Finite Group Databasebeta

← Prev (#89691) Next (#89693) →

C₂×C₄×C₇⋊C₈ #89692

order	448
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

C₂×C₄×C₇⋊C₈

Splittings

C₇ ⋊ C₂×C₄×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄²	D₇
commutator, hypocenter	C₇	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₂×C₄²×C₇	C₂
socle, prime radical	C₂³×C₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₄²

162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 21 points:

(1,2)(8,9)(10,11)(12,13)(14,15,16,18,17,19,20,21);(3,4,5,6)(7,8,10,12,13,11,9);(1,2)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11725425, C₇⋊#1273, #11725473, …
maximal normal subgroup of	#11725425, C₇⋊#1273, #11725473, …
maximal quotient of	C₄²×C₇⋊C₈, #11727142, C₄×C₄⋊(C₇⋊C₈), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 454);