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C₃×C₄×C₈ #839

order	96
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×#2021 (order 256) · Inn(G) = C₁ · Out(G) = C₂×#2021
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₄

Aliases

C₃×C₄×C₈

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₄	C₂²×C₃
socle, prime radical	C₂²×C₃	C₂×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

center of	C₃×#1311, C₃×#1312, C₃×C₈×M₁₆, …
derived subgroup of	#10681907, #10681908, #10681909, …
Frattini subgroup of	C₈×C₉×C₁₆, C₉×#1311, C₉×#1312, …
Fitting subgroup of	C₈×C₃⋊C₈, #4454, #4455, …
maximal subgroup of	C₈×C₃⋊C₈, #4454, #4455, …
maximal normal subgroup of	C₈×C₃⋊C₈, #4454, #4455, …
maximal quotient of	C₃×C₈², C₃×#322, C₃×#336, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 46);