x 13, $C_2\times C_6$ x 6, $D_6$ x 3, $C_3:C_4$

Order 9: $C_3^2$ x 3

Order 8: $C_2\times C_4$ x 6, $C_8$ x 3, $C_2^3$, $D_4$

Order 6: $C_6$ x 10, $S_3$

Order 4: $C_4$ x 5, $C_2^2$ x 4

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

Order 864: $C_3^2:C_8\times D_6$ ($C_2$) x 2, $(C_3^2\times D_6):C_8$ ($C_2$) x 2, $C_{12}^2:C_6$ ($C_2$), $C_{12}^2.S_3$ ($C_2$), $C_{12}^2.C_6$ ($C_2$)

Order 432: $C_{12}:C_6^2$ ($C_2^2$) x 2, $C_2\times C_3^3:C_8$ ($C_2^2$) x 2, $C_6^2.C_{12}$ ($C_2^2$) x 2, $C_6^2.D_6$ ($C_4$) x 2, $C_{12}:C_6^2$ ($C_4$), $C_3\times C_{12}^2$ ($C_2^2$), $C_6^2.D_6$ ($C_4$)

Order 288: $C_{12}\times D_{12}$ ($S_3$) x 4, $C_{12}^2.C_2$ ($S_3$)

Order 216: $C_3^2\times D_{12}$ ($C_8$) x 4, $C_3^2:C_{24}$ ($D_4$) x 2, $S_3\times C_6^2$ ($C_2\times C_4$) x 2, $C_3\times C_6\times C_{12}$ ($C_2\times C_4$) x 2, $C_6.C_6^2$ ($C_2\times C_4$) x 2, $C_3\times C_6\times C_{12}$ ($C_2^3$)

Order 144: $D_6:C_{12}$ ($C_3:C_4$) x 8, $C_{12}\times D_6$ ($D_6$) x 8, $C_{12}^2$ ($D_6$) x 5, $C_{12}:C_{12}$ ($C_3:C_4$) x 4, $C_6\times D_{12}$ ($C_3:C_4$) x 4, $C_{12}.D_6$ ($D_6$) x 2

Order 108: $C_3^2\times D_6$ ($C_2\times C_8$) x 4, $C_3^2\times C_{12}$ ($C_2\times C_8$) x 2, $C_3\times C_6^2$ ($C_2^2\times C_4$), $C_3^2\times C_{12}$ ($D_4:C_2$), $C_3^2\times C_{12}$ ($C_2\times D_4$)

Order 96: $C_4\times D_{12}$ ($C_3:S_3$)

Order 72: $C_3\times D_{12}$ ($C_3:C_8$) x 16, $C_6\times D_6$ ($C_6:C_4$) x 8, $C_6\times C_{12}$ ($C_6:C_4$) x 8, $C_6:C_{12}$ ($C_6:C_4$) x 8, $C_6\times C_{12}$ ($C_2\times D_6$) x 5, $C_6\times C_{12}$ ($C_4\times S_3$) x 2, $C_3^2:C_8$ ($D_{12}$) x 2

Order 54: $C_3^2\times C_6$ ($\OD_{16}:C_2$), $C_3^2\times C_6$ ($C_2^2\times C_8$), $C_3^2\times C_6$ ($C_4\times D_4$)

Order 48: $C_4\times C_{12}$ ($S_3^2$) x 4, $C_4\times D_6$ ($C_6:S_3$) x 2, $D_6:C_4$ ($C_3^2:C_4$) x 2, $C_2\times D_{12}$ ($C_3^2:C_4$), $C_4\times C_{12}$ ($C_6:S_3$), $C_{12}:C_4$ ($C_3^2:C_4$)

Order 36: $C_6\times S_3$ ($C_6:C_8$) x 16, $C_3\times C_{12}$ ($C_6:C_8$) x 8, $C_3\times C_{12}$ ($D_4:S_3$) x 4, $C_3\times C_{12}$ ($S_3\times D_4$) x 4, $C_6^2$ ($C_6.C_2^3$) x 4, $C_3\times C_{12}$ ($S_3\times C_8$) x 2, $C_3\times C_{12}$ ($D_{12}:C_2$), $C_3\times C_{12}$ ($C_2\times D_{12}$), $C_6^2$ ($C_4\times D_6$)

Order 27: $C_3^3$ ($C_8\times D_4$)

Order 24: $C_2\times C_{12}$ ($C_6.D_6$) x 8, $C_2\times C_{12}$ ($S_3\times D_6$) x 4, $D_{12}$ ($C_3^2:C_8$) x 4, $C_2\times C_{12}$ ($C_6.D_6$) x 2, $C_6:C_4$ ($C_6.D_6$) x 2, $C_2\times D_6$ ($C_6.D_6$) x 2, $C_2\times C_{12}$ ($C_6:D_6$)

Order 18: $C_3\times C_6$ ($C_{12}.C_2^3$) x 4, $C_3\times C_6$ ($C_2^3.D_6$) x 4, $C_3\times C_6$ ($C_{12}.C_2^3$) x 4, $C_3\times C_6$ ($C_4\times D_{12}$), $C_3\times C_6$ ($C_8.D_6$), $C_3\times C_6$ ($C_8\times D_6$)

Order 16: $C_4^2$ ($C_3:S_3^2$)

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

Order 9: $C_3^2$ ($C_4^2.D_6$) x 4, $C_3^2$ ($C_8\times D_{12}$)

Order 8: $C_2\times C_4$ ($C_6.S_3^2$) x 2, $C_2\times C_4$ ($C_6:S_3^2$)

Order 6: $C_6$ ($C_6^2.C_2^3$) x 4, $C_6$ ($C_{12}.(C_4\times S_3)$) x 4, $C_6$ ($S_3\times C_6:C_8$) x 4, $C_6$ ($C_{12}.(C_2\times D_6)$), $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_6^2:C_8$)

Order 4: $C_4$ ($C_{12}.S_3^2$) x 2, $C_2^2$ ($C_6^2.D_6$), $C_4$ ($C_{12}:S_3^2$), $C_4$ ($C_{12}.S_3^2$)

Order 3: $C_3$ ($C_4^2.S_3^2$) x 4, $C_3$ ($C_{12}^2.C_2^2$)

Order 2: $C_2$ ($C_3^2:C_4\times D_{12}$), $C_2$ ($C_{12}.(S_3\times D_6)$), $C_2$ ($C_3^2:C_8\times D_6$)

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

Normal subgroups up to automorphism (quotient in parentheses)

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

Order 864: $C_{12}^2:C_6$ ($C_2$), $C_3^2:C_8\times D_6$ ($C_2$), $C_{12}^2.S_3$ ($C_2$), $C_{12}^2.C_6$ ($C_2$), $(C_3^2\times D_6):C_8$ ($C_2$)

Order 432: $C_6^2.C_{12}$ ($C_2^2$) x 2, $C_{12}:C_6^2$ ($C_4$), $C_{12}:C_6^2$ ($C_2^2$), $C_3\times C_{12}^2$ ($C_2^2$), $C_2\times C_3^3:C_8$ ($C_2^2$), $C_6^2.D_6$ ($C_4$), $C_6^2.D_6$ ($C_4$)

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

Order 216: $C_3\times C_6\times C_{12}$ ($C_2\times C_4$) x 2, $C_3^2:C_{24}$ ($D_4$), $S_3\times C_6^2$ ($C_2\times C_4$), $C_3\times C_6\times C_{12}$ ($C_2^3$), $C_6.C_6^2$ ($C_2\times C_4$), $C_3^2\times D_{12}$ ($C_8$)

Order 144: $C_{12}.D_6$ ($D_6$) x 2, $C_{12}^2$ ($D_6$) x 2, $D_6:C_{12}$ ($C_3:C_4$), $C_{12}:C_{12}$ ($C_3:C_4$), $C_6\times D_{12}$ ($C_3:C_4$), $C_{12}\times D_6$ ($D_6$)

Order 108: $C_3\times C_6^2$ ($C_2^2\times C_4$), $C_3^2\times D_6$ ($C_2\times C_8$), $C_3^2\times C_{12}$ ($C_2\times C_8$), $C_3^2\times C_{12}$ ($D_4:C_2$), $C_3^2\times C_{12}$ ($C_2\times D_4$)

Order 96: $C_4\times D_{12}$ ($C_3:S_3$)

Order 72: $C_6\times C_{12}$ ($C_6:C_4$) x 2, $C_6\times C_{12}$ ($C_4\times S_3$) x 2, $C_6\times C_{12}$ ($C_2\times D_6$) x 2, $C_6\times D_6$ ($C_6:C_4$), $C_6:C_{12}$ ($C_6:C_4$), $C_3\times D_{12}$ ($C_3:C_8$), $C_3^2:C_8$ ($D_{12}$)

Order 54: $C_3^2\times C_6$ ($\OD_{16}:C_2$), $C_3^2\times C_6$ ($C_2^2\times C_8$), $C_3^2\times C_6$ ($C_4\times D_4$)

Order 48: $C_2\times D_{12}$ ($C_3^2:C_4$), $C_4\times D_6$ ($C_6:S_3$), $C_4\times C_{12}$ ($C_6:S_3$), $C_4\times C_{12}$ ($S_3^2$), $D_6:C_4$ ($C_3^2:C_4$), $C_{12}:C_4$ ($C_3^2:C_4$)

Order 36: $C_3\times C_{12}$ ($C_6:C_8$), $C_3\times C_{12}$ ($S_3\times C_8$), $C_3\times C_{12}$ ($D_4:S_3$), $C_3\times C_{12}$ ($S_3\times D_4$), $C_3\times C_{12}$ ($D_{12}:C_2$), $C_3\times C_{12}$ ($C_2\times D_{12}$), $C_6^2$ ($C_6.C_2^3$), $C_6^2$ ($C_4\times D_6$), $C_6\times S_3$ ($C_6:C_8$)

Order 27: $C_3^3$ ($C_8\times D_4$)

Order 24: $C_2\times C_{12}$ ($C_6.D_6$) x 2, $C_2\times C_{12}$ ($C_6.D_6$) x 2, $C_2\times C_{12}$ ($C_6:D_6$), $C_2\times C_{12}$ ($S_3\times D_6$), $C_6:C_4$ ($C_6.D_6$), $D_{12}$ ($C_3^2:C_8$), $C_2\times D_6$ ($C_6.D_6$)

Order 18: $C_3\times C_6$ ($C_4\times D_{12}$), $C_3\times C_6$ ($C_{12}.C_2^3$), $C_3\times C_6$ ($C_2^3.D_6$), $C_3\times C_6$ ($C_{12}.C_2^3$), $C_3\times C_6$ ($C_8.D_6$), $C_3\times C_6$ ($C_8\times D_6$)

Order 16: $C_4^2$ ($C_3:S_3^2$)

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

Order 9: $C_3^2$ ($C_4^2.D_6$), $C_3^2$ ($C_8\times D_{12}$)

Order 8: $C_2\times C_4$ ($C_6.S_3^2$) x 2, $C_2\times C_4$ ($C_6:S_3^2$)

Order 6: $C_6$ ($C_{12}.(C_2\times D_6)$), $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_6^2:C_8$), $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_{12}.(C_4\times S_3)$), $C_6$ ($S_3\times C_6:C_8$)

Order 4: $C_2^2$ ($C_6^2.D_6$), $C_4$ ($C_{12}:S_3^2$), $C_4$ ($C_{12}.S_3^2$), $C_4$ ($C_{12}.S_3^2$)

Order 3: $C_3$ ($C_{12}^2.C_2^2$), $C_3$ ($C_4^2.S_3^2$)

Order 2: $C_2$ ($C_3^2:C_4\times D_{12}$), $C_2$ ($C_{12}.(S_3\times D_6)$), $C_2$ ($C_3^2:C_8\times D_6$)

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

Series

Derived series

$C_{12}^2.D_6$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_1$

Chief series

$C_{12}^2.D_6$

$\rhd$

$C_{12}^2:C_6$

$\rhd$

$C_{12}:C_6^2$

$\rhd$

$C_3^2\times D_{12}$

$\rhd$

$C_3\times D_{12}$

$\rhd$

$C_3\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{12}^2.D_6$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_3^3$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_4^2$

Character theory

Complex character table

See the $216 \times 216$ 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 $128 \times 128$ rational character table (warning: may be slow to load).