Abstract group 704.65: $C_8.D_{44}$

• Label: 704.65
• Order: \( 2^{6} \cdot 11 \)
• Exponent: \( 2^{4} \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 5 \)
• Perm deg.: $43$
• Trans deg.: $352$
• Rank: $2$

Group information

Description:	$C_8.D_{44}$
Order:	\(704\)\(\medspace = 2^{6} \cdot 11 \)
Exponent:	\(176\)\(\medspace = 2^{4} \cdot 11 \)
Automorphism group:	$C_{22}.(C_2^4\times C_{20})$, of order \(7040\)\(\medspace = 2^{7} \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 6, $C_{11}$
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	16	22	44	88	176
Elements	1	47	48	96	10	192	30	40	80	160	704
Conjugacy classes	1	3	4	8	5	12	15	20	40	80	188
Divisions	1	3	3	3	1	2	2	2	2	1	20
Autjugacy classes	1	3	3	3	1	2	2	2	2	1	20

Dimension	1	2	4	10	16	20	40	160
Irr. complex chars.	16	172	0	0	0	0	0	0	188
Irr. rational chars.	4	4	3	2	1	3	2	1	20

Minimal presentations

Permutation degree:	$43$
Transitive degree:	$352$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	160
Arbitrary	2	4	26

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{16}=c^{22}=[b,c]=1, b^{a}=b^{13}c^{11}, c^{a}=b^{8}c^{21} \rangle$
Permutation group:	Degree $43$ $\langle(1,2)(3,12)(4,17)(5,21)(6,10)(7,11)(8,14)(9,20)(13,27)(15,23)(16,26)(18,24) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 306 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 207 & 0 \\ 0 & 191 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{353})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2\times C_{16})$ $\,\rtimes\,$ $D_{11}$	$(C_2\times C_{176})$ $\,\rtimes\,$ $C_2$	$(C_{11}:\OD_{32})$ $\,\rtimes\,$ $C_2$	$C_{11}$ $\,\rtimes\,$ $(\OD_{32}:C_2)$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{44}$ . $C_8$	$C_{88}$ . $D_4$ (2)	$C_8$ . $D_{44}$	$(C_{11}:Q_8)$ . $C_8$	all 20

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

Homology

Abelianization:	$C_{2} \times C_{8} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 352 subgroups in 56 conjugacy classes, 27 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_8$	$G/Z \simeq$ $D_{44}$
Commutator:	$G' \simeq$ $C_{44}$	$G/G' \simeq$ $C_2\times C_8$
Frattini:	$\Phi \simeq$ $C_2\times C_8$	$G/\Phi \simeq$ $D_{22}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{176}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_8.D_{44}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{22}$	$G/\operatorname{soc} \simeq$ $C_2^2:C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\OD_{32}:C_2$
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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Classes of subgroups up to conjugation

Order 704: $C_8.D_{44}$

Order 352: $C_8.D_{22}$, $C_2\times C_{176}$, $C_{11}:\OD_{32}$

Order 176: $C_{11}:\OD_{16}$, $C_{88}:C_2$, $D_{44}:C_2$, $C_8\times D_{11}$, $C_2\times C_{88}$, $C_{176}$, $C_{11}:C_{16}$

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

Order 64: $\OD_{32}:C_2$

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

Order 32: $\OD_{16}:C_2$, $\OD_{32}$, $C_2\times C_{16}$

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

Order 16: $\OD_{16}$ x 2, $C_2\times C_8$ x 2, $C_{16}$ x 2, $D_4:C_2$

Order 11: $C_{11}$

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

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 704: $C_8.D_{44}$

Order 352: $C_8.D_{22}$, $C_2\times C_{176}$, $C_{11}:\OD_{32}$

Order 176: $C_{11}:\OD_{16}$, $C_{88}:C_2$, $D_{44}:C_2$, $C_8\times D_{11}$, $C_2\times C_{88}$, $C_{176}$, $C_{11}:C_{16}$

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

Order 64: $\OD_{32}:C_2$

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

Order 32: $\OD_{16}:C_2$, $\OD_{32}$, $C_2\times C_{16}$

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

Order 16: $\OD_{16}$ x 2, $C_2\times C_8$ x 2, $C_{16}$ x 2, $D_4:C_2$

Order 11: $C_{11}$

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

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 704: $C_8.D_{44}$ ($C_1$)

Order 352: $C_8.D_{22}$ ($C_2$), $C_2\times C_{176}$ ($C_2$), $C_{11}:\OD_{32}$ ($C_2$)

Order 176: $C_{11}:\OD_{16}$ ($C_4$), $D_{44}:C_2$ ($C_4$), $C_2\times C_{88}$ ($C_2^2$)

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

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

Order 32: $C_2\times C_{16}$ ($D_{11}$)

Order 22: $C_{22}$ ($C_2^2:C_8$)

Order 16: $C_2\times C_8$ ($D_{22}$)

Order 11: $C_{11}$ ($\OD_{32}:C_2$)

Order 8: $C_2\times C_4$ ($C_4\times D_{11}$), $C_8$ ($C_{11}:D_4$), $C_8$ ($D_{44}$)

Order 4: $C_2^2$ ($C_{88}:C_2$), $C_4$ ($C_8\times D_{11}$), $C_4$ ($D_{22}:C_4$)

Order 2: $C_2$ ($D_{22}:C_8$)

Order 1: $C_1$ ($C_8.D_{44}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 704: $C_8.D_{44}$ ($C_1$)

Order 352: $C_8.D_{22}$ ($C_2$), $C_2\times C_{176}$ ($C_2$), $C_{11}:\OD_{32}$ ($C_2$)

Order 176: $C_{11}:\OD_{16}$ ($C_4$), $D_{44}:C_2$ ($C_4$), $C_2\times C_{88}$ ($C_2^2$)

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

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

Order 32: $C_2\times C_{16}$ ($D_{11}$)

Order 22: $C_{22}$ ($C_2^2:C_8$)

Order 16: $C_2\times C_8$ ($D_{22}$)

Order 11: $C_{11}$ ($\OD_{32}:C_2$)

Order 8: $C_2\times C_4$ ($C_4\times D_{11}$), $C_8$ ($C_{11}:D_4$), $C_8$ ($D_{44}$)

Order 4: $C_2^2$ ($C_{88}:C_2$), $C_4$ ($C_8\times D_{11}$), $C_4$ ($D_{22}:C_4$)

Order 2: $C_2$ ($D_{22}:C_8$)

Order 1: $C_1$ ($C_8.D_{44}$)

Series

Derived series

$C_8.D_{44}$

$\rhd$

$C_{44}$

$\rhd$

$C_1$

Chief series

$C_8.D_{44}$

$\rhd$

$C_2\times C_{176}$

$\rhd$

$C_2\times C_{88}$

$\rhd$

$C_2\times C_{44}$

$\rhd$

$C_{44}$

$\rhd$

$C_{22}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_8.D_{44}$

$\rhd$

$C_{44}$

$\rhd$

$C_{22}$

$\rhd$

$C_{11}$

Upper central series

$C_1$

$\lhd$

$C_8$

$\lhd$

$C_2\times C_8$

$\lhd$

$C_2\times C_{16}$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

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

Character theory

Complex character table

See the $188 \times 188$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $20 \times 20$ rational character table.