Abstract group 864.3125: $C_6^2.D_{12}$

• Label: 864.3125
• Order: \( 2^{5} \cdot 3^{3} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $25$
• Trans deg.: $48$
• Rank: $3$

Group information

Description:	$C_6^2.D_{12}$
Order:	\(864\)\(\medspace = 2^{5} \cdot 3^{3} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_6^2.C_2^6.C_2$, of order \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \)
Composition factors:	$C_2$ x 5, $C_3$ 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	39	26	64	150	24	368	192	864
Conjugacy classes	1	3	11	5	30	2	49	16	117
Divisions	1	3	7	5	17	2	23	6	64
Autjugacy classes	1	3	7	4	15	1	18	3	52

Dimension	1	2	4	8	16
Irr. complex chars.	24	54	39	0	0	117
Irr. rational chars.	8	18	23	11	4	64

Minimal presentations

Permutation degree:	$25$
Transitive degree:	$48$
Rank:	$3$
Inequivalent generating triples:	$34944$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	16
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c \mid c^{12}=1, a^{6}=c^{6}, b^{12}=c^{6}, b^{a}=b^{11}, c^{a}=c^{7}, c^{b}=c^{11} \rangle$
Permutation group:	Degree $25$ $\langle(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(18,19), (1,3,4,11,5,10,13,14) \!\cdots\! \rangle$
Direct product:	$C_3$ $\, \times\, $ $(C_{12}.D_{12})$
Semidirect product:	$(C_{12}.D_6)$ $\,\rtimes\,$ $S_3$	$(C_{12}.D_6)$ $\,\rtimes\,$ $C_6$	$(C_6.D_{12})$ $\,\rtimes\,$ $C_6$ (2)	$(C_{12}.C_{12})$ $\,\rtimes\,$ $S_3$	all 18
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^2$ . $D_{12}$	$C_{12}$ . $(S_3\times D_6)$	$(C_{12}.D_6)$ . $C_6$	$(C_6\times C_{12})$ . $D_6$ (2)	all 50

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:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 1380 subgroups in 354 conjugacy classes, 88 normal (64 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_6:D_{12}$
Commutator:	$G' \simeq$ $C_3\times C_{12}$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_6\times S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_6\times C_{12}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_6^2.D_{12}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{16}:C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

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

Order 432: $C_3^3:Q_{16}$ x 2, $C_{12}.S_3^2$ x 2, $C_6^2.D_6$, $C_{12}.C_6^2$, $C_{12}.C_6^2$

Order 288: $C_6^2.D_4$, $C_{12}.D_{12}$, $C_{12}.D_{12}$

Order 216: $C_3^3:Q_8$ x 3, $C_3^2:C_{24}$ x 2, $C_3\times C_6\times C_{12}$, $C_6^2:C_6$, $C_3^2:D_{12}$, $C_3:S_3\times C_{12}$, $C_3^3:Q_8$, $C_6.C_6^2$

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

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

Order 96: $Q_{16}:C_6$, $C_{12}.D_4$, $C_4.D_{12}$

Order 72: $C_3^2:Q_8$ x 13, $C_6\times C_{12}$ x 8, $C_3:C_{24}$ x 6, $C_6\wr C_2$ x 4, $C_6:C_{12}$ x 4, $S_3\times C_{12}$ x 4, $Q_8\times C_3^2$ x 3, $C_3\times D_{12}$ x 3, $C_3\times C_{24}$ x 2, $C_6^2:C_2$, $C_3:D_{12}$, $C_{12}:S_3$, $C_3^2:Q_8$

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

Order 48: $C_6\times Q_8$ x 3, $D_{12}:C_2$ x 3, $C_3\times \OD_{16}$ x 3, $C_3:Q_{16}$ x 2, $C_{24}:C_2$ x 2, $C_3\times Q_{16}$ x 2, $C_3\times \SD_{16}$ x 2, $C_3:Q_{16}$ x 2, $Q_8:S_3$ x 2, $D_4:C_6$, $C_6:Q_8$, $C_3:\OD_{16}$

Order 36: $C_3\times C_{12}$ x 16, $C_3:C_{12}$ x 12, $C_6^2$ x 7, $C_6\times S_3$ x 4, $C_3^2:C_4$, $C_6:S_3$

Order 32: $Q_{16}:C_2$

Order 27: $C_3^3$

Order 24: $C_2\times C_{12}$ x 12, $C_3\times Q_8$ x 11, $C_3:Q_8$ x 6, $C_{24}$ x 6, $C_3:D_4$ x 4, $C_4\times S_3$ x 4, $D_{12}$ x 3, $C_3\times D_4$ x 2, $C_3:C_8$ x 2, $C_6:C_4$

Order 18: $C_3\times C_6$ x 16, $C_3\times S_3$ x 4, $C_3:S_3$

Order 16: $Q_{16}$ x 2, $\SD_{16}$ x 2, $\OD_{16}$, $D_4:C_2$, $C_2\times Q_8$

Order 12: $C_{12}$ x 23, $C_2\times C_6$ x 8, $C_3:C_4$ x 6, $D_6$ x 4

Order 9: $C_3^2$ x 7

Order 8: $Q_8$ x 4, $C_2\times C_4$ x 3, $D_4$ x 2, $C_8$ x 2

Order 6: $C_6$ x 17, $S_3$ x 4

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

Order 3: $C_3$ x 7

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

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

Order 432: $C_6^2.D_6$, $C_{12}.C_6^2$, $C_{12}.C_6^2$, $C_3^3:Q_{16}$, $C_{12}.S_3^2$

Order 288: $C_6^2.D_4$, $C_{12}.D_{12}$, $C_{12}.D_{12}$

Order 216: $C_3^3:Q_8$ x 2, $C_3^2:C_{24}$, $C_3\times C_6\times C_{12}$, $C_6^2:C_6$, $C_3^2:D_{12}$, $C_3:S_3\times C_{12}$, $C_3^3:Q_8$, $C_6.C_6^2$

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

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

Order 96: $Q_{16}:C_6$, $C_{12}.D_4$, $C_4.D_{12}$

Order 72: $C_3^2:Q_8$ x 9, $C_6\times C_{12}$ x 8, $C_6\wr C_2$ x 3, $C_6:C_{12}$ x 3, $C_3\times D_{12}$ x 3, $S_3\times C_{12}$ x 3, $C_3:C_{24}$ x 3, $Q_8\times C_3^2$ x 2, $C_6^2:C_2$, $C_3:D_{12}$, $C_{12}:S_3$, $C_3^2:Q_8$, $C_3\times C_{24}$

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

Order 48: $C_6\times Q_8$ x 3, $D_{12}:C_2$ x 3, $C_3\times \OD_{16}$ x 3, $C_3:Q_{16}$, $C_{24}:C_2$, $D_4:C_6$, $C_6:Q_8$, $C_3\times Q_{16}$, $C_3\times \SD_{16}$, $C_3:Q_{16}$, $Q_8:S_3$, $C_3:\OD_{16}$

Order 36: $C_3\times C_{12}$ x 15, $C_6^2$ x 7, $C_3:C_{12}$ x 6, $C_6\times S_3$ x 3, $C_3^2:C_4$, $C_6:S_3$

Order 32: $Q_{16}:C_2$

Order 27: $C_3^3$

Order 24: $C_2\times C_{12}$ x 11, $C_3\times Q_8$ x 7, $C_3:Q_8$ x 5, $C_3:D_4$ x 3, $D_{12}$ x 3, $C_4\times S_3$ x 3, $C_{24}$ x 3, $C_3\times D_4$ x 2, $C_6:C_4$, $C_3:C_8$

Order 18: $C_3\times C_6$ x 14, $C_3\times S_3$ x 3, $C_3:S_3$

Order 16: $Q_{16}$, $\SD_{16}$, $\OD_{16}$, $D_4:C_2$, $C_2\times Q_8$

Order 12: $C_{12}$ x 18, $C_2\times C_6$ x 8, $C_3:C_4$ x 4, $D_6$ x 3

Order 9: $C_3^2$ x 7

Order 8: $Q_8$ x 3, $C_2\times C_4$ x 3, $D_4$ x 2, $C_8$

Order 6: $C_6$ x 15, $S_3$ x 3

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

Order 3: $C_3$ x 7

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 864: $C_6^2.D_{12}$ ($C_1$)

Order 432: $C_3^3:Q_{16}$ ($C_2$) x 2, $C_{12}.S_3^2$ ($C_2$) x 2, $C_6^2.D_6$ ($C_2$), $C_{12}.C_6^2$ ($C_2$), $C_{12}.C_6^2$ ($C_2$)

Order 288: $C_{12}.D_{12}$ ($C_3$)

Order 216: $C_3^2:C_{24}$ ($C_2^2$) x 2, $C_3^3:Q_8$ ($C_2^2$) x 2, $C_3\times C_6\times C_{12}$ ($C_2^2$), $C_3^2:D_{12}$ ($C_2^2$), $C_3^3:Q_8$ ($C_2^2$)

Order 144: $C_3^2:Q_{16}$ ($C_6$) x 2, $C_6.D_{12}$ ($C_6$) x 2, $C_{12}.C_{12}$ ($C_6$), $C_{12}.C_{12}$ ($S_3$), $C_{12}.D_6$ ($C_6$), $C_{12}.D_6$ ($C_6$), $C_{12}.D_6$ ($S_3$)

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

Order 72: $C_6\times C_{12}$ ($D_6$) x 2, $C_3^2:Q_8$ ($C_2\times C_6$) x 2, $C_3^2:Q_8$ ($D_6$) x 2, $C_3:C_{24}$ ($C_2\times C_6$) x 2, $C_3:C_{24}$ ($D_6$) x 2, $C_6\times C_{12}$ ($C_2\times C_6$), $C_3:D_{12}$ ($C_2\times C_6$), $C_3^2:Q_8$ ($C_2\times C_6$)

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

Order 48: $C_6:Q_8$ ($C_3\times S_3$), $C_3:\OD_{16}$ ($C_3\times S_3$)

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

Order 27: $C_3^3$ ($Q_{16}:C_2$)

Order 24: $C_2\times C_{12}$ ($C_6\times S_3$) x 2, $C_3:Q_8$ ($C_6\times S_3$) x 2, $C_3:C_8$ ($C_6\times S_3$) x 2, $C_2\times C_{12}$ ($S_3^2$)

Order 18: $C_3\times C_6$ ($C_6\times D_4$), $C_3\times C_6$ ($C_6:D_4$), $C_3\times C_6$ ($C_2\times D_{12}$)

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

Order 9: $C_3^2$ ($Q_{16}:C_6$), $C_3^2$ ($C_{12}.D_4$), $C_3^2$ ($C_4.D_{12}$)

Order 8: $C_2\times C_4$ ($C_3\times S_3^2$)

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

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

Order 3: $C_3$ ($C_6^2.D_4$), $C_3$ ($C_{12}.D_{12}$), $C_3$ ($C_{12}.D_{12}$)

Order 2: $C_2$ ($C_6^2.D_6$)

Order 1: $C_1$ ($C_6^2.D_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 864: $C_6^2.D_{12}$ ($C_1$)

Order 432: $C_6^2.D_6$ ($C_2$), $C_{12}.C_6^2$ ($C_2$), $C_{12}.C_6^2$ ($C_2$), $C_3^3:Q_{16}$ ($C_2$), $C_{12}.S_3^2$ ($C_2$)

Order 288: $C_{12}.D_{12}$ ($C_3$)

Order 216: $C_3^2:C_{24}$ ($C_2^2$), $C_3\times C_6\times C_{12}$ ($C_2^2$), $C_3^2:D_{12}$ ($C_2^2$), $C_3^3:Q_8$ ($C_2^2$), $C_3^3:Q_8$ ($C_2^2$)

Order 144: $C_{12}.C_{12}$ ($C_6$), $C_{12}.C_{12}$ ($S_3$), $C_3^2:Q_{16}$ ($C_6$), $C_6.D_{12}$ ($C_6$), $C_{12}.D_6$ ($C_6$), $C_{12}.D_6$ ($C_6$), $C_{12}.D_6$ ($S_3$)

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

Order 72: $C_6\times C_{12}$ ($D_6$) x 2, $C_6\times C_{12}$ ($C_2\times C_6$), $C_3:D_{12}$ ($C_2\times C_6$), $C_3^2:Q_8$ ($C_2\times C_6$), $C_3^2:Q_8$ ($C_2\times C_6$), $C_3^2:Q_8$ ($D_6$), $C_3:C_{24}$ ($C_2\times C_6$), $C_3:C_{24}$ ($D_6$)

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

Order 48: $C_6:Q_8$ ($C_3\times S_3$), $C_3:\OD_{16}$ ($C_3\times S_3$)

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

Order 27: $C_3^3$ ($Q_{16}:C_2$)

Order 24: $C_2\times C_{12}$ ($C_6\times S_3$) x 2, $C_2\times C_{12}$ ($S_3^2$), $C_3:Q_8$ ($C_6\times S_3$), $C_3:C_8$ ($C_6\times S_3$)

Order 18: $C_3\times C_6$ ($C_6\times D_4$), $C_3\times C_6$ ($C_6:D_4$), $C_3\times C_6$ ($C_2\times D_{12}$)

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

Order 9: $C_3^2$ ($Q_{16}:C_6$), $C_3^2$ ($C_{12}.D_4$), $C_3^2$ ($C_4.D_{12}$)

Order 8: $C_2\times C_4$ ($C_3\times S_3^2$)

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

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

Order 3: $C_3$ ($C_6^2.D_4$), $C_3$ ($C_{12}.D_{12}$), $C_3$ ($C_{12}.D_{12}$)

Order 2: $C_2$ ($C_6^2.D_6$)

Order 1: $C_1$ ($C_6^2.D_{12}$)

Series

Derived series

$C_6^2.D_{12}$

$\rhd$

$C_3\times C_{12}$

$\rhd$

$C_1$

Chief series

$C_6^2.D_{12}$

$\rhd$

$C_{12}.C_6^2$

$\rhd$

$C_{12}.C_{12}$

$\rhd$

$C_6\times C_{12}$

$\rhd$

$C_3\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6^2.D_{12}$

$\rhd$

$C_3\times C_{12}$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_2\times C_{12}$

Supergroups

This group is a maximal subgroup of 64 larger groups in the database.

This group is a maximal quotient of 24 larger groups in the database.

Character theory

Complex character table

See the $117 \times 117$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $64 \times 64$ rational character table.