Abstract group 352.58: $C_2\times C_{176}$

• Label: 352.58
• Order: \( 2^{5} \cdot 11 \)
• Exponent: \( 2^{4} \cdot 11 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{6} \cdot 5 \)
• Perm deg.: $29$
• Trans deg.: $352$
• Rank: $2$

Group information

Description:	$C_2\times C_{176}$
Order:	\(352\)\(\medspace = 2^{5} \cdot 11 \)
Exponent:	\(176\)\(\medspace = 2^{4} \cdot 11 \)
Automorphism group:	$C_{10}.C_2^5$, of order \(320\)\(\medspace = 2^{6} \cdot 5 \)
Composition factors:	$C_2$ x 5, $C_{11}$
Nilpotency class:	$1$
Derived length:	$1$

This group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.

Group statistics

Order	1	2	4	8	11	16	22	44	88	176
Elements	1	3	4	8	10	16	30	40	80	160	352
Conjugacy classes	1	3	4	8	10	16	30	40	80	160	352
Divisions	1	3	2	2	1	2	3	2	2	2	20
Autjugacy classes	1	2	2	2	1	1	2	2	2	1	16

Dimension	1	2	4	8	10	20	40	80
Irr. complex chars.	352	0	0	0	0	0	0	0	352
Irr. rational chars.	4	2	2	2	4	2	2	2	20

Minimal presentations

Permutation degree:	$29$
Transitive degree:	$352$
Rank:	$2$
Inequivalent generating pairs:	$144$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	2	3	18

Constructions

Presentation:	Abelian group $\langle a, b \mid a^{2}=b^{176}=1 \rangle$
Permutation group:	Degree $29$ $\langle(3,18,10,14,6,16,8,12,4,17,9,13,5,15,7,11), (1,2), (19,29,28,27,26,25,24,23,22,21,20) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 97 & 0 \\ 0 & 233 \end{array}\right), \left(\begin{array}{rr} 352 & 0 \\ 0 & 352 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{353})$
Direct product:	$C_2$ $\, \times\, $ $C_{16}$ $\, \times\, $ $C_{11}$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{88}$ . $C_4$	$C_{44}$ . $C_8$	$C_8$ . $C_{44}$	$C_4$ . $C_{88}$	all 16
Aut. group:	$\Aut(C_{391})$	$\Aut(C_{782})$

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

Homology

Primary decomposition:	$C_{2} \times C_{16} \times C_{11}$
Schur multiplier:	$C_{2}$
Commutator length:	$0$

Subgroups

There are 28 subgroups, all normal (20 characteristic).

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

Special subgroups

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

Order 176: $C_{176}$ x 2, $C_2\times C_{88}$

Order 88: $C_{88}$ x 2, $C_2\times C_{44}$

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

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

Order 22: $C_{22}$ x 3

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

Order 11: $C_{11}$

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

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 352: $C_2\times C_{176}$

Order 176: $C_2\times C_{88}$, $C_{176}$

Order 88: $C_{88}$ x 2, $C_2\times C_{44}$

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

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

Order 22: $C_{22}$ x 2

Order 16: $C_2\times C_8$, $C_{16}$

Order 11: $C_{11}$

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

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 352: $C_2\times C_{176}$ ($C_1$)

Order 176: $C_{176}$ ($C_2$) x 2, $C_2\times C_{88}$ ($C_2$)

Order 88: $C_2\times C_{44}$ ($C_4$), $C_{88}$ ($C_2^2$), $C_{88}$ ($C_4$)

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

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

Order 22: $C_{22}$ ($C_{16}$) x 2, $C_{22}$ ($C_2\times C_8$)

Order 16: $C_{16}$ ($C_{22}$) x 2, $C_2\times C_8$ ($C_{22}$)

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

Order 8: $C_2\times C_4$ ($C_{44}$), $C_8$ ($C_2\times C_{22}$), $C_8$ ($C_{44}$)

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

Order 2: $C_2$ ($C_{176}$) x 2, $C_2$ ($C_2\times C_{88}$)

Order 1: $C_1$ ($C_2\times C_{176}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 352: $C_2\times C_{176}$ ($C_1$)

Order 176: $C_2\times C_{88}$ ($C_2$), $C_{176}$ ($C_2$)

Order 88: $C_2\times C_{44}$ ($C_4$), $C_{88}$ ($C_2^2$), $C_{88}$ ($C_4$)

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

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

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

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

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

Order 8: $C_2\times C_4$ ($C_{44}$), $C_8$ ($C_2\times C_{22}$), $C_8$ ($C_{44}$)

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

Order 2: $C_2$ ($C_2\times C_{88}$), $C_2$ ($C_{176}$)

Order 1: $C_1$ ($C_2\times C_{176}$)

Series

Derived series

$C_2\times C_{176}$

$\rhd$

$C_1$

Chief series

$C_2\times C_{176}$

$\rhd$

$C_2\times C_{88}$

$\rhd$

$C_{88}$

$\rhd$

$C_{44}$

$\rhd$

$C_{22}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_2\times C_{176}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{176}$

Supergroups

This group is a maximal subgroup of 60 larger groups in the database.

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

Character theory

Complex character table

See the $352 \times 352$ 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.