C₂²×#62874 #72301079

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

Aliases

direct product

C₂²×#62874

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

481989 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)(15,21)(19,22)(23,27)(26,28);(7,12)(9,11)(13,14)(15,19)(16,24)(17,18)(20,25)(21,22)(23,28)(26,27);(14,18)(15,21)(16,25)(26,28);(5,6,8,10)(7,11,12,9)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,25)(26,27);(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,28)(25,27);(5,7)(6,9)(8,12)(10,11)(13,16)(14,20)(15,23)(17,25)(18,24)(19,26)(21,27)(22,28);(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×#7763450, C₂²×#8069760, 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)(15,21)(19,22)(23,27)(26,28), (7,12)(9,11)(13,14)(15,19)(16,24)(17,18)(20,25)(21,22)(23,28)(26,27), (14,18)(15,21)(16,25)(26,28), (5,6,8,10)(7,11,12,9)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,25)(26,27), (5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,28)(25,27), (5,7)(6,9)(8,12)(10,11)(13,16)(14,20)(15,23)(17,25)(18,24)(19,26)(21,27)(22,28), (3,4), (1,2)(3,4));

C22×#62874 #72301079