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C₂×#6432 #90159

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

Aliases

Splittings

C₇ ⋊ C₂×C₄⋊Q₈

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₇
commutator, cosocle	C₂²×C₇	C₂⁴
Frattini	C₂²	C₂³×D₇
Fitting	C₂×C₄²×C₇	C₂
socle, prime radical	C₂³×C₇	C₂³
hypercenter	C₂×C₄²	D₇
hypocenter	C₇	C₂×C₄⋊Q₈
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 448

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

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

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726385, #11726386, #11726387, …
maximal normal subgroup of	#11726385, #11726386, #11726387, …
maximal quotient of	C₄×#6432, #11730752, #11730753, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 921);