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C₄×C₃⋊#170 #91229

order	480
isoclinism	isoclinic class of ΓL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

C₄×C₃⋊#170

Splittings

derived, Hall	C₃×C₅ ⋊ C₄×C₈
Sylow	C₅ ⋊ C₄×C₃⋊C₈
Sylow	C₃ ⋊ C₄×#170

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₄×C₈
Frattini	C₂²	C₂×ΓL(1,𝔽₁₆)
Fitting	C₂×C₃×C₄×C₅	C₄
socle, prime radical	C₂²×C₃×C₅	C₂×C₄
cosocle	C₂×C₃×Dic₅	C₂²
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃⋊C₈×#170, #11746485, #170⋊(C₃⋊C₈), …
maximal normal subgroup of	C₃⋊C₈×#170, #11746485, #170⋊(C₃⋊C₈), …
maximal quotient of	C₄×C₃⋊#674, #11746730, C₈×C₃⋊#170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 305);