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C₂×#3701 #63712

order	288
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

Splittings

Fitting	C₂×Q₈×C₉ ⋊ C₂
Sylow	C₉ ⋊ C₂×#41

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₉
commutator, cosocle	C₂×C₉	C₂⁴
Frattini	C₂×C₃	C₂³×D₃
Fitting	C₂×Q₈×C₉	C₂
socle	C₂²×C₃	C₂²×D₃
hypercenter	C₂×Q₈	D₉
hypocenter	C₉	C₂×#41
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10588431, #10588483, #10588484, …
maximal normal subgroup of	#10588431, #10588483, #10588484, …
maximal quotient of	C₂×#63452, C₂×#63454, C₂×#63455, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 360);