AGL(1,ℤ/8)×#365 #72342544

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

AGL(1,ℤ/8)×#365

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄×#106
commutator	C₄×C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁵
prime radical	C₂²×AGL(1,ℤ/8)	SD₁₆
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×C₈×AGL(1,ℤ/8) ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(2,3)(6,8);(2,6)(3,8)(4,7)(10,14)(15,16)(17,22)(21,23);(10,15,14,16)(12,20)(13,19)(17,23,22,21)(18,24);(1,2,7,8,5,3,4,6)(9,10,13,22,11,14,19,17)(12,21,20,15,18,23,24,16);(1,3,5,2)(4,6,7,8)(10,16,14,15)(12,20)(13,19)(17,21,22,23)(18,24)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/8)×AGL(1,ℤ/16)
maximal normal subgroup of	AGL(1,ℤ/8)×AGL(1,ℤ/16)

GAP

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

gap> G := Group((2,3)(6,8), (2,6)(3,8)(4,7)(10,14)(15,16)(17,22)(21,23), (10,15,14,16)(12,20)(13,19)(17,23,22,21)(18,24), (1,2,7,8,5,3,4,6)(9,10,13,22,11,14,19,17)(12,21,20,15,18,23,24,16), (1,3,5,2)(4,6,7,8)(10,16,14,15)(12,20)(13,19)(17,21,22,23)(18,24));