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C₂²×C₄×C₃⋊C₈ #82042

order	384
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

C₂²×C₄×C₃⋊C₈

Splittings

C₃ ⋊ C₂²×C₄×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₄²	D₃
commutator, hypocenter	C₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂²×C₃×C₄²	C₂
socle, prime radical	C₂⁴×C₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂⁴
LCS	G ▷ C₃
UCS	C₁ ◁ C₂²×C₄²

760 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 384

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

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃⋊#1273, #10684568, #10684570, …
maximal normal subgroup of	C₂×C₃⋊#1273, #10684568, #10684570, …
maximal quotient of	C₂²×C₈×C₃⋊C₈, C₂²×#4455, C₂×M₁₆×C₃⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 14618);