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C₂×#6438 #90167

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

Aliases

Splittings

Fitting	C₂×C₄²×C₇ ⋊ C₂
Sylow	C₇ ⋊ C₂×#124

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
prime radical	C₂²×D₂₈	C₂
hypercenter	C₂×C₄²	D₇
hypocenter	C₇	C₂×#124
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 224

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

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

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726384, #11726429, #11726431, …
maximal normal subgroup of	#11726384, #11726429, #11726431, …
maximal quotient of	C₄×#6438, #11730753, #11730758, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 929);