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C₂×#890 #5499

order	192
automorphism	Aut(G) order 24576
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

Splittings

C₃ ⋊ C₂×#125

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁴
Frattini	C₂²	C₂³×D₃
Fitting	C₂×C₃×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃	C₂³
hypercenter	C₂×C₄⋊C₄	D₃
hypocenter	C₃	C₂×#125
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 21 points:

(1,2)(3,4,5)(6,8,9,13)(7,10,11,12)(15,19)(16,20);(14,16,17,20)(15,18,19,21);(4,5)(6,7,9,11)(8,12,13,10)(14,15)(16,18)(17,19)(20,21);(1,2)

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#68801, #68802, #68804, …
maximal normal subgroup of	#68801, #68802, #68804, …
maximal quotient of	C₂×#4654, C₂×#4655, C₂×#4661, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1058);