AGL(1,𝔽₅)×#2197 #72242933

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

AGL(1,𝔽₅)×#2197

Splittings

Fitting	C₅×#2197 ⋊ C₄
Sylow	C₅ ⋊ C₄×#2197

Subgroups

	subgroup	G/H
center	C₂	AGL(1,𝔽₅)×B(4,𝔽₂)
commutator	C₂×C₅×D₄	C₂³×C₄
Frattini	C₂×D₄	C₂³×AGL(1,𝔽₅)
Fitting	C₅×#2197	C₄
socle	C₂×C₅	C₄×B(4,𝔽₂)
prime radical	D₅×B(4,𝔽₂)	C₂²
cosocle	C₂×D₄×D₅	C₂⁴
hypercenter	#2197	AGL(1,𝔽₅)
hypocenter	C₅	C₄×#2197
LCS	G ▷ C₂×C₅×D₄ ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #2197

2632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 4 generators on 21 points:

(18,19,20,21);(1,3)(2,7)(4,13)(5,11)(6,14)(8,16)(9,15)(10,12)(17,18,21,19,20);(2,6)(4,12);(1,2)(3,9,11,16)(4,8)(5,6)(7,10,14,13)(12,15)

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#33551
maximal normal subgroup of	AGL(1,𝔽₅)×#33551

GAP

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

gap> G := Group((18,19,20,21), (1,3)(2,7)(4,13)(5,11)(6,14)(8,16)(9,15)(10,12)(17,18,21,19,20), (2,6)(4,12), (1,2)(3,9,11,16)(4,8)(5,6)(7,10,14,13)(12,15));

AGL(1,𝔽5)×#2197 #72242933