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C₂²×C₄×Dic₃ #5782

order	192
isoclinism	isoclinic class of D₃
automorphism	Aut(G) order 294912
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

C₂²×C₄×Dic₃

Splittings

C₃ ⋊ C₂²×C₄²

Subgroups

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

498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 15 points:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₃×#95, #69401, #69402, …
maximal normal subgroup of	Dic₃×#95, #69401, #69402, …
maximal quotient of	C₂²×C₄×C₃⋊C₈, C₂²×#803, C₂×C₄×#208, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1341);