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C₂×#6409 #89996

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	C₂²×C₇×Q₈	C₂
socle, prime radical	C₂³×C₇	D₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂²×Q₈	D₇
hypocenter	C₇	C₂×#103
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₈

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:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11727420, #11727421, #11727422, …
maximal normal subgroup of	#11727420, #11727421, #11727422, …
maximal quotient of	#11729308, C₂×#89324, C₂×Q₈⋊(C₇⋊C₈), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 758);