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C₂×#1161 #6550

order	224
automorphism	Aut(G) order 32256
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

Splittings

Fitting	C₂×C₇×Q₈ ⋊ C₂
Sylow	C₇ ⋊ C₂×#41

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₇
commutator, cosocle	C₂×C₇	C₂⁴
Frattini	C₂	C₂³×D₇
Fitting	C₂×C₇×Q₈	C₂
socle	C₂²×C₇	C₂³
hypercenter	C₂×Q₈	D₇
hypocenter	C₇	C₂×#41
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

164 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 4 generators on 17 points:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#89523, #89575, #89576, …
maximal normal subgroup of	#89523, #89575, #89576, …
maximal quotient of	C₂×#6453, C₂×#6455, C₂×#6456, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 182);