C₂²×#46996 #72293729

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

Aliases

direct product

C₂²×#46996

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

107453 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 28 points:

(5,6)(7,8)(9,12)(10,11)(13,14,17,18)(15,16,20,19)(21,22)(23,25)(24,26)(27,28);(5,6)(7,10)(8,11)(9,12)(13,14,17,18)(15,19,20,16)(21,22,24,26)(23,25,28,27);(5,7,9,11)(6,10,12,8)(13,15,17,20)(14,16,18,19)(21,23)(22,25)(24,28)(26,27);(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,22)(23,27)(24,26)(25,28);(5,8)(6,11)(7,12)(9,10)(13,16)(14,15)(17,19)(18,20)(21,23)(22,25)(24,28)(26,27);(3,4);(1,2)(3,4)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8598543, C₂²×#8932885, C₂×#72232907
maximal normal subgroup of	C₂²×#8598543, C₂²×#8932885, C₂×#72232907
maximal quotient of	C₂²×#4029469, C₂²×#4387614, C₂²×#1082853, …

GAP

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

gap> G := Group((5,6)(7,8)(9,12)(10,11)(13,14,17,18)(15,16,20,19)(21,22)(23,25)(24,26)(27,28), (5,6)(7,10)(8,11)(9,12)(13,14,17,18)(15,19,20,16)(21,22,24,26)(23,25,28,27), (5,7,9,11)(6,10,12,8)(13,15,17,20)(14,16,18,19)(21,23)(22,25)(24,28)(26,27), (5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,22)(23,27)(24,26)(25,28), (5,8)(6,11)(7,12)(9,10)(13,16)(14,15)(17,19)(18,20)(21,23)(22,25)(24,28)(26,27), (3,4), (1,2)(3,4));

C22×#46996 #72293729