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C₂×#4699 #78474

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

Aliases

Splittings

C₃ ⋊ C₂×#494

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×C₄×C₈	C₂
socle, prime radical	C₂³×C₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₃
hypocenter	C₃	C₂×#494
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

552 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 25 points:

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10690492, #10694426, #10694428, …
maximal normal subgroup of	#10690492, #10694426, #10694428, …
maximal quotient of	C₂×#68063, C₂×#68064, C₂×#68065, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11050);