C₂²×#14228 #72320802

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³×C₄²

Aliases

direct product

C₂²×#14228

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

80645 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 20 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#822347, C₂²×#790151, C₂²×#517979, …
maximal normal subgroup of	C₂²×#822347, C₂²×#790151, C₂²×#517979, …
maximal quotient of	C₂²×#154635, C₂²×#154647

GAP

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

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

C22×#14228 #72320802