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C₂×#6384 #89741

order	448
isoclinism	isoclinic class of #1141
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

C₇ ⋊ C₂×#103

Subgroups

	subgroup	G/H
center	C₂³	C₂²⋊D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₂×C₇×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₇	D₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂×C₄⋊C₄	D₇
hypocenter	C₇	C₂×#103
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

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:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726535, #11726537, #11726547, …
maximal normal subgroup of	#11726535, #11726537, #11726547, …
maximal quotient of	C₂×#89247, #11727225, C₄×#6384, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 503);