C₂²×#62821 #72301258

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂²×#62821

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂² ▷ C₁

480141 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

(7,11)(8,12)(13,14)(15,18)(16,19)(17,22)(20,25)(21,24)(23,27)(26,28);(6,10)(7,11)(14,15)(19,21)(20,22)(26,27);(5,6)(7,12)(8,11)(9,10);(5,7,9,11)(6,8,10,12)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27);(5,8)(6,7)(9,12)(10,11)(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,27)(25,28);(13,17,18,25)(14,20,15,22)(16,23,24,28)(19,26,21,27);(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×#10586564
maximal normal subgroup of	C₂²×#10586564

GAP

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

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

C22×#62821 #72301258