C₂²×#36128 #72301380

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂²×#36128

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

136045 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#7985294, C₂²×#7672751, C₂³×#36128
maximal normal subgroup of	C₂²×#7985294, C₂²×#7672751, C₂³×#36128
maximal quotient of	C₂³×#36128

GAP

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

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

C22×#36128 #72301380