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C₂³×C₇⋊C₈ #90471

order	448
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂³×C₇⋊C₈

Splittings

C₇ ⋊ C₂³×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂³×C₄	D₇
commutator, hypocenter	C₇	C₂³×C₈
Frattini	C₄	C₂³×D₇
Fitting	C₂³×C₄×C₇	C₂
socle, prime radical	C₂⁴×C₇	C₄
cosocle	C₄×C₇	C₂⁴
LCS	G ▷ C₇
UCS	C₁ ◁ C₂³×C₄

338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 21 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²⋊C₄×C₇⋊C₈, #11727201, #11727203, …
maximal normal subgroup of	C₂²⋊C₄×C₇⋊C₈, #11727201, #11727203, …
maximal quotient of	C₂²×C₄×C₇⋊C₈, C₂²×C₄⋊(C₇⋊C₈), C₂×#89697, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1233);