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C₂×#6402 #89950

order	448
isoclinism	isoclinic class of #6400
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

Fitting	C₂×C₇×Q₁₆ ⋊ C₂
Sylow	C₇ ⋊ C₂×SD₃₂

Subgroups

	subgroup	G/H
center	C₂²	#1141
commutator, cosocle	C₇×C₈	C₂³
Frattini	C₈	C₂²×D₇
Fitting	C₂×C₇×Q₁₆	C₂
socle	C₂²×C₇	D₈
prime radical	C₂×D₅₆	C₂
hypercenter	C₂×Q₁₆	D₇
hypocenter	C₇	C₂×SD₃₂
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726960, #11726962, #11726977, …
maximal normal subgroup of	#11726960, #11726962, #11726977, …
maximal quotient of	C₂×#89285, C₂×#89286, #11728354, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 712);