Abstract group 352.106: $C_8:D_{22}$

• Label: 352.106
• Order: \( 2^{5} \cdot 11 \)
• Exponent: \( 2^{3} \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 5 \)
• Perm deg.: $19$
• Trans deg.: $88$
• Rank: $3$

Group information

Description:	$C_8:D_{22}$
Order:	\(352\)\(\medspace = 2^{5} \cdot 11 \)
Exponent:	\(88\)\(\medspace = 2^{3} \cdot 11 \)
Automorphism group:	$C_{22}.(C_2^4\times C_{10})$, of order \(3520\)\(\medspace = 2^{6} \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 5, $C_{11}$
Derived length:	$2$

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

Group statistics

Order	1	2	4	8	11	22	44	88
Elements	1	75	68	48	10	90	20	40	352
Conjugacy classes	1	5	3	2	5	15	5	10	46
Divisions	1	5	3	2	1	3	1	1	17
Autjugacy classes	1	5	3	2	1	3	1	1	17

Dimension	1	2	4	10	20	40
Irr. complex chars.	8	22	16	0	0	0	46
Irr. rational chars.	8	2	1	4	1	1	17

Minimal presentations

Permutation degree:	$19$
Transitive degree:	$88$
Rank:	$3$
Inequivalent generating triples:	$4032$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	40
Arbitrary	4	6	14

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{2}=c^{88}=1, b^{a}=bc^{44}, c^{a}=c^{21}, c^{b}=c^{23} \rangle$
Permutation group:	Degree $19$ $\langle(1,2)(4,6)(10,11)(12,13)(14,15)(16,17)(18,19), (3,7)(4,6)(5,8), (1,3)(2,5) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 67 & 31 \\ 0 & 87 \end{array}\right), \left(\begin{array}{rr} 1 & 22 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 43 & 33 \\ 0 & 43 \end{array}\right), \left(\begin{array}{rr} 45 & 11 \\ 0 & 67 \end{array}\right), \left(\begin{array}{rr} 1 & 44 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/88\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_8$ $\,\rtimes\,$ $D_{11}$	$D_4$ $\,\rtimes\,$ $D_{22}$	$C_8$ $\,\rtimes\,$ $D_{22}$	$C_{88}$ $\,\rtimes\,$ $C_2^2$	all 15
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{22}$ . $D_4$	$D_4$ . $D_{22}$	$D_{44}$ . $C_2^2$	$C_{44}$ . $C_2^3$	all 10

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 506 subgroups in 68 conjugacy classes, 27 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_4\times D_{11}$
Commutator:	$G' \simeq$ $C_{44}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_2\times D_{22}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{11}\times D_8$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_8:D_{22}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{22}$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_8:C_2$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

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

Classes of subgroups up to conjugation

Order 352: $C_8:D_{22}$

Order 176: $C_{88}:C_2$, $C_{88}:C_2$, $D_4:D_{11}$, $D_4\times D_{11}$, $C_{11}\times D_8$, $C_{11}:\SD_{16}$, $C_{11}:D_8$

Order 88: $D_4\times C_{11}$ x 2, $C_{11}:D_4$ x 2, $C_{22}:C_4$, $D_{44}$, $C_4\times D_{11}$, $C_{11}:Q_8$, $C_{88}$, $C_2\times D_{22}$, $C_{11}:C_8$

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

Order 32: $D_8:C_2$

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

Order 16: $\SD_{16}$ x 2, $D_8$ x 2, $\OD_{16}$, $D_4:C_2$, $C_2\times D_4$

Order 11: $C_{11}$

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

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

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 352: $C_8:D_{22}$

Order 176: $C_{88}:C_2$, $C_{88}:C_2$, $D_4:D_{11}$, $D_4\times D_{11}$, $C_{11}\times D_8$, $C_{11}:\SD_{16}$, $C_{11}:D_8$

Order 88: $D_4\times C_{11}$ x 2, $C_{11}:D_4$ x 2, $C_{22}:C_4$, $D_{44}$, $C_4\times D_{11}$, $C_{11}:Q_8$, $C_{88}$, $C_2\times D_{22}$, $C_{11}:C_8$

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

Order 32: $D_8:C_2$

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

Order 16: $\SD_{16}$ x 2, $D_8$ x 2, $\OD_{16}$, $D_4:C_2$, $C_2\times D_4$

Order 11: $C_{11}$

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

Order 4: $C_2^2$ x 5, $C_4$ x 3

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 352: $C_8:D_{22}$ ($C_1$)

Order 176: $C_{88}:C_2$ ($C_2$), $C_{88}:C_2$ ($C_2$), $D_4:D_{11}$ ($C_2$), $D_4\times D_{11}$ ($C_2$), $C_{11}\times D_8$ ($C_2$), $C_{11}:\SD_{16}$ ($C_2$), $C_{11}:D_8$ ($C_2$)

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

Order 44: $D_{22}$ ($D_4$), $C_{44}$ ($C_2^3$), $C_{11}:C_4$ ($D_4$)

Order 22: $C_{22}$ ($C_2\times D_4$)

Order 16: $D_8$ ($D_{11}$)

Order 11: $C_{11}$ ($D_8:C_2$)

Order 8: $D_4$ ($D_{22}$) x 2, $C_8$ ($D_{22}$)

Order 4: $C_4$ ($C_2\times D_{22}$)

Order 2: $C_2$ ($D_4\times D_{11}$)

Order 1: $C_1$ ($C_8:D_{22}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 352: $C_8:D_{22}$ ($C_1$)

Order 176: $C_{88}:C_2$ ($C_2$), $C_{88}:C_2$ ($C_2$), $D_4:D_{11}$ ($C_2$), $D_4\times D_{11}$ ($C_2$), $C_{11}\times D_8$ ($C_2$), $C_{11}:\SD_{16}$ ($C_2$), $C_{11}:D_8$ ($C_2$)

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

Order 44: $D_{22}$ ($D_4$), $C_{44}$ ($C_2^3$), $C_{11}:C_4$ ($D_4$)

Order 22: $C_{22}$ ($C_2\times D_4$)

Order 16: $D_8$ ($D_{11}$)

Order 11: $C_{11}$ ($D_8:C_2$)

Order 8: $D_4$ ($D_{22}$) x 2, $C_8$ ($D_{22}$)

Order 4: $C_4$ ($C_2\times D_{22}$)

Order 2: $C_2$ ($D_4\times D_{11}$)

Order 1: $C_1$ ($C_8:D_{22}$)

Series

Derived series

$C_8:D_{22}$

$\rhd$

$C_{44}$

$\rhd$

$C_1$

Chief series

$C_8:D_{22}$

$\rhd$

$D_4:D_{11}$

$\rhd$

$C_4\times D_{11}$

$\rhd$

$C_{44}$

$\rhd$

$C_{22}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_8:D_{22}$

$\rhd$

$C_{44}$

$\rhd$

$C_{22}$

$\rhd$

$C_{11}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

$\lhd$

$D_8$

Supergroups

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

This group is a maximal quotient of 48 larger groups in the database.

Character theory

Complex character table

See the $46 \times 46$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $17 \times 17$ rational character table.