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C₂³×C₄×D₃ #5952

order	192
isoclinism	isoclinic class of D₃
automorphism	Aut(G) order 2064384
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

C₂³×C₄×D₃

Splittings

Fitting	C₂³×C₃×C₄ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂⁴×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂³×C₄	D₃
commutator, hypocenter	C₃	C₂⁴×C₄
Frattini	C₂	C₂⁴×D₃
Fitting	C₂³×C₃×C₄	C₂
socle	C₂⁴×C₃	C₂²
prime radical	C₂⁴×D₃	C₂
cosocle	C₂×C₃	C₂⁵
LCS	G ▷ C₃
UCS	C₁ ◁ C₂³×C₄

1362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 5 generators on 13 points:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	(C₂×D₃)⋊#95, #69955, C₂×#4666, …
maximal normal subgroup of	(C₂×D₃)⋊#95, #69955, C₂×#4666, …
maximal quotient of	C₂²×C₄×Dic₆, C₂×C₄×#235, C₂²×C₄²×D₃, …
automorphism group of	C₅×D₃×C₈, C₅×ΓL(1,𝔽₂₅), C₃×D₃×C₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1511);