C₂²×#63045 #72301078

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

Aliases

direct product

C₂²×#63045

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

485969 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)(7,11)(8,10)(9,12)(13,14)(15,21)(16,24)(17,22)(18,19)(20,25)(23,26)(27,28);(5,6)(7,9)(8,10)(11,12)(14,19)(15,22)(20,24)(23,28);(15,22)(16,25)(17,21)(20,24);(5,6,8,10)(7,11,12,9)(13,15,18,22)(14,17,19,21)(16,23,25,28)(20,26,24,27);(5,6)(7,9)(8,10)(11,12)(13,16)(14,20)(15,23)(17,26)(18,25)(19,24)(21,27)(22,28);(5,7)(6,9)(8,12)(10,11)(13,17)(14,15)(16,26)(18,21)(19,22)(20,23)(24,28)(25,27);(3,4);(1,2)(3,4)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#10586740, C₂²×#10586564
maximal normal subgroup of	C₂²×#10586740, C₂²×#10586564
maximal quotient of	C₂²×#7819605, C₂²×#8062966, C₂²×#7916101, …

GAP

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

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

C22×#63045 #72301078