C₂²×#62992 #72299716

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

Aliases

direct product

C₂²×#62992

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

476013 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)(7,9)(8,11)(10,12)(13,14)(15,21)(16,22)(17,25)(18,19)(20,27)(23,24)(26,28);(5,7,8,12)(6,9,11,10)(13,15)(14,16)(17,23)(18,22)(19,21)(20,24)(25,26)(27,28);(5,6)(7,9)(8,11)(10,12)(14,19)(15,22)(20,25)(23,28);(5,7)(6,10)(8,12)(9,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,27)(26,28);(13,16)(14,15)(17,26)(18,21)(19,22)(20,28)(23,25)(24,27);(13,17)(14,20)(15,23)(16,24)(18,27)(19,25)(21,26)(22,28);(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×#10586570
maximal normal subgroup of	C₂²×#10586570
maximal quotient of	C₂²×#7687902, C₂²×#7699591

GAP

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

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

C22×#62992 #72299716