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C₂×#4504 #71402

order	384
isoclinism	isoclinic class of D₄₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

C₃ ⋊ C₂×#366

Subgroups

	subgroup	G/H
center	C₂³	D₂₄
commutator	C₃×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₂²×C₃×C₁₆	C₂
socle, prime radical	C₂³×C₃	D₈
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₂²×C₁₆	D₃
hypocenter	C₃	C₂×#366
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 25 points:

(3,4,5)(10,11,16,22,13,17,20,23,14,18,24,25,19,21,12,15);(4,5)(6,7,8,9)(11,15)(12,16)(13,19)(17,25)(18,23)(20,24)(21,22);(1,2)

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10682887, #10682888, #10682905, …
maximal normal subgroup of	#10682887, #10682888, #10682905, …
maximal quotient of	C₂×#67500, #10706829, #10706868, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 3978);