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C₂³×C₃⋊C₈ #5780

order	192
isoclinism	isoclinic class of D₃
automorphism	Aut(G) order 258048
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂³×C₃⋊C₈

Splittings

C₃ ⋊ C₂³×C₈

Subgroups

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

338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 17 points:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²⋊C₄×C₃⋊C₈, #69447, #69449, …
maximal normal subgroup of	C₂²⋊C₄×C₃⋊C₈, #69447, #69449, …
maximal quotient of	C₂²×C₄×C₃⋊C₈, C₂²×C₄⋊(C₃⋊C₈), C₂×#4925, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1339);