C₂²×C₂²⋊C₄×#127 #72320580

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²¹×C₄

Aliases

direct product

C₂²×C₂²⋊C₄×#127

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂³	C₂⁶×C₄
Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁵×#127	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1334232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 20 points:

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

Conjugacy classes & character table

560 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×#120×#127
maximal normal subgroup of	C₂²×#120×#127
maximal quotient of	C₂²×C₂²⋊C₄×#392

GAP

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

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

C22×C22⋊C4×#127 #72320580