Abstract group 8.2: $C_2\times C_4$

• Label: 8.2
• Order: \( 2^{3} \)
• Exponent: \( 2^{2} \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{3} \)
• Perm deg.: $6$
• Trans deg.: $8$
• Rank: $2$

Group information

Description:	$C_2\times C_4$
Order:	\(8\)\(\medspace = 2^{3} \)
Exponent:	\(4\)\(\medspace = 2^{2} \)
Automorphism group:	$D_4$, of order \(8\)\(\medspace = 2^{3} \)
Composition factors:	$C_2$ x 3
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
Elements	1	3	4	8
Conjugacy classes	1	3	4	8
Divisions	1	3	2	6
Autjugacy classes	1	2	1	4

Dimension	1	2
Irr. complex chars.	8	0	8
Irr. rational chars.	4	2	6

Minimal presentations

Permutation degree:	$6$
Transitive degree:	$8$
Rank:	$2$
Inequivalent generating pairs:	$3$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	Abelian group $\langle a, b \mid a^{2}=b^{4}=1 \rangle$
Permutation group:	$\langle(3,6,4,5), (1,2), (3,4)(5,6)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrr} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{3}(\Z)$
	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 2 \end{array}\right), \left(\begin{array}{rr} 4 & 0 \\ 0 & 4 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{5})$
Transitive group:	8T2				more information
Direct product:	$C_2$ $\, \times\, $ $C_4$
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:	$C_2^2$ . $C_2$	$C_2$ . $C_2^2$			more information
Aut. group:	$\Aut(C_{15})$	$\Aut(C_{16})$	$\Aut(C_{20})$	$\Aut(C_{30})$

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

Homology

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

Subgroups

There are 8 subgroups, all normal (4 characteristic).

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

Special subgroups

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

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 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 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 8: $C_2\times C_4$ ($C_1$)

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

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

Order 1: $C_1$ ($C_2\times C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 8: $C_2\times C_4$ ($C_1$)

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

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

Order 1: $C_1$ ($C_2\times C_4$)

Series

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

Supergroups

This group is a maximal subgroup of 233 larger groups in the database.

This group is a maximal quotient of 296 larger groups in the database.

Character theory

Complex character table

		1A	2A	2B	2C	4A1	4A-1	4B1	4B-1
	Size	1	1	1	1	1	1	1	1
	2 P	1A	1A	1A	1A	2C	2C	2C	2C
	Type
8.2.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
8.2.1b	R	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$1$
8.2.1c	R	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$
8.2.1d	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
8.2.1e1	C	$1$	$-1$	$1$	$-1$	$-i$	$i$	$i$	$-i$
8.2.1e2	C	$1$	$-1$	$1$	$-1$	$i$	$-i$	$-i$	$i$
8.2.1f1	C	$1$	$1$	$-1$	$-1$	$-i$	$i$	$-i$	$i$
8.2.1f2	C	$1$	$1$	$-1$	$-1$	$i$	$-i$	$i$	$-i$

Rational character table

		1A	2A	2B	2C	4A	4B
	Size	1	1	1	1	2	2
	2 P	1A	1A	1A	1A	2C	2C
8.2.1a		$1$	$1$	$1$	$1$	$1$	$1$
8.2.1b		$1$	$-1$	$-1$	$1$	$-1$	$1$
8.2.1c		$1$	$-1$	$-1$	$1$	$1$	$-1$
8.2.1d		$1$	$1$	$1$	$1$	$-1$	$-1$
8.2.1e		$2$	$-2$	$2$	$-2$	$0$	$0$
8.2.1f		$2$	$2$	$-2$	$-2$	$0$	$0$