C₂²×#14175 #72289259

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁷

Aliases

direct product

C₂²×#14175

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

105397 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#1418385, C₂²×#773762, C₂²×#519563, …
maximal normal subgroup of	C₂²×#1418385, C₂²×#773762, C₂²×#519563, …
maximal quotient of	C₂²×#154634, C₂²×#154635, C₂×#72298509

GAP

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

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

C22×#14175 #72289259