Abstract group 1408.902: $D_8.C_{88}$

• Label: 1408.902
• Order: \( 2^{7} \cdot 11 \)
• Exponent: \( 2^{4} \cdot 11 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 11 \)
• $\card{Z(G)}$: \( 2^{4} \cdot 11 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 5 \)
• Perm deg.: $43$
• Trans deg.: $352$
• Rank: $3$

Group information

Description:	$D_8.C_{88}$
Order:	\(1408\)\(\medspace = 2^{7} \cdot 11 \)
Exponent:	\(176\)\(\medspace = 2^{4} \cdot 11 \)
Automorphism group:	$C_5:((C_2\times C_4).C_2^6)$, of order \(2560\)\(\medspace = 2^{9} \cdot 5 \)
Composition factors:	$C_2$ x 7, $C_{11}$
Nilpotency class:	$3$
Derived length:	$2$

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

Group statistics

Order	1	2	4	8	11	16	22	44	88	176
Elements	1	11	20	32	10	64	110	200	320	640	1408
Conjugacy classes	1	4	9	14	10	28	40	90	140	280	616
Divisions	1	4	6	6	1	5	4	6	6	5	44
Autjugacy classes	1	3	4	5	1	4	3	4	5	4	34

Dimension	1	2	4	8	10	16	20	40	80	160
Irr. complex chars.	352	264	0	0	0	0	0	0	0	0	616
Irr. rational chars.	8	6	5	1	8	2	6	5	1	2	44

Minimal presentations

Permutation degree:	$43$
Transitive degree:	$352$
Rank:	$3$
Inequivalent generating triples:	$357504$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	not computed	160
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=c^{88}=[a,b]=1, b^{8}=c^{28}, c^{a}=c^{23}, c^{b}=c^{23} \rangle$
Permutation group:	Degree $43$ $\langle(1,26,8,31,4,27,6,29,2,25,7,32,3,28,5,30)(9,22,16,18,12,23,14,19,10,21,15,17,11,24,13,20) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 283 & 0 \\ 0 & 237 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{353})$
Direct product:	$C_{11}$ $\, \times\, $ $(C_{16}.D_4)$
Semidirect product:	$(C_8.C_{88})$ $\,\rtimes\,$ $C_2$	$(C_8.C_8)$ $\,\rtimes\,$ $C_{22}$	$(C_4\times C_{176})$ $\,\rtimes\,$ $C_2$	$(C_4\times C_{16})$ $\,\rtimes\,$ $C_{22}$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_8$ . $C_{88}$	$Q_{16}$ . $C_{88}$	$C_{176}$ . $D_4$ (2)	$\SD_{16}$ . $C_{88}$ (2)	all 54

Elements of the group are displayed as matrices in $\GL_{2}(\F_{353})$.

Homology

Abelianization:	$C_{2}^{2} \times C_{88} \simeq C_{2}^{2} \times C_{8} \times C_{11}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 208 subgroups in 146 conjugacy classes, 92 normal (52 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{176}$	$G/Z \simeq$ $D_4$
Commutator:	$G' \simeq$ $C_4$	$G/G' \simeq$ $C_2^2\times C_{88}$
Frattini:	$\Phi \simeq$ $C_2\times C_8$	$G/\Phi \simeq$ $C_2^2\times C_{22}$
Fitting:	$\operatorname{Fit} \simeq$ $D_8.C_{88}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $D_8.C_{88}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{22}$	$G/\operatorname{soc} \simeq$ $C_8\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_{16}.D_4$
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

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 1408: $D_8.C_{88}$

Order 704: $\OD_{32}:C_{22}$ x 2, $\OD_{32}:C_{22}$ x 2, $D_8:C_{44}$, $C_8.C_{88}$, $C_4\times C_{176}$

Order 352: $C_{11}\times \OD_{32}$ x 4, $C_2\times C_{176}$ x 4, $D_4:C_{44}$ x 2, $\OD_{16}:C_{22}$ x 2, $C_8.C_{44}$, $C_4\times C_{88}$, $D_8:C_{22}$

Order 176: $C_{176}$ x 5, $C_{11}\times \OD_{16}$ x 4, $C_2\times C_{88}$ x 4, $D_4:C_{22}$ x 2, $C_{11}\times \SD_{16}$ x 2, $C_{11}\times Q_{16}$, $C_{11}\times D_8$, $C_4\times C_{44}$

Order 128: $C_{16}.D_4$

Order 88: $C_{88}$ x 6, $D_4\times C_{11}$ x 4, $C_2\times C_{44}$ x 4, $Q_8\times C_{11}$ x 2

Order 64: $\OD_{32}:C_2$ x 2, $\OD_{32}:C_2$ x 2, $C_8.C_8$, $C_4\times C_{16}$, $D_8:C_4$

Order 44: $C_{44}$ x 6, $C_2\times C_{22}$ x 3

Order 32: $\OD_{32}$ x 4, $C_2\times C_{16}$ x 4, $\OD_{16}:C_2$ x 2, $C_4\wr C_2$ x 2, $D_8:C_2$, $C_4\times C_8$, $C_8.C_4$

Order 22: $C_{22}$ x 4

Order 16: $C_{16}$ x 5, $\OD_{16}$ x 4, $C_2\times C_8$ x 4, $\SD_{16}$ x 2, $D_4:C_2$ x 2, $Q_{16}$, $D_8$, $C_4^2$

Order 11: $C_{11}$

Order 8: $C_8$ x 6, $D_4$ x 4, $C_2\times C_4$ x 4, $Q_8$ x 2

Order 4: $C_4$ x 6, $C_2^2$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1408: $D_8.C_{88}$

Order 704: $\OD_{32}:C_{22}$, $D_8:C_{44}$, $C_8.C_{88}$, $\OD_{32}:C_{22}$, $C_4\times C_{176}$

Order 352: $C_2\times C_{176}$ x 3, $C_{11}\times \OD_{32}$ x 2, $C_8.C_{44}$, $D_4:C_{44}$, $C_4\times C_{88}$, $D_8:C_{22}$, $\OD_{16}:C_{22}$

Order 176: $C_{176}$ x 4, $C_2\times C_{88}$ x 3, $C_{11}\times \OD_{16}$ x 2, $D_4:C_{22}$, $C_{11}\times Q_{16}$, $C_{11}\times \SD_{16}$, $C_{11}\times D_8$, $C_4\times C_{44}$

Order 128: $C_{16}.D_4$

Order 88: $C_{88}$ x 5, $C_2\times C_{44}$ x 3, $D_4\times C_{11}$ x 2, $Q_8\times C_{11}$

Order 64: $C_8.C_8$, $\OD_{32}:C_2$, $C_4\times C_{16}$, $\OD_{32}:C_2$, $D_8:C_4$

Order 44: $C_{44}$ x 4, $C_2\times C_{22}$ x 2

Order 32: $C_2\times C_{16}$ x 3, $\OD_{32}$ x 2, $D_8:C_2$, $\OD_{16}:C_2$, $C_4\times C_8$, $C_8.C_4$, $C_4\wr C_2$

Order 22: $C_{22}$ x 3

Order 16: $C_{16}$ x 4, $C_2\times C_8$ x 3, $\OD_{16}$ x 2, $Q_{16}$, $\SD_{16}$, $D_8$, $C_4^2$, $D_4:C_2$

Order 11: $C_{11}$

Order 8: $C_8$ x 5, $C_2\times C_4$ x 3, $D_4$ x 2, $Q_8$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1408: $D_8.C_{88}$ ($C_1$)

Order 704: $\OD_{32}:C_{22}$ ($C_2$) x 2, $\OD_{32}:C_{22}$ ($C_2$) x 2, $D_8:C_{44}$ ($C_2$), $C_8.C_{88}$ ($C_2$), $C_4\times C_{176}$ ($C_2$)

Order 352: $C_{11}\times \OD_{32}$ ($C_2^2$) x 2, $C_2\times C_{176}$ ($C_2^2$) x 2, $D_4:C_{44}$ ($C_4$) x 2, $\OD_{16}:C_{22}$ ($C_2^2$) x 2, $C_8.C_{44}$ ($C_4$), $C_4\times C_{88}$ ($C_2^2$), $D_8:C_{22}$ ($C_4$)

Order 176: $D_4:C_{22}$ ($C_2\times C_4$) x 2, $C_{11}\times \SD_{16}$ ($C_8$) x 2, $C_{11}\times \OD_{16}$ ($C_2\times C_4$) x 2, $C_{176}$ ($D_4$) x 2, $C_{11}\times Q_{16}$ ($C_8$), $C_{11}\times D_8$ ($C_8$), $C_2\times C_{88}$ ($C_2^3$), $C_2\times C_{88}$ ($C_2\times C_4$), $C_4\times C_{44}$ ($C_2\times C_4$)

Order 128: $C_{16}.D_4$ ($C_{11}$)

Order 88: $D_4\times C_{11}$ ($C_2\times C_8$) x 2, $C_{88}$ ($C_2\times C_8$) x 2, $Q_8\times C_{11}$ ($C_2\times C_8$) x 2, $C_2\times C_{44}$ ($C_2^2\times C_4$), $C_{88}$ ($D_4:C_2$), $C_{88}$ ($C_2\times D_4$)

Order 64: $\OD_{32}:C_2$ ($C_{22}$) x 2, $\OD_{32}:C_2$ ($C_{22}$) x 2, $C_8.C_8$ ($C_{22}$), $C_4\times C_{16}$ ($C_{22}$), $D_8:C_4$ ($C_{22}$)

Order 44: $C_2\times C_{22}$ ($\OD_{16}:C_2$), $C_{44}$ ($C_2^2\times C_8$), $C_{44}$ ($C_4\times D_4$)

Order 32: $\OD_{16}:C_2$ ($C_2\times C_{22}$) x 2, $\OD_{32}$ ($C_2\times C_{22}$) x 2, $C_2\times C_{16}$ ($C_2\times C_{22}$) x 2, $C_4\wr C_2$ ($C_{44}$) x 2, $D_8:C_2$ ($C_{44}$), $C_4\times C_8$ ($C_2\times C_{22}$), $C_8.C_4$ ($C_{44}$)

Order 22: $C_{22}$ ($C_8\times D_4$)

Order 16: $\SD_{16}$ ($C_{88}$) x 2, $\OD_{16}$ ($C_2\times C_{44}$) x 2, $D_4:C_2$ ($C_2\times C_{44}$) x 2, $C_{16}$ ($D_4\times C_{11}$) x 2, $Q_{16}$ ($C_{88}$), $D_8$ ($C_{88}$), $C_2\times C_8$ ($C_2\times C_{44}$), $C_2\times C_8$ ($C_2^2\times C_{22}$), $C_4^2$ ($C_2\times C_{44}$)

Order 11: $C_{11}$ ($C_{16}.D_4$)

Order 8: $Q_8$ ($C_2\times C_{88}$) x 2, $D_4$ ($C_2\times C_{88}$) x 2, $C_8$ ($C_2\times C_{88}$) x 2, $C_2\times C_4$ ($C_2^2\times C_{44}$), $C_8$ ($D_4:C_{22}$), $C_8$ ($D_4\times C_{22}$)

Order 4: $C_2^2$ ($\OD_{16}:C_{22}$), $C_4$ ($C_2^2\times C_{88}$), $C_4$ ($D_4\times C_{44}$)

Order 2: $C_2$ ($D_4\times C_{88}$)

Order 1: $C_1$ ($D_8.C_{88}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1408: $D_8.C_{88}$ ($C_1$)

Order 704: $\OD_{32}:C_{22}$ ($C_2$), $D_8:C_{44}$ ($C_2$), $C_8.C_{88}$ ($C_2$), $\OD_{32}:C_{22}$ ($C_2$), $C_4\times C_{176}$ ($C_2$)

Order 352: $C_2\times C_{176}$ ($C_2^2$) x 2, $C_{11}\times \OD_{32}$ ($C_2^2$), $C_8.C_{44}$ ($C_4$), $D_4:C_{44}$ ($C_4$), $C_4\times C_{88}$ ($C_2^2$), $D_8:C_{22}$ ($C_4$), $\OD_{16}:C_{22}$ ($C_2^2$)

Order 176: $C_{176}$ ($D_4$) x 2, $D_4:C_{22}$ ($C_2\times C_4$), $C_{11}\times Q_{16}$ ($C_8$), $C_{11}\times \SD_{16}$ ($C_8$), $C_{11}\times D_8$ ($C_8$), $C_{11}\times \OD_{16}$ ($C_2\times C_4$), $C_2\times C_{88}$ ($C_2^3$), $C_2\times C_{88}$ ($C_2\times C_4$), $C_4\times C_{44}$ ($C_2\times C_4$)

Order 128: $C_{16}.D_4$ ($C_{11}$)

Order 88: $C_{88}$ ($C_2\times C_8$) x 2, $D_4\times C_{11}$ ($C_2\times C_8$), $C_2\times C_{44}$ ($C_2^2\times C_4$), $C_{88}$ ($D_4:C_2$), $C_{88}$ ($C_2\times D_4$), $Q_8\times C_{11}$ ($C_2\times C_8$)

Order 64: $C_8.C_8$ ($C_{22}$), $\OD_{32}:C_2$ ($C_{22}$), $C_4\times C_{16}$ ($C_{22}$), $\OD_{32}:C_2$ ($C_{22}$), $D_8:C_4$ ($C_{22}$)

Order 44: $C_2\times C_{22}$ ($\OD_{16}:C_2$), $C_{44}$ ($C_2^2\times C_8$), $C_{44}$ ($C_4\times D_4$)

Order 32: $C_2\times C_{16}$ ($C_2\times C_{22}$) x 2, $D_8:C_2$ ($C_{44}$), $\OD_{16}:C_2$ ($C_2\times C_{22}$), $C_4\times C_8$ ($C_2\times C_{22}$), $\OD_{32}$ ($C_2\times C_{22}$), $C_8.C_4$ ($C_{44}$), $C_4\wr C_2$ ($C_{44}$)

Order 22: $C_{22}$ ($C_8\times D_4$)

Order 16: $C_{16}$ ($D_4\times C_{11}$) x 2, $Q_{16}$ ($C_{88}$), $\SD_{16}$ ($C_{88}$), $D_8$ ($C_{88}$), $\OD_{16}$ ($C_2\times C_{44}$), $C_2\times C_8$ ($C_2\times C_{44}$), $C_2\times C_8$ ($C_2^2\times C_{22}$), $C_4^2$ ($C_2\times C_{44}$), $D_4:C_2$ ($C_2\times C_{44}$)

Order 11: $C_{11}$ ($C_{16}.D_4$)

Order 8: $C_8$ ($C_2\times C_{88}$) x 2, $Q_8$ ($C_2\times C_{88}$), $D_4$ ($C_2\times C_{88}$), $C_2\times C_4$ ($C_2^2\times C_{44}$), $C_8$ ($D_4:C_{22}$), $C_8$ ($D_4\times C_{22}$)

Order 4: $C_2^2$ ($\OD_{16}:C_{22}$), $C_4$ ($C_2^2\times C_{88}$), $C_4$ ($D_4\times C_{44}$)

Order 2: $C_2$ ($D_4\times C_{88}$)

Order 1: $C_1$ ($D_8.C_{88}$)

Series

Derived series

$D_8.C_{88}$

$\rhd$

$C_4$

$\rhd$

$C_1$

Chief series

$D_8.C_{88}$

$\rhd$

$C_4\times C_{176}$

$\rhd$

$C_4\times C_{88}$

$\rhd$

$C_2\times C_{88}$

$\rhd$

$C_2\times C_{44}$

$\rhd$

$C_{44}$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$D_8.C_{88}$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_{176}$

$\lhd$

$C_2\times C_{176}$

$\lhd$

$D_8.C_{88}$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

The $616 \times 616$ character table is not available for this group.

Rational character table

See the $44 \times 44$ rational character table.