Abstract group 1728.23380: $C_{12}^2.D_6$

• Label: 1728.23380
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• Perm deg.: $25$
• Trans deg.: $192$
• 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:	$C_6^2.C_2^6.C_2^4$, of order \(36864\)\(\medspace = 2^{12} \cdot 3^{2} \)
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	3	26	204	78	48	984	384	1728
Conjugacy classes	1	3	11	11	33	4	112	32	207
Divisions	1	3	7	10	21	2	48	6	98
Autjugacy classes	1	3	7	7	21	1	33	3	76

Dimension	1	2	4	8	16
Irr. complex chars.	24	102	81	0	0	207
Irr. rational chars.	8	20	34	26	10	98

Minimal presentations

Permutation degree:	$25$
Transitive degree:	$192$
Rank:	$3$
Inequivalent generating triples:	$34944$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	6	8	12

Constructions

Presentation:	$\langle a, b, c \mid b^{24}=c^{12}=1, a^{6}=c^{6}, b^{a}=b^{11}, c^{a}=c^{7}, c^{b}=c^{11} \rangle$
Permutation group:	Degree $25$ $\langle(1,2)(3,5)(4,6)(10,11,13,14,12,15,16,17)(18,19,22,24)(20,21,25,23), (8,9) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 46 & 15 \\ 15 & 31 \end{array}\right), \left(\begin{array}{rr} 31 & 30 \\ 30 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 80 \\ 20 & 71 \end{array}\right), \left(\begin{array}{rr} 71 & 0 \\ 0 & 71 \end{array}\right), \left(\begin{array}{rr} 0 & 83 \\ 23 & 40 \end{array}\right), \left(\begin{array}{rr} 61 & 0 \\ 0 & 61 \end{array}\right), \left(\begin{array}{rr} 0 & 11 \\ 59 & 40 \end{array}\right), \left(\begin{array}{rr} 49 & 25 \\ 40 & 59 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/90\Z)$
Direct product:	$C_3$ $\, \times\, $ $(C_4^2.S_3^2)$
Semidirect product:	$(C_3^3:Q_8)$ $\,\rtimes\,$ $Q_8$ (2)	$C_3^3$ $\,\rtimes\,$ $(Q_8:Q_8)$	$C_3^2$ $\,\rtimes\,$ $(C_4^2.D_6)$	$C_3^2$ $\,\rtimes\,$ $(C_4^2.D_6)$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{12}^2$ . $D_6$ (2)	$(C_6:C_{24})$ . $D_6$ (2)	$C_6$ . $(C_{24}:D_6)$	$(C_{12}.D_{12})$ . $S_3$	all 100

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

Homology

Abelianization:	$C_{2}^{2} \times C_{6} \simeq C_{2}^{3} \times C_{3}$
Schur multiplier:	$C_{2}$
Commutator length:	$2$

Subgroups

There are 2408 subgroups in 590 conjugacy classes, 148 normal (104 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:	$G' \simeq$ $C_6\times C_{12}$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_6\times S_3^2$
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:	$\operatorname{soc} \simeq$ $C_3\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_8:Q_8$
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: $C_6^2.D_{12}$ x 2, $C_6^2.D_{12}$ x 2, $C_{12}^2.C_6$, $C_{12}^2.C_6$, $C_{12}^2.C_6$

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

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

Order 288: $C_{12}.D_{12}$ x 3, $C_{12}^2.C_2$ x 3, $C_{12}:C_{24}$ x 3, $C_{24}:C_{12}$ x 2, $C_6^2.D_4$ x 2, $C_{12}.D_{12}$ x 2, $C_6^2.D_4$ x 2, $C_6^2.D_4$ x 2, $C_{12}.D_{12}$ x 2, $C_{12}^2.C_2$, $C_{12}^2.C_2$, $C_{12}:C_{24}$

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

Order 192: $C_{12}.\SD_{16}$, $C_4^2.D_6$, $C_4^2.D_6$

Order 144: $C_{12}:C_{12}$ x 13, $C_{12}.D_6$ x 7, $C_{12}^2$ x 7, $C_6:C_{24}$ x 6, $C_6^2.C_2^2$ x 4, $C_{12}:C_{12}$ x 4, $C_{12}:C_{12}$ x 3, $C_6\times C_{24}$ x 2, $C_6.D_{12}$, $C_6^2.C_2^2$, $C_3^2:C_4^2$, $C_4.C_6^2$, $C_{12}.D_6$

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

Order 96: $C_{12}:Q_8$ x 3, $C_4:C_{24}$ x 3, $C_{12}:Q_8$ x 3, $C_8:C_{12}$ x 2, $Q_8:C_{12}$ x 2, $C_{24}:C_4$ x 2, $C_6.Q_{16}$ x 2, $C_4.D_{12}$ x 2, $C_{12}.Q_8$ x 2, $C_{12}:Q_8$, $Q_8\times C_{12}$, $C_{12}:C_8$

Order 72: $C_6\times C_{12}$ x 23, $C_3^2:Q_8$ x 18, $C_6:C_{12}$ x 16, $C_3:C_{24}$ x 6, $C_3^2:Q_8$ x 3, $Q_8\times C_3^2$ x 2, $C_6.D_6$ x 2, $C_3\times C_{24}$ x 2

Order 64: $Q_8:Q_8$

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

Order 48: $C_4:C_{12}$ x 12, $C_4\times C_{12}$ x 8, $C_2\times C_{24}$ x 6, $C_{12}:C_4$ x 6, $C_6\times Q_8$ x 5, $C_6:Q_8$ x 4, $C_6.D_4$ x 4, $C_{12}:C_4$ x 4, $C_6:C_8$ x 2

Order 36: $C_3\times C_{12}$ x 39, $C_3:C_{12}$ x 20, $C_6^2$ x 7, $C_3^2:C_4$ x 3

Order 32: $C_8:C_4$ x 2, $Q_8:C_4$ x 2, $C_4:Q_8$, $C_4\times Q_8$, $C_4:C_8$

Order 27: $C_3^3$

Order 24: $C_2\times C_{12}$ x 31, $C_3:Q_8$ x 12, $C_3\times Q_8$ x 11, $C_6:C_4$ x 10, $C_{24}$ x 6, $C_3:C_8$ x 2

Order 18: $C_3\times C_6$ x 21

Order 16: $C_4:C_4$ x 5, $C_2\times C_8$ x 2, $C_4^2$ x 2, $C_2\times Q_8$ x 2

Order 12: $C_{12}$ x 48, $C_3:C_4$ x 14, $C_2\times C_6$ x 7

Order 9: $C_3^2$ x 7

Order 8: $C_2\times C_4$ x 7, $Q_8$ x 5, $C_8$ x 2

Order 6: $C_6$ x 21

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

Order 3: $C_3$ x 7

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

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

Order 864: $C_{12}^2.C_6$, $C_{12}^2.C_6$, $C_{12}^2.C_6$, $C_6^2.D_{12}$, $C_6^2.D_{12}$

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

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

Order 288: $C_{12}.D_{12}$ x 3, $C_{12}^2.C_2$ x 3, $C_{12}:C_{24}$ x 3, $C_{12}^2.C_2$, $C_{12}^2.C_2$, $C_{12}:C_{24}$, $C_{24}:C_{12}$, $C_6^2.D_4$, $C_{12}.D_{12}$, $C_6^2.D_4$, $C_6^2.D_4$, $C_{12}.D_{12}$

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

Order 192: $C_{12}.\SD_{16}$, $C_4^2.D_6$, $C_4^2.D_6$

Order 144: $C_{12}:C_{12}$ x 9, $C_{12}^2$ x 7, $C_{12}.D_6$ x 6, $C_6^2.C_2^2$ x 3, $C_{12}:C_{12}$ x 3, $C_6:C_{24}$ x 3, $C_{12}:C_{12}$ x 2, $C_6.D_{12}$, $C_6^2.C_2^2$, $C_3^2:C_4^2$, $C_4.C_6^2$, $C_{12}.D_6$, $C_6\times C_{24}$

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

Order 96: $C_{12}:Q_8$ x 3, $C_4:C_{24}$ x 3, $C_{12}:Q_8$ x 3, $C_{12}:Q_8$, $C_8:C_{12}$, $Q_8:C_{12}$, $C_{24}:C_4$, $C_6.Q_{16}$, $C_4.D_{12}$, $Q_8\times C_{12}$, $C_{12}.Q_8$, $C_{12}:C_8$

Order 72: $C_6\times C_{12}$ x 22, $C_6:C_{12}$ x 9, $C_3^2:Q_8$ x 9, $C_3:C_{24}$ x 3, $C_6.D_6$ x 2, $C_3^2:Q_8$ x 2, $Q_8\times C_3^2$, $C_3\times C_{24}$

Order 64: $Q_8:Q_8$

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

Order 48: $C_4:C_{12}$ x 8, $C_4\times C_{12}$ x 8, $C_{12}:C_4$ x 5, $C_6\times Q_8$ x 4, $C_6:Q_8$ x 4, $C_2\times C_{24}$ x 3, $C_6.D_4$ x 3, $C_{12}:C_4$ x 3, $C_6:C_8$

Order 36: $C_3\times C_{12}$ x 29, $C_3:C_{12}$ x 9, $C_6^2$ x 7, $C_3^2:C_4$ x 2

Order 32: $C_4:Q_8$, $C_4\times Q_8$, $C_8:C_4$, $C_4:C_8$, $Q_8:C_4$

Order 27: $C_3^3$

Order 24: $C_2\times C_{12}$ x 26, $C_6:C_4$ x 7, $C_3:Q_8$ x 7, $C_3\times Q_8$ x 5, $C_{24}$ x 3, $C_3:C_8$

Order 18: $C_3\times C_6$ x 21

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

Order 12: $C_{12}$ x 33, $C_2\times C_6$ x 7, $C_3:C_4$ x 7

Order 9: $C_3^2$ x 7

Order 8: $C_2\times C_4$ x 6, $Q_8$ x 3, $C_8$

Order 6: $C_6$ x 21

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

Order 3: $C_3$ x 7

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

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

Order 576: $C_4^2.S_3^2$ ($C_3$)

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

Order 288: $C_6^2.D_4$ ($C_6$) x 2, $C_{12}.D_{12}$ ($C_6$) x 2, $C_{12}^2.C_2$ ($C_6$), $C_{12}.D_{12}$ ($C_6$), $C_{12}.D_{12}$ ($S_3$), $C_{12}:C_{24}$ ($C_6$), $C_{12}:C_{24}$ ($S_3$)

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

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

Order 108: $C_3^2\times C_{12}$ ($\SD_{16}$) x 2, $C_3\times C_6^2$ ($C_2\times D_4$), $C_3^2\times C_{12}$ ($D_4:C_2$), $C_3^2\times C_{12}$ ($C_2\times Q_8$)

Order 96: $C_{12}:Q_8$ ($C_3\times S_3$), $C_{12}:C_8$ ($C_3\times S_3$)

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

Order 54: $C_3^2\times C_6$ ($Q_{16}:C_2$), $C_3^2\times C_6$ ($C_2\times \SD_{16}$), $C_3^2\times C_6$ ($C_2^2:Q_8$)

Order 48: $C_6:C_8$ ($C_6\times S_3$) x 2, $C_4\times C_{12}$ ($C_6\times S_3$) x 2, $C_{12}:C_4$ ($C_6\times S_3$) x 2, $C_4\times C_{12}$ ($S_3^2$)

Order 36: $C_3\times C_{12}$ ($C_{24}:C_2$) x 2, $C_3\times C_{12}$ ($S_3\times Q_8$) x 2, $C_3\times C_{12}$ ($C_3\times \SD_{16}$) x 2, $C_3\times C_{12}$ ($C_3:\SD_{16}$) x 2, $C_3\times C_{12}$ ($D_4:C_6$), $C_3\times C_{12}$ ($C_6\times Q_8$), $C_3\times C_{12}$ ($D_{12}:C_2$), $C_3\times C_{12}$ ($D_4:S_3$), $C_6^2$ ($C_6\times D_4$), $C_6^2$ ($C_6:D_4$), $C_6^2$ ($C_2\times D_{12}$)

Order 27: $C_3^3$ ($Q_8:Q_8$)

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

Order 18: $C_3\times C_6$ ($Q_{16}:C_6$), $C_3\times C_6$ ($C_6\times \SD_{16}$), $C_3\times C_6$ ($C_{12}.D_4$), $C_3\times C_6$ ($D_6:Q_8$), $C_3\times C_6$ ($C_{12}.D_4$), $C_3\times C_6$ ($C_6:\SD_{16}$), $C_3\times C_6$ ($C_4.D_{12}$), $C_3\times C_6$ ($C_8:D_6$), $C_3\times C_6$ ($D_6:Q_8$)

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

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

Order 9: $C_3^2$ ($C_{12}.\SD_{16}$), $C_3^2$ ($C_4^2.D_6$), $C_3^2$ ($C_4^2.D_6$)

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

Order 6: $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_6^2.D_4$), $C_6$ ($C_6^2.D_4$), $C_6$ ($C_{12}.D_{12}$), $C_6$ ($C_{24}:D_6$), $C_6$ ($C_{12}.D_{12}$), $C_6$ ($C_{12}.D_{12}$), $C_6$ ($C_{12}.D_{12}$), $C_6$ ($C_{12}.D_{12}$)

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

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

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

Order 576: $C_4^2.S_3^2$ ($C_3$)

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

Order 288: $C_{12}^2.C_2$ ($C_6$), $C_{12}.D_{12}$ ($C_6$), $C_{12}.D_{12}$ ($S_3$), $C_{12}:C_{24}$ ($C_6$), $C_{12}:C_{24}$ ($S_3$), $C_6^2.D_4$ ($C_6$), $C_{12}.D_{12}$ ($C_6$)

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

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

Order 108: $C_3\times C_6^2$ ($C_2\times D_4$), $C_3^2\times C_{12}$ ($\SD_{16}$), $C_3^2\times C_{12}$ ($D_4:C_2$), $C_3^2\times C_{12}$ ($C_2\times Q_8$)

Order 96: $C_{12}:Q_8$ ($C_3\times S_3$), $C_{12}:C_8$ ($C_3\times S_3$)

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

Order 54: $C_3^2\times C_6$ ($Q_{16}:C_2$), $C_3^2\times C_6$ ($C_2\times \SD_{16}$), $C_3^2\times C_6$ ($C_2^2:Q_8$)

Order 48: $C_4\times C_{12}$ ($C_6\times S_3$) x 2, $C_6:C_8$ ($C_6\times S_3$), $C_4\times C_{12}$ ($S_3^2$), $C_{12}:C_4$ ($C_6\times S_3$)

Order 36: $C_3\times C_{12}$ ($S_3\times Q_8$) x 2, $C_3\times C_{12}$ ($C_{24}:C_2$), $C_3\times C_{12}$ ($D_4:C_6$), $C_3\times C_{12}$ ($C_6\times Q_8$), $C_3\times C_{12}$ ($D_{12}:C_2$), $C_3\times C_{12}$ ($D_4:S_3$), $C_3\times C_{12}$ ($C_3\times \SD_{16}$), $C_3\times C_{12}$ ($C_3:\SD_{16}$), $C_6^2$ ($C_6\times D_4$), $C_6^2$ ($C_6:D_4$), $C_6^2$ ($C_2\times D_{12}$)

Order 27: $C_3^3$ ($Q_8:Q_8$)

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

Order 18: $C_3\times C_6$ ($Q_{16}:C_6$), $C_3\times C_6$ ($C_6\times \SD_{16}$), $C_3\times C_6$ ($C_{12}.D_4$), $C_3\times C_6$ ($D_6:Q_8$), $C_3\times C_6$ ($C_{12}.D_4$), $C_3\times C_6$ ($C_6:\SD_{16}$), $C_3\times C_6$ ($C_4.D_{12}$), $C_3\times C_6$ ($C_8:D_6$), $C_3\times C_6$ ($D_6:Q_8$)

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

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

Order 9: $C_3^2$ ($C_{12}.\SD_{16}$), $C_3^2$ ($C_4^2.D_6$), $C_3^2$ ($C_4^2.D_6$)

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

Order 6: $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_6^2.D_4$), $C_6$ ($C_6^2.D_4$), $C_6$ ($C_{12}.D_{12}$), $C_6$ ($C_{24}:D_6$), $C_6$ ($C_{12}.D_{12}$), $C_6$ ($C_{12}.D_{12}$), $C_6$ ($C_{12}.D_{12}$), $C_6$ ($C_{12}.D_{12}$)

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

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

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

Series

Derived series

$C_{12}^2.D_6$

$\rhd$

$C_6\times C_{12}$

$\rhd$

$C_1$

Chief series

$C_{12}^2.D_6$

$\rhd$

$C_{12}^2.C_6$

$\rhd$

$C_{12}:C_{24}$

$\rhd$

$C_{12}^2$

$\rhd$

$C_6\times C_{12}$

$\rhd$

$C_3\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{12}^2.D_6$

$\rhd$

$C_6\times C_{12}$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2\times C_6$

$\lhd$

$C_4\times C_{12}$

Character theory

Complex character table

See the $207 \times 207$ 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 $98 \times 98$ rational character table.