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C₂²×#237 #5956

order	192
automorphism	Aut(G) order 147456
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

C₂²×#237

Splittings

Fitting	C₂²×C₃×D₄ ⋊ C₂
Sylow	C₃ ⋊ C₂²×#41

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₃
commutator, cosocle	C₂×C₃	C₂⁵
Frattini	C₂	C₂⁴×D₃
Fitting	C₂²×C₃×D₄	C₂
socle	C₂³×C₃	C₂³
hypercenter	C₂²×D₄	D₃
hypocenter	C₃	C₂²×#41
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 15 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#70027, #70080, #70199, …
maximal normal subgroup of	#70027, #70080, #70199, …
maximal quotient of	C₂²×#877, C₂²×#67⋊C₄, C₂²×#878, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1515);