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C₂³×D₃×C₈ #87145

order	384
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂¹⁰

Aliases

C₂³×D₃×C₈

Splittings

Fitting	C₂³×C₃×C₈ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂⁴×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂³×C₈	D₃
commutator, hypocenter	C₃	C₂⁴×C₈
Frattini	C₄	C₂⁴×D₃
Fitting	C₂³×C₃×C₈	C₂
socle	C₂⁴×C₃	C₂×C₄
prime radical	C₂⁴×D₃	C₄
cosocle	C₃×C₄	C₂⁵
LCS	G ▷ C₃
UCS	C₁ ◁ C₂³×C₈

1976 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)(5,6)(8,9);(7,8,9)(10,11,12,14,13,15,16,17);(5,6);(3,4);(1,2)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10700409, #10700435, #10700441, …
maximal normal subgroup of	#10700409, #10700435, #10700441, …
maximal quotient of	C₂²×C₄×D₃×C₈, C₂³×C₈×Dic₃, C₂²×D₃×C₂²⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 19721);