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C₃×C₂²⋊C₁₆ #4595

order	192
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂²×#525 (order 256) · Inn(G) = C₂² · Out(G) = C₂²×#41
homology	Abelianization (H₁) = C₂×C₃×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

C₃×C₂²⋊C₁₆

Subgroups

	subgroup	G/H
center	C₂×C₃×C₈	C₂²
commutator	C₂	C₂×C₃×C₁₆
Frattini, cosocle	C₂×C₈	C₂²×C₃
socle	C₂²×C₃	C₂×C₈
prime radical	C₂³×C₃	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₈ ◁ G

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 2 generators on 23 points:

(1,2)(3,4)(5,6,7,10,8,11,13,16,9,12,14,17,15,18,19,20)(21,22,23);(2,4)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#67522, #67523, #67524, …
maximal subgroup of	#67522, #67523, #67524, …
maximal normal subgroup of	#67522, #67523, #67524, …
maximal quotient of	C₃×#1314, C₃×#1322, C₃×D₄⋊C₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 154);