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C₂×#827 #5148

order	192
isoclinism	isoclinic class of #826
automorphism	Aut(G) = D₃×#10576928 (order 3072) · Inn(G) = #213 · Out(G) = C₂²×#41
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

C₃ ⋊ C₂×SD₃₂

Subgroups

	subgroup	G/H
center	C₂²	#213
commutator, cosocle	C₃×C₈	C₂³
Frattini	C₈	C₂²×D₃
Fitting, prime radical	C₂×C₃×D₈	C₂
socle	C₂²×C₃	D₈
hypercenter	C₂×D₈	D₃
hypocenter	C₃	C₂×SD₃₂
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 21 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#69175, #69176, #69193, …
maximal normal subgroup of	#69175, #69176, #69193, …
maximal quotient of	C₂×#4490, C₂×#4492, #70602, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 707);