Finite Group Databasebeta

← Prev (#10) Next (#12) →

C₂×C₄ #11

order	8
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = D₄ (order 8) · Inn(G) = C₁ · Out(G) = D₄
holomorph	G ⋊ Aut(G) = B(4,𝔽₂)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂

Aliases

abelian	C₂×C₄
multiplicative	(ℤ/15)ˣ, (ℤ/16)ˣ, (ℤ/20)ˣ, (ℤ/30)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂²
socle, prime radical	C₂²	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₂ ▷ C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 6
transitive degree	μ_t(G) = 8

A minimal faithful permutation representation — 2 generators on 6 points:

(1,2,3,4);(5,6)

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈
size	1	1	1	1	1	1	1	1
order	1	4	2	4	2	4	2	4
χ₁	1	1	1	1	1	1	1	1
χ₂	1	$i$	-1	$- i$	1	$i$	-1	$- i$
χ₃	1	-1	1	-1	1	-1	1	-1
χ₄	1	$- i$	-1	$i$	1	$- i$	-1	$i$
χ₅	1	1	1	1	-1	-1	-1	-1
χ₆	1	$i$	-1	$- i$	-1	$- i$	1	$i$
χ₇	1	-1	1	-1	-1	1	-1	1
χ₈	1	$- i$	-1	$i$	-1	$i$	1	$- i$

Here i is the imaginary unit (i² = −1).

Relationships

center of	#97, C₂²⋊C₈, C₄⋊C₈, …
derived subgroup of	#327, #328, #329, …
Frattini subgroup of	C₄×C₈, #97, C₂²⋊C₈, …
Fitting subgroup of	C₄×SL(2,𝔽₅), C₂×C₄×A₅, C₂×#6732, …
socle of	#72262003, C₂×#72258201, #72281475, …
maximal subgroup of	C₄², C₂²⋊C₄, C₄⋊C₄, …
maximal normal subgroup of	C₄², C₂²⋊C₄, C₄⋊C₄, …
maximal quotient of	C₄², C₂²⋊C₄, C₄⋊C₄, …
automorphism group of	C₃×C₅, C₁₆, C₄×C₅, …
outer automorphism group of	C₃×C₅, C₁₆, C₄×C₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(8, 2);