Abstract group 88.8: $C_2\times C_{44}$

• Label: 88.8
• Order: \( 2^{3} \cdot 11 \)
• Exponent: \( 2^{2} \cdot 11 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{4} \cdot 5 \)
• Perm deg.: $17$
• Trans deg.: $88$
• Rank: $2$

Group information

Description:	$C_2\times C_{44}$
Order:	\(88\)\(\medspace = 2^{3} \cdot 11 \)
Exponent:	\(44\)\(\medspace = 2^{2} \cdot 11 \)
Automorphism group:	$D_4\times C_{10}$, of order \(80\)\(\medspace = 2^{4} \cdot 5 \)
Composition factors:	$C_2$ x 3, $C_{11}$
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	11	22	44
Elements	1	3	4	10	30	40	88
Conjugacy classes	1	3	4	10	30	40	88
Divisions	1	3	2	1	3	2	12
Autjugacy classes	1	2	1	1	2	1	8

Dimension	1	2	10	20
Irr. complex chars.	88	0	0	0	88
Irr. rational chars.	4	2	4	2	12

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$88$
Rank:	$2$
Inequivalent generating pairs:	$36$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	2	3	12

Constructions

Presentation:	Abelian group $\langle a, b \mid a^{2}=b^{44}=1 \rangle$
Permutation group:	Degree $17$ $\langle(3,6,4,5), (1,2), (7,17,16,15,14,13,12,11,10,9,8), (3,4)(5,6)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 34 \end{array}\right), \left(\begin{array}{rr} 81 & 0 \\ 0 & 11 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{89})$
Direct product:	$C_2$ $\, \times\, $ $C_4$ $\, \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:	$C_{22}$ . $C_2^2$	$C_2^2$ . $C_{22}$	$(C_2\times C_{22})$ . $C_2$	$C_2$ . $(C_2\times C_{22})$	more information
Aut. group:	$\Aut(C_{115})$	$\Aut(C_{230})$

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

Homology

Primary decomposition:	$C_{2} \times C_{4} \times C_{11}$
Schur multiplier:	$C_{2}$
Commutator length:	$0$

Subgroups

There are 16 subgroups, all normal (8 characteristic).

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

Special subgroups

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

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

Order 22: $C_{22}$ x 3

Order 11: $C_{11}$

Order 8: $C_2\times C_4$

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 88: $C_2\times C_{44}$

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

Order 22: $C_{22}$ x 2

Order 11: $C_{11}$

Order 8: $C_2\times C_4$

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 88: $C_2\times C_{44}$ ($C_1$)

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

Order 22: $C_{22}$ ($C_4$) x 2, $C_{22}$ ($C_2^2$)

Order 11: $C_{11}$ ($C_2\times C_4$)

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

Order 4: $C_4$ ($C_{22}$) x 2, $C_2^2$ ($C_{22}$)

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

Order 1: $C_1$ ($C_2\times C_{44}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 88: $C_2\times C_{44}$ ($C_1$)

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

Order 22: $C_{22}$ ($C_2^2$), $C_{22}$ ($C_4$)

Order 11: $C_{11}$ ($C_2\times C_4$)

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

Order 4: $C_2^2$ ($C_{22}$), $C_4$ ($C_{22}$)

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

Order 1: $C_1$ ($C_2\times C_{44}$)

Series

Derived series	$C_2\times C_{44}$	$\rhd$	$C_1$
Chief series	$C_2\times C_{44}$	$\rhd$	$C_2\times C_{22}$	$\rhd$	$C_{22}$	$\rhd$	$C_{11}$	$\rhd$	$C_1$
Lower central series	$C_2\times C_{44}$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_2\times C_{44}$

Supergroups

This group is a maximal subgroup of 62 larger groups in the database.

This group is a maximal quotient of 47 larger groups in the database.

Character theory

Complex character table

See the $88 \times 88$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	4A	4B	11A	22A	22B	22C	44A	44B
	Size	1	1	1	1	2	2	10	10	10	10	20	20
	2 P	1A	1A	1A	1A	2C	2C	11A	11A	11A	11A	22C	22C
	11 P	1A	2A	2B	2C	4A	4B	11A	22A	22B	22C	44A	44B
88.8.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
88.8.1b		$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$
88.8.1c		$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$
88.8.1d		$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$
88.8.1e		$2$	$-2$	$2$	$-2$	$0$	$0$	$2$	$-2$	$-2$	$-2$	$0$	$0$
88.8.1f		$2$	$2$	$-2$	$-2$	$0$	$0$	$2$	$2$	$2$	$2$	$0$	$0$
88.8.1g		$10$	$10$	$10$	$10$	$10$	$10$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$
88.8.1h		$10$	$-10$	$-10$	$10$	$-10$	$10$	$-1$	$1$	$1$	$1$	$1$	$-1$
88.8.1i		$10$	$-10$	$-10$	$10$	$10$	$-10$	$-1$	$1$	$1$	$1$	$-1$	$1$
88.8.1j		$10$	$10$	$10$	$10$	$-10$	$-10$	$-1$	$-1$	$-1$	$-1$	$1$	$1$
88.8.1k		$20$	$-20$	$20$	$-20$	$0$	$0$	$-2$	$2$	$2$	$2$	$0$	$0$
88.8.1l		$20$	$20$	$-20$	$-20$	$0$	$0$	$-2$	$-2$	$-2$	$-2$	$0$	$0$