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C₂×#6400 #89918

order	448
isoclinism	isoclinic class of #6400
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

Fitting	C₂×C₇×D₈ ⋊ C₂
Sylow	C₇ ⋊ C₂×D₁₆

Subgroups

	subgroup	G/H
center	C₂²	#1141
commutator, cosocle	C₇×C₈	C₂³
Frattini	C₈	C₂²×D₇
Fitting	C₂×C₇×D₈	C₂
socle	C₂²×C₇	D₈
hypercenter	C₂×D₈	D₇
hypocenter	C₇	C₂×D₁₆
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

98 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 25 points:

(3,4,6,8,9,7,5)(11,15)(12,19)(13,20)(16,17)(18,24)(21,25)(22,23);(4,5)(6,7)(8,9)(10,11)(12,15)(13,22)(14,18)(16,19)(17,20)(21,24)(23,25);(1,2)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726928, #11726931, #11726945, …
maximal normal subgroup of	#11726928, #11726931, #11726945, …
maximal quotient of	C₂×#89284, C₂×#89286, #11728353, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 680);