Abstract group 576.8418: $D_6^2:C_2^2$

• Label: 576.8418
• Order: \( 2^{6} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $12$
• Trans deg.: $24$
• Rank: $4$

Group information

Description:	$D_6^2:C_2^2$
Order:	\(576\)\(\medspace = 2^{6} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^2.C_2^6.C_2^5$, of order \(18432\)\(\medspace = 2^{11} \cdot 3^{2} \)
Composition factors:	$C_2$ x 6, $C_3$ x 2
Derived length:	$3$

This group is nonabelian, monomial (hence solvable), and rational.

Group statistics

Order	1	2	3	4	6	12
Elements	1	151	8	168	200	48	576
Conjugacy classes	1	21	2	6	22	2	54
Divisions	1	21	2	6	22	2	54
Autjugacy classes	1	8	2	2	8	1	22

Dimension	1	2	4	8
Irr. complex chars.	16	12	24	2	54
Irr. rational chars.	16	12	24	2	54

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$24$
Rank:	$4$
Inequivalent generating quadruples:	$1834560$

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:	${\langle a, b, c, d, e \mid a^{2}=b^{4}=c^{2}=d^{6}=e^{6}=[a,c]=[a,e]=[b,c]= \!\cdots\! \rangle}$
Permutation group:	Degree $12$ $\langle(4,6)(7,8)(9,10)(11,12), (1,2)(3,4)(5,6)(7,9)(8,10), (11,12), (9,10), (3,5)(4,6), (7,8)(9,10), (2,4,6), (1,3,5)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrr} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrr} -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrr} -1 & -1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{5}(\Z)$
Transitive group:	24T1475	24T1476			more information
Direct product:	$C_2$ $\, \times\, $ $(D_6\wr C_2)$
Semidirect product:	$D_6^2$ $\,\rtimes\,$ $C_2^2$ (2)	$(S_3\times D_6)$ $\,\rtimes\,$ $D_4$	$(C_6:D_6)$ $\,\rtimes\,$ $D_4$ (2)	$(S_3\times D_6)$ $\,\rtimes\,$ $C_2^3$	all 26
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_6^2$ . $C_2^2$	$(S_3\times D_6)$ . $C_2^3$	$(C_6:S_3)$ . $C_2^4$	$(C_6:D_6)$ . $C_2^3$ (2)	all 11
Aut. group:	$\Aut(C_{12}.D_6)$	$\Aut(C_{12}.D_6)$	$\Aut(C_{12}.D_6)$	$\Aut(C_4\times S_3^2)$	all 12

Elements of the group are displayed as permutations of degree 12.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 7506 subgroups in 1154 conjugacy classes, 113 normal (19 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 576: $D_6^2:C_2^2$

Order 288: $D_6\wr C_2$ x 8, $C_6^2.D_4$ x 2, $C_6^2:D_4$ x 2, $D_6^2:C_2$, $C_2\times C_6^2:C_4$, $C_2\times D_6^2$

Order 144: $D_6^2$ x 18, $S_3^2:C_2^2$ x 16, $S_3^2:C_4$ x 8, $D_6:D_6$ x 6, $C_6^2:C_4$ x 4, $C_6^2:C_4$ x 2, $C_6^2:C_2^2$, $C_6^2:C_2^2$, $C_6^2:C_2^2$, $C_6:D_{12}$, $C_2^2.S_3^2$

Order 96: $C_2^3\times D_6$, $D_4\times D_6$

Order 72: $S_3\times D_6$ x 69, $\SOPlus(4,2)$ x 16, $C_6\times D_6$ x 15, $C_6:D_6$ x 10, $C_2\times C_3^2:C_4$ x 8, $C_6\wr C_2$ x 4, $C_3:D_{12}$ x 4, $C_6.D_6$ x 3, $C_2\times C_6^2$, $C_6:C_{12}$

Order 64: $C_2^3:D_4$

Order 48: $C_2^2\times D_6$ x 32, $S_3\times D_4$ x 8, $C_6:D_4$ x 2, $C_2^3\times C_6$, $C_6\times D_4$, $C_2\times D_{12}$, $C_4\times D_6$

Order 36: $S_3^2$ x 42, $C_6\times S_3$ x 32, $C_6:S_3$ x 18, $C_6^2$ x 5, $C_3^2:C_4$ x 4, $C_3:C_{12}$ x 2

Order 32: $C_2^2\wr C_2$ x 8, $C_2^2\times D_4$ x 3, $C_2^3:C_4$ x 3, $C_2^5$

Order 24: $C_2\times D_6$ x 159, $C_2^2\times C_6$ x 17, $C_3:D_4$ x 8, $D_{12}$ x 4, $C_4\times S_3$ x 4, $C_3\times D_4$ x 4, $C_2\times C_{12}$, $C_6:C_4$

Order 18: $C_3\times S_3$ x 10, $C_3:S_3$ x 6, $C_3\times C_6$ x 5

Order 16: $C_2\times D_4$ x 24, $C_2^4$ x 20, $C_2^2:C_4$ x 12, $C_2^2\times C_4$ x 3

Order 12: $D_6$ x 150, $C_2\times C_6$ x 44, $C_{12}$ x 2, $C_3:C_4$ x 2

Order 9: $C_3^2$

Order 8: $C_2^3$ x 95, $D_4$ x 24, $C_2\times C_4$ x 12

Order 6: $S_3$ x 24, $C_6$ x 22

Order 4: $C_2^2$ x 97, $C_4$ x 6

Order 3: $C_3$ x 2

Order 2: $C_2$ x 21

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 576: $D_6^2:C_2^2$

Order 288: $D_6^2:C_2$, $C_2\times C_6^2:C_4$, $D_6\wr C_2$, $C_6^2.D_4$, $C_2\times D_6^2$, $C_6^2:D_4$

Order 144: $D_6^2$ x 6, $S_3^2:C_2^2$ x 2, $D_6:D_6$ x 2, $C_6^2:C_2^2$, $C_6^2:C_2^2$, $C_6^2:C_4$, $C_6^2:C_2^2$, $C_6:D_{12}$, $C_2^2.S_3^2$, $C_6^2:C_4$, $S_3^2:C_4$

Order 96: $C_2^3\times D_6$, $D_4\times D_6$

Order 72: $S_3\times D_6$ x 11, $C_6:D_6$ x 5, $C_6\times D_6$ x 4, $C_2\times C_3^2:C_4$ x 2, $C_6.D_6$ x 2, $C_2\times C_6^2$, $\SOPlus(4,2)$, $C_6\wr C_2$, $C_6:C_{12}$, $C_3:D_{12}$

Order 64: $C_2^3:D_4$

Order 48: $C_2^2\times D_6$ x 9, $C_6:D_4$ x 2, $S_3\times D_4$ x 2, $C_2^3\times C_6$, $C_6\times D_4$, $C_2\times D_{12}$, $C_4\times D_6$

Order 36: $C_6:S_3$ x 7, $S_3^2$ x 6, $C_6\times S_3$ x 5, $C_6^2$ x 3, $C_3^2:C_4$, $C_3:C_{12}$

Order 32: $C_2^2\times D_4$ x 2, $C_2^3:C_4$ x 2, $C_2^5$, $C_2^2\wr C_2$

Order 24: $C_2\times D_6$ x 25, $C_2^2\times C_6$ x 6, $C_3:D_4$ x 2, $C_4\times S_3$ x 2, $C_2\times C_{12}$, $C_6:C_4$, $D_{12}$, $C_3\times D_4$

Order 18: $C_3\times C_6$ x 3, $C_3:S_3$ x 3, $C_3\times S_3$ x 2

Order 16: $C_2^4$ x 8, $C_2\times D_4$ x 6, $C_2^2:C_4$ x 2, $C_2^2\times C_4$ x 2

Order 12: $D_6$ x 25, $C_2\times C_6$ x 11, $C_{12}$, $C_3:C_4$

Order 9: $C_3^2$

Order 8: $C_2^3$ x 21, $C_2\times C_4$ x 5, $D_4$ x 3

Order 6: $C_6$ x 8, $S_3$ x 8

Order 4: $C_2^2$ x 21, $C_4$ x 2

Order 3: $C_3$ x 2

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 576: $D_6^2:C_2^2$ ($C_1$)

Order 288: $D_6\wr C_2$ ($C_2$) x 8, $C_6^2.D_4$ ($C_2$) x 2, $C_6^2:D_4$ ($C_2$) x 2, $D_6^2:C_2$ ($C_2$), $C_2\times C_6^2:C_4$ ($C_2$), $C_2\times D_6^2$ ($C_2$)

Order 144: $S_3^2:C_2^2$ ($C_2^2$) x 8, $S_3^2:C_4$ ($C_2^2$) x 8, $D_6^2$ ($C_2^2$) x 7, $D_6:D_6$ ($C_2^2$) x 4, $C_6^2:C_4$ ($C_2^2$) x 4, $C_6^2:C_4$ ($C_2^2$) x 2, $C_6^2:C_2^2$ ($C_2^2$), $C_2^2.S_3^2$ ($C_2^2$)

Order 72: $S_3\times D_6$ ($D_4$) x 8, $S_3\times D_6$ ($C_2^3$) x 6, $C_2\times C_3^2:C_4$ ($C_2^3$) x 4, $C_6:D_6$ ($C_2^3$) x 3, $C_6:D_6$ ($D_4$) x 3, $C_6.D_6$ ($C_2^3$) x 2, $C_2\times C_6^2$ ($D_4$)

Order 36: $S_3^2$ ($C_2\times D_4$) x 8, $C_6:S_3$ ($C_2\times D_4$) x 7, $C_6^2$ ($C_2\times D_4$) x 3, $C_6:S_3$ ($C_2^4$)

Order 18: $C_3\times C_6$ ($C_2^2\wr C_2$) x 2, $C_3:S_3$ ($C_2^2\times D_4$) x 2, $C_3:S_3$ ($C_2^2\wr C_2$) x 2, $C_3\times C_6$ ($C_2^2\times D_4$)

Order 9: $C_3^2$ ($C_2^3:D_4$)

Order 8: $C_2^3$ ($\SOPlus(4,2)$)

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

Order 2: $C_2$ ($D_6\wr C_2$) x 2, $C_2$ ($C_6^2:D_4$)

Order 1: $C_1$ ($D_6^2:C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 576: $D_6^2:C_2^2$ ($C_1$)

Order 288: $D_6^2:C_2$ ($C_2$), $C_2\times C_6^2:C_4$ ($C_2$), $D_6\wr C_2$ ($C_2$), $C_6^2.D_4$ ($C_2$), $C_2\times D_6^2$ ($C_2$), $C_6^2:D_4$ ($C_2$)

Order 144: $D_6^2$ ($C_2^2$) x 3, $C_6^2:C_2^2$ ($C_2^2$), $C_6^2:C_4$ ($C_2^2$), $S_3^2:C_2^2$ ($C_2^2$), $D_6:D_6$ ($C_2^2$), $C_2^2.S_3^2$ ($C_2^2$), $C_6^2:C_4$ ($C_2^2$), $S_3^2:C_4$ ($C_2^2$)

Order 72: $C_6:D_6$ ($C_2^3$) x 2, $C_6:D_6$ ($D_4$) x 2, $S_3\times D_6$ ($C_2^3$) x 2, $C_2\times C_6^2$ ($D_4$), $S_3\times D_6$ ($D_4$), $C_2\times C_3^2:C_4$ ($C_2^3$), $C_6.D_6$ ($C_2^3$)

Order 36: $C_6:S_3$ ($C_2\times D_4$) x 3, $C_6^2$ ($C_2\times D_4$) x 2, $C_6:S_3$ ($C_2^4$), $S_3^2$ ($C_2\times D_4$)

Order 18: $C_3\times C_6$ ($C_2^2\times D_4$), $C_3\times C_6$ ($C_2^2\wr C_2$), $C_3:S_3$ ($C_2^2\times D_4$), $C_3:S_3$ ($C_2^2\wr C_2$)

Order 9: $C_3^2$ ($C_2^3:D_4$)

Order 8: $C_2^3$ ($\SOPlus(4,2)$)

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

Order 2: $C_2$ ($D_6\wr C_2$), $C_2$ ($C_6^2:D_4$)

Order 1: $C_1$ ($D_6^2:C_2^2$)

Series

Derived series

$D_6^2:C_2^2$

$\rhd$

$C_6:S_3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$D_6^2:C_2^2$

$\rhd$

$C_2\times D_6^2$

$\rhd$

$C_6^2:C_2^2$

$\rhd$

$C_6:D_6$

$\rhd$

$C_6:S_3$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$D_6^2:C_2^2$

$\rhd$

$C_6:S_3$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^3$

Supergroups

This group is a maximal subgroup of 45 larger groups in the database.

This group is a maximal quotient of 31 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $54 \times 54$ rational character table. Alternatively, you may search for characters of this group with desired properties.