Abstract group 864.4378: $C_6.D_6^2$

• Label: 864.4378
• Order: \( 2^{5} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $17$
• Trans deg.: $24$
• Rank: $4$

Group information

Description:	$C_6.D_6^2$
Order:	\(864\)\(\medspace = 2^{5} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^3.C_2^6.C_2^2$, of order \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \)
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	12
Elements	1	159	26	96	246	336	864
Conjugacy classes	1	9	7	10	24	21	72
Divisions	1	9	7	8	24	16	65
Autjugacy classes	1	6	5	6	14	10	42

Dimension	1	2	4	8
Irr. complex chars.	16	28	22	6	72
Irr. rational chars.	16	24	14	11	65

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$24$
Rank:	$4$
Inequivalent generating quadruples:	$22145760$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	8
Arbitrary	6	8	8

Constructions

Presentation:	$\langle a, b, c, d \mid b^{6}=c^{12}=d^{6}=[a,c]=1, a^{2}=c^{6}, b^{a}=b^{5}, d^{a}=c^{6}d, c^{b}=c^{5}, d^{b}=c^{6}d, d^{c}=d^{5} \rangle$
Permutation group:	Degree $17$ $\langle(1,2)(3,4)(12,13)(14,16)(15,17), (1,3,2,4)(5,7,6,8)(12,13)(14,17)(15,16) \!\cdots\! \rangle$
Transitive group:	24T2624				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(D_6.D_6)$ $\,\rtimes\,$ $S_3$ (2)	$(D_6.D_6)$ $\,\rtimes\,$ $S_3$	$(D_6:S_3)$ $\,\rtimes\,$ $D_6$	$(C_6.D_6)$ $\,\rtimes\,$ $D_6$	all 44
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $D_6^2$ (3)	$C_2^2$ . $S_3^3$	$D_6$ . $(S_3\times D_6)$ (2)	$C_2$ . $(C_2\times S_3^3)$	all 22

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 6448 subgroups in 850 conjugacy classes, 142 normal (46 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

There are too many subgroups to show. See a full page version of the diagram.

Classes of subgroups up to automorphism

There are too many subgroups to show. See a full page version of the diagram.

Normal subgroups

There are too many subgroups to show. See a full page version of the diagram.

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 864: $C_6.D_6^2$

Order 432: $C_6^2:D_6$ x 2, $C_6^2.D_6$ x 2, $C_2.S_3^3$ x 2, $C_2.S_3^3$ x 2, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_2.S_3^3$, $C_2.S_3^3$, $C_2.S_3^3$, $D_6:S_3^2$

Order 288: $D_4:S_3^2$ x 2, $D_{12}:D_6$

Order 216: $C_6.S_3^2$ x 6, $C_6.S_3^2$ x 4, $C_3^3:Q_8$ x 3, $C_6:S_3^2$ x 2, $C_6^2:C_6$ x 2, $C_6^2:C_6$ x 2, $C_6.S_3^2$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:D_{12}$ x 2, $C_6.S_3^2$ x 2, $C_6^2:S_3$, $C_6^2.C_6$, $C_6.C_6^2$, $C_3^3:Q_8$, $C_3^2:D_{12}$, $C_3^3:D_4$

Order 144: $D_6:D_6$ x 8, $D_6.D_6$ x 8, $C_2^2.S_3^2$ x 5, $C_{12}.D_6$ x 4, $C_4\times S_3^2$ x 4, $C_{12}.D_6$ x 4, $C_{12}.D_6$ x 3, $C_{12}:D_6$ x 2, $D_{12}:C_6$ x 2, $D_6.D_6$ x 2, $C_{12}.D_6$ x 2, $D_{12}:S_3$ x 2, $C_{12}:D_6$, $C_{12}:D_6$

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

Order 96: $D_4:D_6$ x 3

Order 72: $C_6.D_6$ x 22, $C_6\wr C_2$ x 16, $C_3:D_{12}$ x 16, $C_6:C_{12}$ x 10, $C_6:D_6$ x 9, $S_3\times C_{12}$ x 8, $C_6.D_6$ x 8, $S_3\times D_6$ x 6, $C_{12}:S_3$ x 6, $C_3^2:Q_8$ x 6, $C_3^2:Q_8$ x 6, $C_6^2:C_2$ x 4, $D_4\times C_3^2$ x 2, $C_3:D_{12}$ x 2, $C_3\times D_{12}$ x 2, $D_6:S_3$ x 2, $C_6\times C_{12}$, $C_6.D_6$

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

Order 48: $S_3\times D_4$ x 11, $C_4\times D_6$ x 10, $D_4:S_3$ x 9, $D_{12}:C_2$ x 9, $D_4:C_6$ x 3, $D_{12}:C_2$ x 3, $S_3\times Q_8$ x 3

Order 36: $C_6:S_3$ x 38, $C_3:C_{12}$ x 24, $C_6\times S_3$ x 12, $S_3^2$ x 12, $C_6^2$ x 9, $C_3\times C_{12}$ x 4, $C_3^2:C_4$ x 4

Order 32: $D_4:C_2^2$

Order 27: $C_3^3$

Order 24: $C_4\times S_3$ x 36, $C_3:D_4$ x 22, $C_2\times D_6$ x 15, $D_{12}$ x 11, $C_3\times D_4$ x 11, $C_2\times C_{12}$ x 10, $C_6:C_4$ x 10, $C_3:Q_8$ x 9, $C_3\times Q_8$ x 3

Order 18: $C_3:S_3$ x 25, $C_3\times C_6$ x 18, $C_3\times S_3$ x 12

Order 16: $D_4:C_2$ x 8, $C_2\times D_4$ x 3, $C_2^2\times C_4$ x 3, $C_2\times Q_8$

Order 12: $D_6$ x 56, $C_{12}$ x 16, $C_3:C_4$ x 16, $C_2\times C_6$ x 15

Order 9: $C_3^2$ x 7

Order 8: $C_2\times C_4$ x 16, $D_4$ x 12, $Q_8$ x 4, $C_2^3$ x 3

Order 6: $S_3$ x 31, $C_6$ x 24

Order 4: $C_2^2$ x 13, $C_4$ x 8

Order 3: $C_3$ x 7

Order 2: $C_2$ x 9

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 864: $C_6.D_6^2$

Order 432: $C_6^2.D_6$, $C_6^2:D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_2.S_3^3$, $C_2.S_3^3$, $C_2.S_3^3$, $C_2.S_3^3$, $D_6:S_3^2$, $C_2.S_3^3$

Order 288: $D_4:S_3^2$, $D_{12}:D_6$

Order 216: $C_6.S_3^2$ x 5, $C_3^3:Q_8$ x 2, $C_6.S_3^2$ x 2, $C_6^2:S_3$, $C_6:S_3^2$, $C_6^2:C_6$, $C_6^2.C_6$, $C_6^2:C_6$, $C_6.C_6^2$, $C_3^3:Q_8$, $C_3^2:D_{12}$, $C_6.S_3^2$, $C_3^2:D_{12}$, $C_3^2:D_{12}$, $C_3^2:D_{12}$, $C_3^3:D_4$, $C_6.S_3^2$

Order 144: $D_6:D_6$ x 4, $D_6.D_6$ x 4, $C_2^2.S_3^2$ x 3, $C_{12}.D_6$ x 2, $D_6.D_6$ x 2, $C_4\times S_3^2$ x 2, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_{12}:D_6$, $C_{12}:D_6$, $D_{12}:C_6$, $C_{12}:D_6$, $D_{12}:S_3$

Order 108: $C_3^2:D_6$ x 3, $C_3^2:C_{12}$ x 3, $C_3^2:C_{12}$ x 3, $C_3\times C_6^2$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3:S_3^2$

Order 96: $D_4:D_6$ x 2

Order 72: $C_6.D_6$ x 13, $C_6\wr C_2$ x 8, $C_3:D_{12}$ x 8, $C_6:D_6$ x 6, $C_6:C_{12}$ x 6, $C_{12}:S_3$ x 5, $S_3\times C_{12}$ x 4, $C_3^2:Q_8$ x 4, $C_6.D_6$ x 4, $S_3\times D_6$ x 3, $C_3^2:Q_8$ x 3, $C_6^2:C_2$ x 2, $D_6:S_3$ x 2, $D_4\times C_3^2$, $C_6\times C_{12}$, $C_6.D_6$, $C_3:D_{12}$, $C_3\times D_{12}$

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

Order 48: $S_3\times D_4$ x 6, $C_4\times D_6$ x 6, $D_4:S_3$ x 5, $D_{12}:C_2$ x 5, $D_4:C_6$ x 2, $D_{12}:C_2$ x 2, $S_3\times Q_8$ x 2

Order 36: $C_6:S_3$ x 17, $C_3:C_{12}$ x 14, $C_6^2$ x 6, $C_6\times S_3$ x 6, $C_3\times C_{12}$ x 3, $C_3^2:C_4$ x 3, $S_3^2$ x 3

Order 32: $D_4:C_2^2$

Order 27: $C_3^3$

Order 24: $C_4\times S_3$ x 22, $C_3:D_4$ x 11, $C_2\times D_6$ x 9, $C_2\times C_{12}$ x 6, $C_6:C_4$ x 6, $D_{12}$ x 6, $C_3\times D_4$ x 6, $C_3:Q_8$ x 5, $C_3\times Q_8$ x 2

Order 18: $C_3\times C_6$ x 11, $C_3:S_3$ x 11, $C_3\times S_3$ x 6

Order 16: $D_4:C_2$ x 6, $C_2\times D_4$ x 2, $C_2^2\times C_4$ x 2, $C_2\times Q_8$

Order 12: $D_6$ x 23, $C_{12}$ x 10, $C_3:C_4$ x 10, $C_2\times C_6$ x 9

Order 9: $C_3^2$ x 5

Order 8: $C_2\times C_4$ x 11, $D_4$ x 7, $Q_8$ x 3, $C_2^3$ x 2

Order 6: $C_6$ x 14, $S_3$ x 14

Order 4: $C_2^2$ x 7, $C_4$ x 6

Order 3: $C_3$ x 5

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 864: $C_6.D_6^2$ ($C_1$)

Order 432: $C_6^2:D_6$ ($C_2$) x 2, $C_6^2.D_6$ ($C_2$) x 2, $C_2.S_3^3$ ($C_2$) x 2, $C_2.S_3^3$ ($C_2$) x 2, $C_6^2.D_6$ ($C_2$), $C_6^2.D_6$ ($C_2$), $C_6^2.D_6$ ($C_2$), $C_2.S_3^3$ ($C_2$), $C_2.S_3^3$ ($C_2$), $C_2.S_3^3$ ($C_2$), $D_6:S_3^2$ ($C_2$)

Order 216: $C_6.S_3^2$ ($C_2^2$) x 6, $C_6.S_3^2$ ($C_2^2$) x 4, $C_3^3:Q_8$ ($C_2^2$) x 3, $C_6:S_3^2$ ($C_2^2$) x 2, $C_6^2:C_6$ ($C_2^2$) x 2, $C_6^2:C_6$ ($C_2^2$) x 2, $C_6.S_3^2$ ($C_2^2$) x 2, $C_3^2:D_{12}$ ($C_2^2$) x 2, $C_3^2:D_{12}$ ($C_2^2$) x 2, $C_3^2:D_{12}$ ($C_2^2$) x 2, $C_6.S_3^2$ ($C_2^2$) x 2, $C_6^2:S_3$ ($C_2^2$), $C_6^2.C_6$ ($C_2^2$), $C_6.C_6^2$ ($C_2^2$), $C_3^3:Q_8$ ($C_2^2$), $C_3^2:D_{12}$ ($C_2^2$), $C_3^3:D_4$ ($C_2^2$)

Order 144: $D_6.D_6$ ($S_3$) x 2, $D_6.D_6$ ($S_3$)

Order 108: $C_3^2:C_{12}$ ($C_2^3$) x 4, $C_3^2:C_{12}$ ($C_2^3$) x 4, $C_3^2:D_6$ ($C_2^3$) x 2, $C_3^2:D_6$ ($C_2^3$) x 2, $C_3^2\times D_6$ ($C_2^3$) x 2, $C_3\times C_6^2$ ($C_2^3$)

Order 72: $C_6\wr C_2$ ($D_6$) x 4, $C_6.D_6$ ($D_6$) x 4, $C_3^2:Q_8$ ($D_6$) x 3, $C_6^2:C_2$ ($D_6$) x 2, $C_6:C_{12}$ ($D_6$) x 2, $C_3:D_{12}$ ($D_6$) x 2, $C_6.D_6$ ($D_6$) x 2, $C_6.D_6$ ($D_6$), $D_6:S_3$ ($D_6$)

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

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

Order 27: $C_3^3$ ($D_4:C_2^2$)

Order 24: $C_3:D_4$ ($S_3^2$) x 2, $C_6:C_4$ ($S_3^2$)

Order 18: $C_3\times C_6$ ($C_2^2\times D_6$) x 3

Order 12: $C_3:C_4$ ($S_3\times D_6$) x 4, $C_2\times C_6$ ($S_3\times D_6$) x 3, $D_6$ ($S_3\times D_6$) x 2

Order 9: $C_3^2$ ($D_4:D_6$) x 3

Order 6: $C_6$ ($D_6^2$) x 3

Order 4: $C_2^2$ ($S_3^3$)

Order 3: $C_3$ ($D_4:S_3^2$) x 2, $C_3$ ($D_{12}:D_6$)

Order 2: $C_2$ ($C_2\times S_3^3$)

Order 1: $C_1$ ($C_6.D_6^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 864: $C_6.D_6^2$ ($C_1$)

Order 432: $C_6^2.D_6$ ($C_2$), $C_6^2:D_6$ ($C_2$), $C_6^2.D_6$ ($C_2$), $C_6^2.D_6$ ($C_2$), $C_6^2.D_6$ ($C_2$), $C_2.S_3^3$ ($C_2$), $C_2.S_3^3$ ($C_2$), $C_2.S_3^3$ ($C_2$), $C_2.S_3^3$ ($C_2$), $D_6:S_3^2$ ($C_2$), $C_2.S_3^3$ ($C_2$)

Order 216: $C_6.S_3^2$ ($C_2^2$) x 5, $C_3^3:Q_8$ ($C_2^2$) x 2, $C_6.S_3^2$ ($C_2^2$) x 2, $C_6^2:S_3$ ($C_2^2$), $C_6:S_3^2$ ($C_2^2$), $C_6^2:C_6$ ($C_2^2$), $C_6^2.C_6$ ($C_2^2$), $C_6^2:C_6$ ($C_2^2$), $C_6.C_6^2$ ($C_2^2$), $C_3^3:Q_8$ ($C_2^2$), $C_3^2:D_{12}$ ($C_2^2$), $C_6.S_3^2$ ($C_2^2$), $C_3^2:D_{12}$ ($C_2^2$), $C_3^2:D_{12}$ ($C_2^2$), $C_3^2:D_{12}$ ($C_2^2$), $C_3^3:D_4$ ($C_2^2$), $C_6.S_3^2$ ($C_2^2$)

Order 144: $D_6.D_6$ ($S_3$), $D_6.D_6$ ($S_3$)

Order 108: $C_3^2:C_{12}$ ($C_2^3$) x 3, $C_3^2:C_{12}$ ($C_2^3$) x 3, $C_3^2:D_6$ ($C_2^3$) x 2, $C_3\times C_6^2$ ($C_2^3$), $C_3^2:D_6$ ($C_2^3$), $C_3^2\times D_6$ ($C_2^3$)

Order 72: $C_6\wr C_2$ ($D_6$) x 2, $C_3^2:Q_8$ ($D_6$) x 2, $C_6.D_6$ ($D_6$) x 2, $C_6^2:C_2$ ($D_6$), $C_6.D_6$ ($D_6$), $C_6:C_{12}$ ($D_6$), $C_3:D_{12}$ ($D_6$), $D_6:S_3$ ($D_6$), $C_6.D_6$ ($D_6$)

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

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

Order 27: $C_3^3$ ($D_4:C_2^2$)

Order 24: $C_3:D_4$ ($S_3^2$), $C_6:C_4$ ($S_3^2$)

Order 18: $C_3\times C_6$ ($C_2^2\times D_6$) x 2

Order 12: $C_3:C_4$ ($S_3\times D_6$) x 3, $C_2\times C_6$ ($S_3\times D_6$) x 2, $D_6$ ($S_3\times D_6$)

Order 9: $C_3^2$ ($D_4:D_6$) x 2

Order 6: $C_6$ ($D_6^2$) x 2

Order 4: $C_2^2$ ($S_3^3$)

Order 3: $C_3$ ($D_4:S_3^2$), $C_3$ ($D_{12}:D_6$)

Order 2: $C_2$ ($C_2\times S_3^3$)

Order 1: $C_1$ ($C_6.D_6^2$)

Series

Derived series

$C_6.D_6^2$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_1$

Chief series

$C_6.D_6^2$

$\rhd$

$C_2.S_3^3$

$\rhd$

$C_6.S_3^2$

$\rhd$

$C_3^2:C_{12}$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6.D_6^2$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_3^3$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 25 larger groups in the database.

This group is a maximal quotient of 290 larger groups in the database.

Character theory

Complex character table

See the $72 \times 72$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $65 \times 65$ rational character table.