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D₄×C₃⋊D₃×AGL(1,𝔽₅) #72229474

order	2880
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₃

Aliases

D₄×C₃⋊D₃×AGL(1,𝔽₅)

Splittings

Fitting	C₃²×C₅×D₄ ⋊ C₂×C₄
Hall	C₃²×C₅ ⋊ C₂×C₄×D₄
Sylow	C₃² ⋊ C₂×D₄×AGL(1,𝔽₅)
Sylow	C₅ ⋊ C₄×D₄×C₃⋊D₃

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×C₃⋊D₃×AGL(1,𝔽₅)
commutator, socle	C₂×C₃²×C₅	C₂³×C₄
Fitting	C₃²×C₅×D₄	C₂×C₄
prime radical	D₄×D₅×C₃⋊D₃	C₂
cosocle	C₂×C₃²×D₅	C₂⁴
hypercenter	D₄	C₃⋊D₃×AGL(1,𝔽₅)
hypocenter	C₃²×C₅	C₂×C₄×D₄
LCS	G ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂ ◁ D₄

2556 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 180

A minimal faithful permutation representation — 4 generators on 15 points:

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₄×C₃⋊D₃×AGL(1,𝔽₅), D₃²×D₄×AGL(1,𝔽₅), AGL(1,𝔽₅)×#63930, …
maximal normal subgroup of	C₂×D₄×C₃⋊D₃×AGL(1,𝔽₅), D₃²×D₄×AGL(1,𝔽₅), AGL(1,𝔽₅)×#63930, …
maximal quotient of	C₂×D₄×C₃⋊D₃×AGL(1,𝔽₅), AGL(1,𝔽₅)×#64146, #72230349, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,5,3,4)(6,7,8,9)(13,14,15), (11,12)(14,15), (2,3,4,5)(10,11,12), (7,9));