Abstract group 1056.923: $D_{12}:D_{22}$

• Label: 1056.923
• Order: \( 2^{5} \cdot 3 \cdot 11 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 5 \)
• Perm deg.: $22$
• Trans deg.: $264$
• Rank: $4$

Group information

Description:	$D_{12}:D_{22}$
Order:	\(1056\)\(\medspace = 2^{5} \cdot 3 \cdot 11 \)
Exponent:	\(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \)
Automorphism group:	$C_{66}.C_{10}.C_2^6$, of order \(42240\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 5, $C_3$, $C_{11}$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Group statistics

Order	1	2	3	4	6	11	12	22	33	44	66	132
Elements	1	203	2	180	94	10	96	270	20	40	60	80	1056
Conjugacy classes	1	10	1	6	4	5	5	35	5	10	15	20	117
Divisions	1	10	1	6	4	1	4	6	1	2	2	2	40
Autjugacy classes	1	5	1	4	3	1	3	3	1	2	2	2	28

Dimension	1	2	4	8	10	20	40	80
Irr. complex chars.	16	48	53	0	0	0	0	0	117
Irr. rational chars.	16	8	1	1	8	4	1	1	40

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$264$
Rank:	$4$
Inequivalent generating quadruples:	$8714160$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	80
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{12}=d^{22}=[a,b]=[a,d]=[b,c]=[c,d]=1, c^{a}=c^{11}, d^{b}=c^{6}d^{21} \rangle$
Permutation group:	Degree $22$ $\langle(4,8)(6,7)(13,14)(15,16)(17,18)(19,20)(21,22), (1,2)(3,5)(4,6)(7,8)(10,11) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_{44}$ $\,\rtimes\,$ $D_6$	$D_{12}$ $\,\rtimes\,$ $D_{22}$ (2)	$D_{132}$ $\,\rtimes\,$ $C_2^2$	$(D_{44}:C_2)$ $\,\rtimes\,$ $S_3$	all 37
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{66}$ . $C_2^3$	$C_{66}$ . $C_2^4$	$C_{132}$ . $C_2^3$ (2)	$C_4$ . $(S_3\times D_{22})$ (2)	all 20

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 3488 subgroups in 332 conjugacy classes, 108 normal (36 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_6\times D_{22}$
Commutator:	$G' \simeq$ $C_{66}$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_6\times D_{22}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{132}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $D_{12}:D_{22}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{66}$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4:C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$

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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Subgroup information

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Classes of subgroups up to conjugation

Order 1056: $D_{12}:D_{22}$

Order 528: $D_6:D_{22}$ x 4, $C_{12}:D_{22}$ x 2, $D_{11}\times D_{12}$ x 2, $D_{12}:D_{11}$ x 2, $D_{12}:D_{11}$ x 2, $D_{132}:C_2$, $C_{22}\times D_{12}$, $D_{44}:C_6$

Order 352: $D_4:D_{22}$

Order 264: $C_{33}:D_4$ x 4, $C_6.D_{22}$ x 4, $S_3\times D_{22}$ x 4, $C_{11}\times D_{12}$ x 4, $C_{11}:D_{12}$ x 4, $D_6\times C_{22}$ x 2, $C_{33}:D_4$ x 2, $C_4\times D_{33}$ x 2, $C_{33}:D_4$ x 2, $C_{12}\times D_{11}$ x 2, $C_2\times C_{132}$, $D_{132}$, $C_{33}:Q_8$, $C_3\times D_{44}$, $C_{33}:Q_8$

Order 176: $C_{22}:D_4$ x 4, $D_4:D_{11}$ x 4, $D_4\times D_{11}$ x 4, $D_{44}:C_2$ x 2, $D_4\times C_{22}$

Order 132: $S_3\times C_{22}$ x 6, $S_3\times D_{11}$ x 4, $D_{66}$ x 2, $C_3\times D_{22}$ x 2, $C_{132}$ x 2, $C_{33}:C_4$ x 2, $C_{11}:C_{12}$ x 2, $C_2\times C_{66}$

Order 96: $D_4:D_6$

Order 88: $C_{11}:D_4$ x 12, $D_4\times C_{11}$ x 4, $C_{22}:C_4$ x 4, $C_4\times D_{11}$ x 4, $C_2\times D_{22}$ x 4, $D_{44}$ x 2, $C_{11}:Q_8$ x 2, $C_2^2\times C_{22}$ x 2, $C_2\times C_{44}$

Order 66: $S_3\times C_{11}$ x 4, $C_{66}$ x 2, $D_{33}$ x 2, $C_3\times D_{11}$ x 2

Order 48: $S_3\times D_4$ x 6, $D_{12}:C_2$ x 3, $C_2\times D_{12}$ x 3, $D_{12}:C_2$ x 2, $D_4:C_6$

Order 44: $D_{22}$ x 8, $C_2\times C_{22}$ x 7, $C_{11}:C_4$ x 4, $C_{44}$ x 2

Order 33: $C_{33}$

Order 32: $D_4:C_2^2$

Order 24: $D_{12}$ x 9, $C_3:D_4$ x 6, $C_4\times S_3$ x 6, $C_2\times D_6$ x 6, $C_2\times C_{12}$ x 3, $C_3\times D_4$ x 3, $C_3:Q_8$, $C_3\times Q_8$

Order 22: $C_{22}$ x 6, $D_{11}$ x 4

Order 16: $C_2\times D_4$ x 9, $D_4:C_2$ x 6

Order 12: $D_6$ x 12, $C_{12}$ x 4, $C_2\times C_6$ x 3, $C_3:C_4$ x 2

Order 11: $C_{11}$

Order 8: $D_4$ x 18, $C_2\times C_4$ x 9, $C_2^3$ x 6, $Q_8$ x 2

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

Order 4: $C_2^2$ x 15, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 10

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1056: $D_{12}:D_{22}$

Order 528: $D_{132}:C_2$, $C_{22}\times D_{12}$, $D_{44}:C_6$, $D_6:D_{22}$, $C_{12}:D_{22}$, $D_{11}\times D_{12}$, $D_{12}:D_{11}$, $D_{12}:D_{11}$

Order 352: $D_4:D_{22}$

Order 264: $C_{11}\times D_{12}$ x 2, $C_{33}:D_4$, $C_6.D_{22}$, $D_6\times C_{22}$, $S_3\times D_{22}$, $C_2\times C_{132}$, $C_{33}:D_4$, $D_{132}$, $C_4\times D_{33}$, $C_{33}:Q_8$, $C_{33}:D_4$, $C_3\times D_{44}$, $C_{12}\times D_{11}$, $C_{33}:Q_8$, $C_{11}:D_{12}$

Order 176: $D_4:D_{11}$ x 2, $D_4\times D_{11}$ x 2, $D_{44}:C_2$ x 2, $D_4\times C_{22}$, $C_{22}:D_4$

Order 132: $S_3\times C_{22}$ x 2, $C_{132}$ x 2, $D_{66}$, $C_3\times D_{22}$, $S_3\times D_{11}$, $C_{33}:C_4$, $C_{11}:C_{12}$, $C_2\times C_{66}$

Order 96: $D_4:D_6$

Order 88: $C_{11}:D_4$ x 4, $D_4\times C_{11}$ x 2, $D_{44}$ x 2, $C_4\times D_{11}$ x 2, $C_{11}:Q_8$ x 2, $C_2\times C_{44}$, $C_{22}:C_4$, $C_2^2\times C_{22}$, $C_2\times D_{22}$

Order 66: $C_{66}$ x 2, $D_{33}$, $C_3\times D_{11}$, $S_3\times C_{11}$

Order 48: $S_3\times D_4$ x 2, $D_{12}:C_2$ x 2, $C_2\times D_{12}$ x 2, $D_4:C_6$, $D_{12}:C_2$

Order 44: $C_2\times C_{22}$ x 3, $D_{22}$ x 3, $C_{44}$ x 2, $C_{11}:C_4$ x 2

Order 33: $C_{33}$

Order 32: $D_4:C_2^2$

Order 24: $D_{12}$ x 4, $C_2\times C_{12}$ x 2, $C_3:D_4$ x 2, $C_4\times S_3$ x 2, $C_2\times D_6$ x 2, $C_3\times D_4$ x 2, $C_3:Q_8$, $C_3\times Q_8$

Order 22: $C_{22}$ x 3, $D_{11}$ x 2

Order 16: $D_4:C_2$ x 4, $C_2\times D_4$ x 4

Order 12: $D_6$ x 4, $C_{12}$ x 3, $C_2\times C_6$ x 2, $C_3:C_4$

Order 11: $C_{11}$

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

Order 6: $C_6$ x 3, $S_3$ x 2

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

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1056: $D_{12}:D_{22}$ ($C_1$)

Order 528: $D_6:D_{22}$ ($C_2$) x 4, $C_{12}:D_{22}$ ($C_2$) x 2, $D_{11}\times D_{12}$ ($C_2$) x 2, $D_{12}:D_{11}$ ($C_2$) x 2, $D_{12}:D_{11}$ ($C_2$) x 2, $D_{132}:C_2$ ($C_2$), $C_{22}\times D_{12}$ ($C_2$), $D_{44}:C_6$ ($C_2$)

Order 264: $C_{33}:D_4$ ($C_2^2$) x 4, $C_6.D_{22}$ ($C_2^2$) x 4, $S_3\times D_{22}$ ($C_2^2$) x 4, $C_{11}\times D_{12}$ ($C_2^2$) x 4, $C_{11}:D_{12}$ ($C_2^2$) x 4, $D_6\times C_{22}$ ($C_2^2$) x 2, $C_{33}:D_4$ ($C_2^2$) x 2, $C_4\times D_{33}$ ($C_2^2$) x 2, $C_{33}:D_4$ ($C_2^2$) x 2, $C_{12}\times D_{11}$ ($C_2^2$) x 2, $C_2\times C_{132}$ ($C_2^2$), $D_{132}$ ($C_2^2$), $C_{33}:Q_8$ ($C_2^2$), $C_3\times D_{44}$ ($C_2^2$), $C_{33}:Q_8$ ($C_2^2$)

Order 176: $D_{44}:C_2$ ($S_3$)

Order 132: $S_3\times C_{22}$ ($C_2^3$) x 4, $D_{66}$ ($C_2^3$) x 2, $C_3\times D_{22}$ ($C_2^3$) x 2, $C_{132}$ ($C_2^3$) x 2, $C_{33}:C_4$ ($C_2^3$) x 2, $C_{11}:C_{12}$ ($C_2^3$) x 2, $C_2\times C_{66}$ ($C_2^3$)

Order 88: $C_{11}:D_4$ ($D_6$) x 2, $C_4\times D_{11}$ ($D_6$) x 2, $C_2\times C_{44}$ ($D_6$), $D_{44}$ ($D_6$), $C_{11}:Q_8$ ($D_6$)

Order 66: $C_{66}$ ($C_2^4$)

Order 48: $C_2\times D_{12}$ ($D_{11}$)

Order 44: $D_{22}$ ($C_2\times D_6$) x 2, $C_{44}$ ($C_2\times D_6$) x 2, $C_{11}:C_4$ ($C_2\times D_6$) x 2, $C_2\times C_{22}$ ($C_2\times D_6$)

Order 33: $C_{33}$ ($D_4:C_2^2$)

Order 24: $D_{12}$ ($D_{22}$) x 4, $C_2\times D_6$ ($D_{22}$) x 2, $C_2\times C_{12}$ ($D_{22}$)

Order 22: $C_{22}$ ($C_2^2\times D_6$)

Order 12: $D_6$ ($C_2\times D_{22}$) x 4, $C_{12}$ ($C_2\times D_{22}$) x 2, $C_2\times C_6$ ($C_2\times D_{22}$)

Order 11: $C_{11}$ ($D_4:D_6$)

Order 8: $C_2\times C_4$ ($S_3\times D_{11}$)

Order 6: $C_6$ ($C_2^2\times D_{22}$)

Order 4: $C_4$ ($S_3\times D_{22}$) x 2, $C_2^2$ ($S_3\times D_{22}$)

Order 3: $C_3$ ($D_4:D_{22}$)

Order 2: $C_2$ ($D_6\times D_{22}$)

Order 1: $C_1$ ($D_{12}:D_{22}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1056: $D_{12}:D_{22}$ ($C_1$)

Order 528: $D_{132}:C_2$ ($C_2$), $C_{22}\times D_{12}$ ($C_2$), $D_{44}:C_6$ ($C_2$), $D_6:D_{22}$ ($C_2$), $C_{12}:D_{22}$ ($C_2$), $D_{11}\times D_{12}$ ($C_2$), $D_{12}:D_{11}$ ($C_2$), $D_{12}:D_{11}$ ($C_2$)

Order 264: $C_{11}\times D_{12}$ ($C_2^2$) x 2, $C_{33}:D_4$ ($C_2^2$), $C_6.D_{22}$ ($C_2^2$), $D_6\times C_{22}$ ($C_2^2$), $S_3\times D_{22}$ ($C_2^2$), $C_2\times C_{132}$ ($C_2^2$), $C_{33}:D_4$ ($C_2^2$), $D_{132}$ ($C_2^2$), $C_4\times D_{33}$ ($C_2^2$), $C_{33}:Q_8$ ($C_2^2$), $C_{33}:D_4$ ($C_2^2$), $C_3\times D_{44}$ ($C_2^2$), $C_{12}\times D_{11}$ ($C_2^2$), $C_{33}:Q_8$ ($C_2^2$), $C_{11}:D_{12}$ ($C_2^2$)

Order 176: $D_{44}:C_2$ ($S_3$)

Order 132: $C_{132}$ ($C_2^3$) x 2, $D_{66}$ ($C_2^3$), $S_3\times C_{22}$ ($C_2^3$), $C_3\times D_{22}$ ($C_2^3$), $C_{33}:C_4$ ($C_2^3$), $C_{11}:C_{12}$ ($C_2^3$), $C_2\times C_{66}$ ($C_2^3$)

Order 88: $C_2\times C_{44}$ ($D_6$), $C_{11}:D_4$ ($D_6$), $D_{44}$ ($D_6$), $C_4\times D_{11}$ ($D_6$), $C_{11}:Q_8$ ($D_6$)

Order 66: $C_{66}$ ($C_2^4$)

Order 48: $C_2\times D_{12}$ ($D_{11}$)

Order 44: $C_{44}$ ($C_2\times D_6$) x 2, $C_2\times C_{22}$ ($C_2\times D_6$), $D_{22}$ ($C_2\times D_6$), $C_{11}:C_4$ ($C_2\times D_6$)

Order 33: $C_{33}$ ($D_4:C_2^2$)

Order 24: $D_{12}$ ($D_{22}$) x 2, $C_2\times C_{12}$ ($D_{22}$), $C_2\times D_6$ ($D_{22}$)

Order 22: $C_{22}$ ($C_2^2\times D_6$)

Order 12: $C_{12}$ ($C_2\times D_{22}$) x 2, $C_2\times C_6$ ($C_2\times D_{22}$), $D_6$ ($C_2\times D_{22}$)

Order 11: $C_{11}$ ($D_4:D_6$)

Order 8: $C_2\times C_4$ ($S_3\times D_{11}$)

Order 6: $C_6$ ($C_2^2\times D_{22}$)

Order 4: $C_4$ ($S_3\times D_{22}$) x 2, $C_2^2$ ($S_3\times D_{22}$)

Order 3: $C_3$ ($D_4:D_{22}$)

Order 2: $C_2$ ($D_6\times D_{22}$)

Order 1: $C_1$ ($D_{12}:D_{22}$)

Series

Derived series

$D_{12}:D_{22}$

$\rhd$

$C_{66}$

$\rhd$

$C_1$

Chief series

$D_{12}:D_{22}$

$\rhd$

$D_{11}\times D_{12}$

$\rhd$

$S_3\times D_{22}$

$\rhd$

$S_3\times C_{22}$

$\rhd$

$C_{66}$

$\rhd$

$C_{33}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$D_{12}:D_{22}$

$\rhd$

$C_{66}$

$\rhd$

$C_{33}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2\times C_4$

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 $40 \times 40$ rational character table.