Finite Group Databasebeta

← Prev (#95) Next (#97) →

C₄×C₈ #96

order	32
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = #2021 (order 128) · Inn(G) = C₁ · Out(G) = #2021
holomorph	G ⋊ Aut(G) = #72217072
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₄

Aliases

C₄×C₈

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₄	C₂²
socle, prime radical	C₂²	C₂×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

22 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 2 generators on 12 points:

(1,2,3,4,5,6,7,8);(9,10,11,12)

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

center of	#1311, #1312, C₈×M₁₆, …
derived subgroup of	#7092, #7093, #7094, …
Frattini subgroup of	C₈×C₁₆, #1311, #1312, …
Fitting subgroup of	C₄×C₈×A₅, C₄×#91145, C₈×#6732, …
maximal subgroup of	C₈², #322, D₄⋊C₈, …
maximal normal subgroup of	C₈², #322, D₄⋊C₈, …
maximal quotient of	C₈², #322, #336, …
outer automorphism group of	C₅×D₁₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32, 3);