Abstract group 16.5: $C_2\times C_8$

• Label: 16.5
• Order: \( 2^{4} \)
• Exponent: \( 2^{3} \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{4} \)
• Perm deg.: $10$
• Trans deg.: $16$
• Rank: $2$

Group information

Description:	$C_2\times C_8$
Order:	\(16\)\(\medspace = 2^{4} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \)
Composition factors:	$C_2$ x 4
Nilpotency class:	$1$
Derived length:	$1$

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

Group statistics

Order	1	2	4	8
Elements	1	3	4	8	16
Conjugacy classes	1	3	4	8	16
Divisions	1	3	2	2	8
Autjugacy classes	1	2	2	1	6

Dimension	1	2	4
Irr. complex chars.	16	0	0	16
Irr. rational chars.	4	2	2	8

Minimal presentations

Permutation degree:	$10$
Transitive degree:	$16$
Rank:	$2$
Inequivalent generating pairs:	$6$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	2	3	5

Constructions

Presentation:	Abelian group $\langle a, b \mid a^{2}=b^{8}=1 \rangle$
Permutation group:	Degree $10$ $\langle(3,10,6,8,4,9,5,7), (1,2), (3,6,4,5)(7,10,8,9), (3,4)(5,6)(7,8)(9,10)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrr} 0 & -1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & -1 & -1 & 1 & 0 \\ 1 & -1 & -1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{5}(\Z)$
	$\left\langle \left(\begin{array}{rr} 13 & 0 \\ 0 & 4 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{17})$
Transitive group:	16T5				more information
Direct product:	$C_2$ $\, \times\, $ $C_8$
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:	$C_4$ . $C_4$	$C_2^2$ . $C_4$	$C_4$ . $C_2^2$	$(C_2\times C_4)$ . $C_2$	all 5
Aut. group:	$\Aut(C_{32})$

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

Homology

Primary decomposition:	$C_{2} \times C_{8}$
Schur multiplier:	$C_{2}$
Commutator length:	$0$

Subgroups

There are 11 subgroups, all normal (7 characteristic).

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

Special subgroups

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

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 16: $C_2\times C_8$

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 16: $C_2\times C_8$

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

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 16: $C_2\times C_8$ ($C_1$)

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

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

Order 2: $C_2$ ($C_8$) x 2, $C_2$ ($C_2\times C_4$)

Order 1: $C_1$ ($C_2\times C_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 16: $C_2\times C_8$ ($C_1$)

Order 8: $C_2\times C_4$ ($C_2$), $C_8$ ($C_2$)

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

Order 2: $C_2$ ($C_2\times C_4$), $C_2$ ($C_8$)

Order 1: $C_1$ ($C_2\times C_8$)

Series

Derived series	$C_2\times C_8$	$\rhd$	$C_1$
Chief series	$C_2\times C_8$	$\rhd$	$C_8$	$\rhd$	$C_4$	$\rhd$	$C_2$	$\rhd$	$C_1$
Lower central series	$C_2\times C_8$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_2\times C_8$

Supergroups

This group is a maximal subgroup of 197 larger groups in the database.

This group is a maximal quotient of 200 larger groups in the database.

Character theory

Complex character table

		1A	2A	2B	2C	4A1	4A-1	4B1	4B-1	8A1	8A-1	8A3	8A-3	8B1	8B-1	8B3	8B-3
	Size	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
	2 P	1A	1A	1A	1A	2C	2C	2C	2C	4B1	4B-1	4B-1	4B1	4B1	4B-1	4B-1	4B1
	Type
16.5.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
16.5.1b	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$
16.5.1c	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$-1$
16.5.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$
16.5.1e1	C	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$-i$	$-i$	$-i$	$i$	$-i$	$i$	$i$	$i$
16.5.1e2	C	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$i$	$i$	$i$	$-i$	$i$	$-i$	$-i$	$-i$
16.5.1f1	C	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-i$	$i$	$-i$	$i$	$i$	$i$	$-i$	$-i$
16.5.1f2	C	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$i$	$-i$	$i$	$-i$	$-i$	$-i$	$i$	$i$
16.5.1g1	C	$1$	$-1$	$-1$	$1$	$-{\zeta }_8^2$	${\zeta }_8^2$	$-{\zeta }_8^2$	${\zeta }_8^2$	${\zeta }_8^3$	${\zeta }_8$	$-{\zeta }_8^3$	$-{\zeta }_8$	$-{\zeta }_8$	${\zeta }_8$	$-{\zeta }_8^3$	${\zeta }_8^3$
16.5.1g2	C	$1$	$-1$	$-1$	$1$	${\zeta }_8^2$	$-{\zeta }_8^2$	${\zeta }_8^2$	$-{\zeta }_8^2$	$-{\zeta }_8$	$-{\zeta }_8^3$	${\zeta }_8$	${\zeta }_8^3$	${\zeta }_8^3$	$-{\zeta }_8^3$	${\zeta }_8$	$-{\zeta }_8$
16.5.1g3	C	$1$	$-1$	$-1$	$1$	$-{\zeta }_8^2$	${\zeta }_8^2$	$-{\zeta }_8^2$	${\zeta }_8^2$	$-{\zeta }_8^3$	$-{\zeta }_8$	${\zeta }_8^3$	${\zeta }_8$	${\zeta }_8$	$-{\zeta }_8$	${\zeta }_8^3$	$-{\zeta }_8^3$
16.5.1g4	C	$1$	$-1$	$-1$	$1$	${\zeta }_8^2$	$-{\zeta }_8^2$	${\zeta }_8^2$	$-{\zeta }_8^2$	${\zeta }_8$	${\zeta }_8^3$	$-{\zeta }_8$	$-{\zeta }_8^3$	$-{\zeta }_8^3$	${\zeta }_8^3$	$-{\zeta }_8$	${\zeta }_8$
16.5.1h1	C	$1$	$1$	$-1$	$-1$	$-{\zeta }_8^2$	${\zeta }_8^2$	${\zeta }_8^2$	$-{\zeta }_8^2$	${\zeta }_8^3$	$-{\zeta }_8$	$-{\zeta }_8^3$	$-{\zeta }_8$	${\zeta }_8$	${\zeta }_8$	${\zeta }_8^3$	$-{\zeta }_8^3$
16.5.1h2	C	$1$	$1$	$-1$	$-1$	${\zeta }_8^2$	$-{\zeta }_8^2$	$-{\zeta }_8^2$	${\zeta }_8^2$	$-{\zeta }_8$	${\zeta }_8^3$	${\zeta }_8$	${\zeta }_8^3$	$-{\zeta }_8^3$	$-{\zeta }_8^3$	$-{\zeta }_8$	${\zeta }_8$
16.5.1h3	C	$1$	$1$	$-1$	$-1$	$-{\zeta }_8^2$	${\zeta }_8^2$	${\zeta }_8^2$	$-{\zeta }_8^2$	$-{\zeta }_8^3$	${\zeta }_8$	${\zeta }_8^3$	${\zeta }_8$	$-{\zeta }_8$	$-{\zeta }_8$	$-{\zeta }_8^3$	${\zeta }_8^3$
16.5.1h4	C	$1$	$1$	$-1$	$-1$	${\zeta }_8^2$	$-{\zeta }_8^2$	$-{\zeta }_8^2$	${\zeta }_8^2$	${\zeta }_8$	$-{\zeta }_8^3$	$-{\zeta }_8$	$-{\zeta }_8^3$	${\zeta }_8^3$	${\zeta }_8^3$	${\zeta }_8$	$-{\zeta }_8$

Rational character table

		1A	2A	2B	2C	4A	4B	8A	8B
	Size	1	1	1	1	2	2	4	4
	2 P	1A	1A	1A	1A	2C	2C	4B	4B
16.5.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
16.5.1b		$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$
16.5.1c		$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$
16.5.1d		$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
16.5.1e		$2$	$-2$	$2$	$-2$	$-2$	$2$	$0$	$0$
16.5.1f		$2$	$2$	$2$	$2$	$-2$	$-2$	$0$	$0$
16.5.1g		$4$	$-4$	$-4$	$4$	$0$	$0$	$0$	$0$
16.5.1h		$4$	$4$	$-4$	$-4$	$0$	$0$	$0$	$0$