Abstract group 1728.24191: $C_6^3.D_4$

• Label: 1728.24191
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \)
• Perm deg.: $29$
• Trans deg.: $96$
• Rank: $3$

Group information

Description:	$C_6^3.D_4$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_6^2.C_2^6.C_2^5$, of order \(73728\)\(\medspace = 2^{13} \cdot 3^{2} \)
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	7	26	200	182	48	880	384	1728
Conjugacy classes	1	5	11	12	67	4	102	32	234
Divisions	1	5	7	10	39	2	46	8	118
Autjugacy classes	1	3	7	5	21	1	25	3	66

Dimension	1	2	4	8	16
Irr. complex chars.	48	108	78	0	0	234
Irr. rational chars.	8	28	52	22	8	118

Minimal presentations

Permutation degree:	$29$
Transitive degree:	$96$
Rank:	$3$
Inequivalent generating triples:	$17472$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	5	10	12

Constructions

Presentation:	$\langle a, b, c, d \mid b^{6}=c^{24}=d^{6}=[a,b]=[a,d]=[b,d]=1, a^{2}=c^{12}, c^{a}=c^{23}d^{3}, c^{b}=c^{13}, d^{c}=d^{5} \rangle$
Permutation group:	Degree $29$ $\langle(1,2,6,10)(3,11,12,16)(4,9,5,8)(7,13,15,14)(20,21)(22,23)(25,26), (1,3,4,13,6,12,5,14) \!\cdots\! \rangle$
Direct product:	$C_3$ $\, \times\, $ $(C_{12}.(C_4\times D_6))$
Semidirect product:	$(C_6^2.D_4)$ $\,\rtimes\,$ $C_6$ (4)	$(C_{12}.D_6)$ $\,\rtimes\,$ $C_{12}$ (4)	$(C_6^2.D_{12})$ $\,\rtimes\,$ $C_2$ (4)	$(C_{12}.C_6^2)$ $\,\rtimes\,$ $C_4$ (4)	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^3$ . $D_4$	$(C_{12}.D_6)$ . $D_6$ (2)	$C_{12}$ . $(D_6.D_6)$	$C_{12}$ . $(D_6:D_6)$	all 116

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

Homology

Abelianization:	$C_{2}^{2} \times C_{12} \simeq C_{2}^{2} \times C_{4} \times C_{3}$
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 3240 subgroups in 954 conjugacy classes, 224 normal (84 characteristic).

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

Special subgroups

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

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

Order 576: $(C_6\times Q_8):C_{12}$, $C_3\times C_2^3.D_{12}$, $C_{12}.(C_4\times D_6)$

Order 432: $C_{12}.C_6^2$ x 9, $C_6^2.D_6$ x 2, $C_{12}.C_6^2$ x 2, $C_{12}.C_6^2$ x 2, $C_6^2\times C_{12}$, $C_6^2:C_{12}$, $C_6^2:C_{12}$, $C_6^2.D_6$

Order 288: $C_6^2.D_4$ x 4, $C_6^2.D_4$ x 4, $C_6^2.D_4$ x 4, $C_6^2:Q_8$ x 3, $C_6^2.C_2^3$ x 3, $C_2\times C_{12}.C_{12}$ x 3, $C_8:C_6^2$, $C_6^2.C_2^3$, $Q_8\times C_6^2$

Order 216: $C_3^3:Q_8$ x 10, $C_3\times C_6\times C_{12}$ x 6, $C_6.C_6^2$ x 6, $C_3^2:C_{24}$ x 2, $C_6^2.C_6$ x 2, $C_6^3$

Order 192: $(C_2\times Q_8):C_{12}$, $(C_6\times Q_8):C_4$, $C_2^3.D_{12}$

Order 144: $C_{12}.D_6$ x 30, $C_4.C_6^2$ x 9, $C_{12}.C_{12}$ x 8, $C_2\times C_6\times C_{12}$ x 8, $C_{12}:C_{12}$ x 6, $C_6:C_{24}$ x 6, $C_6^2:C_4$ x 4, $C_{12}:C_{12}$ x 4, $C_6^2:C_4$ x 4, $C_6.D_{12}$ x 2, $C_3^2\times \OD_{16}$ x 2, $C_6\times C_{24}$ x 2, $C_3^2:C_4^2$, $C_6^2:C_4$

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

Order 96: $Q_8:C_{12}$ x 4, $C_6.Q_{16}$ x 4, $C_6.Q_{16}$ x 4, $C_6.C_2^4$ x 3, $C_6\times \OD_{16}$ x 3, $C_2^3.D_6$ x 3, $C_6.C_2^4$, $C_4^2:C_6$, $C_6:\OD_{16}$

Order 72: $C_6\times C_{12}$ x 48, $C_3^2:Q_8$ x 36, $C_6:C_{12}$ x 32, $Q_8\times C_3^2$ x 10, $C_3:C_{24}$ x 8, $C_2\times C_6^2$ x 7, $C_6.D_6$ x 2, $C_3\times C_{24}$ x 2

Order 64: $C_2^3.D_4$

Order 54: $C_3^2\times C_6$ x 5

Order 48: $C_6\times Q_8$ x 30, $C_2^2\times C_{12}$ x 11, $C_6:Q_8$ x 9, $C_3\times \OD_{16}$ x 8, $C_2\times C_{24}$ x 6, $C_{12}:C_4$ x 6, $C_6.D_4$ x 4, $C_{12}:C_4$ x 4, $C_6:C_8$ x 2, $C_4:C_{12}$ x 2, $C_3:\OD_{16}$ x 2, $C_6.C_2^3$, $C_2^2:C_{12}$, $C_4\times C_{12}$

Order 36: $C_6^2$ x 39, $C_3\times C_{12}$ x 32, $C_3:C_{12}$ x 24, $C_3^2:C_4$ x 2

Order 32: $Q_8:C_4$ x 4, $C_2^2\times Q_8$, $C_2\times \OD_{16}$, $C_4^2:C_2$

Order 27: $C_3^3$

Order 24: $C_2\times C_{12}$ x 68, $C_3\times Q_8$ x 36, $C_6:C_4$ x 14, $C_3:Q_8$ x 10, $C_{24}$ x 8, $C_2^2\times C_6$ x 7, $C_3:C_8$ x 2

Order 18: $C_3\times C_6$ x 39

Order 16: $C_2\times Q_8$ x 9, $\OD_{16}$ x 2, $C_2\times C_8$ x 2, $C_4:C_4$ x 2, $C_2^2\times C_4$ x 2, $C_2^2:C_4$, $C_4^2$

Order 12: $C_{12}$ x 46, $C_2\times C_6$ x 39, $C_3:C_4$ x 12

Order 9: $C_3^2$ x 7

Order 8: $C_2\times C_4$ x 14, $Q_8$ x 10, $C_8$ x 2, $C_2^3$

Order 6: $C_6$ x 39

Order 4: $C_4$ x 10, $C_2^2$ x 5

Order 3: $C_3$ x 7

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1728: $C_6^3.D_4$

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

Order 576: $(C_6\times Q_8):C_{12}$, $C_3\times C_2^3.D_{12}$, $C_{12}.(C_4\times D_6)$

Order 432: $C_{12}.C_6^2$ x 4, $C_6^2\times C_{12}$, $C_6^2:C_{12}$, $C_6^2:C_{12}$, $C_6^2.D_6$, $C_6^2.D_6$, $C_{12}.C_6^2$, $C_{12}.C_6^2$

Order 288: $C_6^2:Q_8$ x 3, $C_6^2.C_2^3$ x 3, $C_2\times C_{12}.C_{12}$ x 3, $C_8:C_6^2$, $C_6^2.C_2^3$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.D_4$, $Q_8\times C_6^2$

Order 216: $C_3\times C_6\times C_{12}$ x 4, $C_3^3:Q_8$ x 3, $C_6.C_6^2$ x 2, $C_3^2:C_{24}$, $C_6^3$, $C_6^2.C_6$

Order 192: $(C_2\times Q_8):C_{12}$, $(C_6\times Q_8):C_4$, $C_2^3.D_{12}$

Order 144: $C_{12}.D_6$ x 12, $C_2\times C_6\times C_{12}$ x 8, $C_4.C_6^2$ x 4, $C_6^2:C_4$ x 3, $C_{12}:C_{12}$ x 3, $C_{12}:C_{12}$ x 3, $C_{12}.C_{12}$ x 3, $C_6:C_{24}$ x 3, $C_6^2:C_4$ x 3, $C_6.D_{12}$, $C_3^2:C_4^2$, $C_3^2\times \OD_{16}$, $C_6\times C_{24}$, $C_6^2:C_4$

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

Order 96: $C_6.C_2^4$ x 3, $C_6\times \OD_{16}$ x 3, $C_2^3.D_6$ x 3, $Q_8:C_{12}$, $C_6.Q_{16}$, $C_6.Q_{16}$, $C_6.C_2^4$, $C_4^2:C_6$, $C_6:\OD_{16}$

Order 72: $C_6\times C_{12}$ x 30, $C_6:C_{12}$ x 9, $C_3^2:Q_8$ x 9, $C_2\times C_6^2$ x 7, $Q_8\times C_3^2$ x 3, $C_3:C_{24}$ x 3, $C_6.D_6$, $C_3\times C_{24}$

Order 64: $C_2^3.D_4$

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

Order 48: $C_6\times Q_8$ x 12, $C_2^2\times C_{12}$ x 10, $C_6:Q_8$ x 4, $C_3\times \OD_{16}$ x 3, $C_2\times C_{24}$ x 3, $C_6.D_4$ x 3, $C_{12}:C_4$ x 3, $C_{12}:C_4$ x 3, $C_6:C_8$, $C_6.C_2^3$, $C_4:C_{12}$, $C_2^2:C_{12}$, $C_4\times C_{12}$, $C_3:\OD_{16}$

Order 36: $C_3\times C_{12}$ x 22, $C_6^2$ x 21, $C_3:C_{12}$ x 6, $C_3^2:C_4$

Order 32: $C_2^2\times Q_8$, $C_2\times \OD_{16}$, $C_4^2:C_2$, $Q_8:C_4$

Order 27: $C_3^3$

Order 24: $C_2\times C_{12}$ x 35, $C_3\times Q_8$ x 9, $C_2^2\times C_6$ x 7, $C_6:C_4$ x 5, $C_3:Q_8$ x 3, $C_{24}$ x 3, $C_3:C_8$

Order 18: $C_3\times C_6$ x 21

Order 16: $C_2\times Q_8$ x 4, $C_2^2\times C_4$ x 2, $\OD_{16}$, $C_2\times C_8$, $C_4:C_4$, $C_2^2:C_4$, $C_4^2$

Order 12: $C_{12}$ x 25, $C_2\times C_6$ x 21, $C_3:C_4$ x 4

Order 9: $C_3^2$ x 7

Order 8: $C_2\times C_4$ x 7, $Q_8$ x 3, $C_2^3$, $C_8$

Order 6: $C_6$ x 21

Order 4: $C_4$ x 5, $C_2^2$ x 3

Order 3: $C_3$ x 7

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1728: $C_6^3.D_4$ ($C_1$)

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

Order 576: $C_{12}.(C_4\times D_6)$ ($C_3$)

Order 432: $C_{12}.C_6^2$ ($C_4$) x 4, $C_{12}.C_6^2$ ($C_2^2$) x 2, $C_6^2.D_6$ ($C_2^2$) x 2, $C_{12}.C_6^2$ ($C_2^2$) x 2, $C_6^2\times C_{12}$ ($C_2^2$)

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

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

Order 144: $C_{12}.D_6$ ($C_{12}$) x 4, $C_{12}.D_6$ ($C_3:C_4$) x 4, $C_6.D_{12}$ ($C_2\times C_6$) x 2, $C_6:C_{24}$ ($C_2\times C_6$) x 2, $C_6:C_{24}$ ($D_6$) x 2, $C_2\times C_6\times C_{12}$ ($D_6$) x 2, $C_{12}.D_6$ ($C_2\times C_6$) x 2, $C_{12}.D_6$ ($D_6$) x 2, $C_2\times C_6\times C_{12}$ ($C_2\times C_6$)

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

Order 96: $C_6.C_2^4$ ($C_3\times S_3$), $C_6:\OD_{16}$ ($C_3\times S_3$)

Order 72: $C_6\times C_{12}$ ($C_3:D_4$) x 5, $C_3^2:Q_8$ ($C_2\times C_{12}$) x 4, $C_3^2:Q_8$ ($C_6:C_4$) x 4, $C_6\times C_{12}$ ($C_3\times D_4$) x 3, $C_6\times C_{12}$ ($C_2\times C_{12}$) x 2, $C_6\times C_{12}$ ($C_6:C_4$) 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, $C_2\times C_6^2$ ($C_3:D_4$), $C_2\times C_6^2$ ($D_{12}$), $C_2\times C_6^2$ ($C_3\times D_4$), $C_6\times C_{12}$ ($D_{12}$), $C_6\times C_{12}$ ($C_2^2\times C_6$)

Order 54: $C_3^2\times C_6$ ($Q_{16}:C_2$) x 2, $C_3^2\times C_6$ ($C_2^3:C_4$)

Order 48: $C_6:Q_8$ ($C_3:C_{12}$) x 4, $C_6:C_8$ ($C_6\times S_3$) x 2, $C_2^2\times C_{12}$ ($C_6\times S_3$) x 2, $C_6:Q_8$ ($C_6\times S_3$) x 2, $C_2^2\times C_{12}$ ($S_3^2$)

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

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

Order 24: $C_2\times C_{12}$ ($C_6\wr C_2$) x 5, $C_3:Q_8$ ($C_6:C_{12}$) x 4, $C_2\times C_{12}$ ($C_6\times D_6$) x 2, $C_2\times C_{12}$ ($C_6:C_{12}$) x 2, $C_2\times C_{12}$ ($S_3\times C_{12}$) x 2, $C_2\times C_{12}$ ($D_6:S_3$) x 2, $C_2\times C_{12}$ ($C_6.D_6$) x 2, $C_2\times C_{12}$ ($S_3\times D_6$), $C_2\times C_{12}$ ($C_3\times D_{12}$), $C_2\times C_{12}$ ($C_3:D_{12}$), $C_2^2\times C_6$ ($C_6\wr C_2$), $C_2^2\times C_6$ ($C_3\times D_{12}$), $C_2^2\times C_6$ ($C_3:D_{12}$)

Order 18: $C_3\times C_6$ ($Q_{16}:C_6$) x 2, $C_3\times C_6$ ($C_{12}.D_4$) x 2, $C_3\times C_6$ ($C_4.D_{12}$) x 2, $C_3\times C_6$ ($C_2^3:C_{12}$), $C_3\times C_6$ ($C_2^3.D_6$), $C_3\times C_6$ ($C_2^2.D_{12}$)

Order 16: $C_2^2\times C_4$ ($C_3\times S_3^2$)

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

Order 9: $C_3^2$ ($(C_2\times Q_8):C_{12}$), $C_3^2$ ($(C_6\times Q_8):C_4$), $C_3^2$ ($C_2^3.D_{12}$)

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

Order 6: $C_6$ ($C_6^2.D_4$) x 2, $C_6$ ($C_{12}.D_{12}$) x 2, $C_6$ ($C_{12}.D_{12}$) x 2, $C_6$ ($C_6^2.D_4$), $C_6$ ($C_6^2.D_4$), $C_6$ ($C_6^2.D_4$)

Order 4: $C_2^2$ ($C_6^2.D_6$) x 2, $C_4$ ($C_6^2.D_6$) x 2, $C_2^2$ ($C_6^2.D_6$), $C_4$ ($C_6^2.D_6$), $C_4$ ($C_6^2.D_6$)

Order 3: $C_3$ ($(C_6\times Q_8):C_{12}$), $C_3$ ($C_3\times C_2^3.D_{12}$), $C_3$ ($C_{12}.(C_4\times D_6)$)

Order 2: $C_2$ ($C_6^2.D_{12}$) x 2, $C_2$ ($C_6^2.D_{12}$)

Order 1: $C_1$ ($C_6^3.D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1728: $C_6^3.D_4$ ($C_1$)

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

Order 576: $C_{12}.(C_4\times D_6)$ ($C_3$)

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

Order 288: $C_6^2:Q_8$ ($C_6$), $C_6^2:Q_8$ ($S_3$), $C_6^2.C_2^3$ ($C_6$), $C_2\times C_{12}.C_{12}$ ($C_6$), $C_2\times C_{12}.C_{12}$ ($S_3$), $C_6^2.D_4$ ($C_6$)

Order 216: $C_3\times C_6\times C_{12}$ ($D_4$) x 2, $C_6^3$ ($D_4$), $C_3\times C_6\times C_{12}$ ($C_2^3$), $C_3\times C_6\times C_{12}$ ($C_2\times C_4$), $C_3^3:Q_8$ ($C_2\times C_4$)

Order 144: $C_2\times C_6\times C_{12}$ ($D_6$) x 2, $C_6.D_{12}$ ($C_2\times C_6$), $C_6:C_{24}$ ($C_2\times C_6$), $C_6:C_{24}$ ($D_6$), $C_2\times C_6\times C_{12}$ ($C_2\times C_6$), $C_{12}.D_6$ ($C_2\times C_6$), $C_{12}.D_6$ ($D_6$), $C_{12}.D_6$ ($C_{12}$), $C_{12}.D_6$ ($C_3:C_4$)

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

Order 96: $C_6.C_2^4$ ($C_3\times S_3$), $C_6:\OD_{16}$ ($C_3\times S_3$)

Order 72: $C_6\times C_{12}$ ($C_3:D_4$) x 3, $C_6\times C_{12}$ ($C_2\times D_6$) x 2, $C_6\times C_{12}$ ($C_3\times D_4$) x 2, $C_2\times C_6^2$ ($C_3:D_4$), $C_2\times C_6^2$ ($D_{12}$), $C_2\times C_6^2$ ($C_3\times D_4$), $C_6\times C_{12}$ ($C_2\times C_{12}$), $C_6\times C_{12}$ ($C_6:C_4$), $C_6\times C_{12}$ ($D_{12}$), $C_6\times C_{12}$ ($C_4\times S_3$), $C_6\times C_{12}$ ($C_2^2\times C_6$), $C_3^2:Q_8$ ($C_2\times C_{12}$), $C_3^2:Q_8$ ($C_6:C_4$)

Order 54: $C_3^2\times C_6$ ($Q_{16}:C_2$), $C_3^2\times C_6$ ($C_2^3:C_4$)

Order 48: $C_2^2\times C_{12}$ ($C_6\times S_3$) x 2, $C_6:C_8$ ($C_6\times S_3$), $C_2^2\times C_{12}$ ($S_3^2$), $C_6:Q_8$ ($C_3:C_{12}$), $C_6:Q_8$ ($C_6\times S_3$)

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

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

Order 24: $C_2\times C_{12}$ ($C_6\wr C_2$) x 3, $C_2\times C_{12}$ ($C_6\times D_6$) x 2, $C_2\times C_{12}$ ($S_3\times D_6$), $C_2\times C_{12}$ ($C_6:C_{12}$), $C_2\times C_{12}$ ($C_3\times D_{12}$), $C_2\times C_{12}$ ($S_3\times C_{12}$), $C_2\times C_{12}$ ($C_3:D_{12}$), $C_2\times C_{12}$ ($D_6:S_3$), $C_2\times C_{12}$ ($C_6.D_6$), $C_3:Q_8$ ($C_6:C_{12}$), $C_2^2\times C_6$ ($C_6\wr C_2$), $C_2^2\times C_6$ ($C_3\times D_{12}$), $C_2^2\times C_6$ ($C_3:D_{12}$)

Order 18: $C_3\times C_6$ ($Q_{16}:C_6$), $C_3\times C_6$ ($C_2^3:C_{12}$), $C_3\times C_6$ ($C_2^3.D_6$), $C_3\times C_6$ ($C_{12}.D_4$), $C_3\times C_6$ ($C_2^2.D_{12}$), $C_3\times C_6$ ($C_4.D_{12}$)

Order 16: $C_2^2\times C_4$ ($C_3\times S_3^2$)

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

Order 9: $C_3^2$ ($(C_2\times Q_8):C_{12}$), $C_3^2$ ($(C_6\times Q_8):C_4$), $C_3^2$ ($C_2^3.D_{12}$)

Order 8: $C_2^3$ ($C_3^2:D_{12}$), $C_2\times C_4$ ($C_6\times S_3^2$), $C_2\times C_4$ ($C_3^2:D_{12}$), $C_2\times C_4$ ($C_3^3:D_4$), $C_2\times C_4$ ($C_6.S_3^2$)

Order 6: $C_6$ ($C_6^2.D_4$), $C_6$ ($C_6^2.D_4$), $C_6$ ($C_6^2.D_4$), $C_6$ ($C_{12}.D_{12}$), $C_6$ ($C_6^2.D_4$), $C_6$ ($C_{12}.D_{12}$)

Order 4: $C_2^2$ ($C_6^2.D_6$), $C_2^2$ ($C_6^2.D_6$), $C_4$ ($C_6^2.D_6$), $C_4$ ($C_6^2.D_6$), $C_4$ ($C_6^2.D_6$)

Order 3: $C_3$ ($(C_6\times Q_8):C_{12}$), $C_3$ ($C_3\times C_2^3.D_{12}$), $C_3$ ($C_{12}.(C_4\times D_6)$)

Order 2: $C_2$ ($C_6^2.D_{12}$), $C_2$ ($C_6^2.D_{12}$)

Order 1: $C_1$ ($C_6^3.D_4$)

Series

Derived series

$C_6^3.D_4$

$\rhd$

$C_3\times C_{12}$

$\rhd$

$C_1$

Chief series

$C_6^3.D_4$

$\rhd$

$C_6^2.D_{12}$

$\rhd$

$C_{12}.C_6^2$

$\rhd$

$C_6:C_{24}$

$\rhd$

$C_6\times C_{12}$

$\rhd$

$C_6^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6^3.D_4$

$\rhd$

$C_3\times C_{12}$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2\times C_6$

$\lhd$

$C_2^2\times C_{12}$

Character theory

Complex character table

See the $234 \times 234$ 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 $118 \times 118$ rational character table.