C₂²×#62862 #72281007

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

Aliases

direct product

C₂²×#62862

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

486493 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)(9,12)(13,14)(15,16)(17,25)(18,19)(20,23)(21,26)(22,27)(24,28);(5,6,8,10)(7,9,11,12)(14,19)(16,20)(21,25)(24,27);(5,7)(6,9)(8,11)(10,12)(13,15,18,23)(14,20,19,16)(17,22,26,28)(21,27,25,24);(5,6)(7,9)(8,10)(11,12)(13,15)(14,16)(17,22)(18,23)(19,20)(21,24)(25,27)(26,28);(3,4)(13,16)(14,15)(17,24)(18,20)(19,23)(21,22)(25,28)(26,27);(1,2)(3,4);(13,17)(14,21)(15,22)(16,24)(18,26)(19,25)(20,27)(23,28);(13,16)(14,15)(17,24)(18,20)(19,23)(21,22)(25,28)(26,27)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#6499892, #72285197, C₂²×#10586564
maximal normal subgroup of	C₂²×#6499892, #72285197, C₂²×#10586564
maximal quotient of	C₂²×#7682069, C₂²×#8898065, C₂²×#7915509, …

GAP

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

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

C22×#62862 #72281007