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C₂×C₄×C₃⋊C₁₆ #69739

order	384
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂²×C₄

Aliases

C₂×C₄×C₃⋊C₁₆

Splittings

C₃ ⋊ C₂×C₄×C₁₆

Subgroups

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

216 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:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10682014, #10682015, #10682026, …
maximal normal subgroup of	#10682014, #10682015, #10682026, …
maximal quotient of	C₂×C₈×C₃⋊C₁₆, C₂×#67438, M₁₆×C₃⋊C₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 2315);