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C₂²×C₇⋊C₁₆ #89868

order	448
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

C₂²×C₇⋊C₁₆

Splittings

C₇ ⋊ C₂²×C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₈	D₇
commutator, hypocenter	C₇	C₂²×C₁₆
Frattini	C₈	C₂²×D₇
Fitting	C₂²×C₇×C₈	C₂
socle, prime radical	C₂³×C₇	C₈
cosocle	C₇×C₈	C₂³
LCS	G ▷ C₇
UCS	C₁ ◁ C₂²×C₈

98 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2)(3,4)(6,7)(8,9)(10,11)(12,13,14,17,15,18,20,23,16,19,21,24,22,25,26,27);(3,4)(5,6,8,10,11,9,7);(1,2)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11725463, #11725477, #11725528, …
maximal normal subgroup of	#11725463, #11725477, #11725528, …
maximal quotient of	C₂×C₄×C₇⋊C₁₆, C₂×C₄⋊(C₇⋊C₁₆), #11727498, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 630);