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C₃×C₄×#168 #91004

order	480
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

C₃×C₄×#168

Splittings

derived, Sylow	C₅ ⋊ C₃×C₄×C₈
Hall	C₃×C₅ ⋊ C₄×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₄²	D₅
commutator, hypocenter	C₅	C₃×C₄×C₈
Frattini	C₂×C₄	C₂×C₃×D₅
Fitting	C₃×C₄²×C₅	C₂
socle, prime radical	C₂²×C₃×C₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂²×C₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₃×C₄²

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 2 generators on 20 points:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃⋊C₈×#168, #11745831, #11745832, …
maximal normal subgroup of	C₃⋊C₈×#168, #11745831, #11745832, …
maximal quotient of	C₃×C₈×#168, C₃×#64675, C₃×C₄×#672, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 80);