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C₂×#5827 #87259

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

Splittings

C₃ ⋊ C₂×#520

Subgroups

	subgroup	G/H
center	C₂³	C₂³×D₃
commutator	C₂×C₃	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₃
Fitting, prime radical	C₂²×C₃×#41	C₂
socle	C₂³×C₃	C₂⁴
cosocle	C₂²×C₃	C₂⁵
hypercenter	C₂²×#41	D₃
hypocenter	C₃	C₂×#520
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×#41

1640 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 5 generators on 17 points:

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10700624, #10700627, #10700666, …
maximal normal subgroup of	#10700624, #10700627, #10700666, …
maximal quotient of	C₂×#82223, C₂×#82317, C₂×#82450, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 19835);