C₂²×#8934366 #72296038

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂²×#8934366

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

1132664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 8 generators on 28 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72236189, C₂³×#8934366
maximal normal subgroup of	C₂²×#72236189, C₂³×#8934366
maximal quotient of	C₂²×#72231761, C₂³×#8934366

GAP

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

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

C22×#8934366 #72296038