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C₂×#6374 #89675

order	448
isoclinism	isoclinic class of D₁₁₂
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

Fitting	C₂×C₇×C₁₆ ⋊ C₂
Sylow	C₇ ⋊ C₂×SD₃₂

Subgroups

	subgroup	G/H
center	C₂²	D₅₆
commutator, cosocle	C₇×C₈	C₂³
Frattini	C₈	C₂²×D₇
Fitting	C₂×C₇×C₁₆	C₂
socle	C₂²×C₇	D₈
prime radical	C₂×D₅₆	C₂
hypercenter	C₂×C₁₆	D₇
hypocenter	C₇	C₂×SD₃₂
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

90 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)(10,11,16,22,21,25,12,15,14,18,19,24,13,17,20,23);(1,2)(4,5)(6,7)(8,9)(11,15)(12,19)(13,21)(16,20)(17,24)(18,23)(22,25);(1,2)

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726815, #11726816, C₄×#6374, …
maximal normal subgroup of	#11726815, #11726816, C₄×#6374, …
maximal quotient of	#11726800, #11726812, #11726815, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 437);