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C₂×C₃×C₄² #954

order	96
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×#72209884 (order 3072) · Inn(G) = C₁ · Out(G) = C₂×#72209884
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂²×C₄

Aliases

abelian	C₂×C₃×C₄²
multiplicative	(ℤ/195)ˣ, (ℤ/208)ˣ, (ℤ/260)ˣ, (ℤ/390)ˣ

Subgroups

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

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

center of	C₃×#1273, C₂×C₃×#322, C₃×C₄×#97, …
derived subgroup of	#10682693, #10682694, #10682695, …
Frattini subgroup of	C₉×#1270, C₉×#1271, C₉×#1272, …
Fitting subgroup of	#4523, #4524, C₂×C₄×C₃⋊C₈, …
maximal subgroup of	#4523, #4524, C₃×#336, …
maximal normal subgroup of	#4523, #4524, C₃×#336, …
maximal quotient of	C₃×C₄³, C₂×C₃×#95, C₃×#376, …
automorphism group of	C₃×C₅×C₁₃, C₁₃×C₁₆, C₄×C₅×C₁₃, …
outer automorphism group of	C₃×C₅×C₁₃, C₁₃×C₁₆, C₄×C₅×C₁₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 161);