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C₂²×#217 #5839

order	192
isoclinism	isoclinic class of Dic₆
automorphism	Aut(G) order 294912
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

C₂²×#217

Splittings

C₃ ⋊ C₂²×C₂²⋊C₄

Subgroups

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

674 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 96

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

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#69495, #69496, C₂×#4526, …
maximal normal subgroup of	#69495, #69496, C₂×#4526, …
maximal quotient of	C₂²×#830, C₂×#5207, C₂²×C₃⋊#95, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1398);