C₂²×#62990 #72301263

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

Aliases

direct product

C₂²×#62990

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

480313 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:

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

Conjugacy classes & character table

340 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((5,6,9,12)(7,10,8,11)(13,14)(15,20)(16,21)(17,22)(18,19)(23,24)(25,27)(26,28), (5,6)(7,10)(8,11)(9,12)(14,19)(17,20)(21,27)(24,26), (5,7,9,8)(6,10,12,11)(15,22)(17,20)(23,28)(24,26), (5,8,9,7)(6,11,12,10)(13,15,18,22)(14,20,19,17)(16,23,25,28)(21,24,27,26), (13,16)(14,21)(15,23)(17,26)(18,25)(19,27)(20,24)(22,28), (13,17,18,20)(14,22,19,15)(16,24,25,26)(21,23,27,28), (3,4), (1,2)(3,4));

C22×#62990 #72301263