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C₂×#6437 #90166

order	448
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

Splittings

Fitting	C₂×C₄²×C₇ ⋊ C₂
Sylow	C₇ ⋊ C₂×#127

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₇
commutator, cosocle	C₂²×C₇	C₂⁴
Frattini	C₂²	C₂³×D₇
Fitting	C₂×C₄²×C₇	C₂
socle	C₂³×C₇	C₂³
hypercenter	C₂×C₄²	D₇
hypocenter	C₇	C₂×#127
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 4 generators on 17 points:

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

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726383, #11726428, #11726430, …
maximal normal subgroup of	#11726383, #11726428, #11726430, …
maximal quotient of	C₄×#6437, #11730756, #11730758, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 928);