C₂²×#62918 #72301255

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

Aliases

direct product

C₂²×#62918

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

481269 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)(7,8)(9,12)(10,11)(14,18)(15,21)(20,24)(22,27);(5,7)(6,10)(8,12)(9,11)(16,23)(20,24)(22,27)(26,28);(5,7)(6,8)(9,11)(10,12)(15,21)(16,23)(19,25)(20,24);(5,7)(6,8)(9,11)(10,12)(13,14)(15,19)(16,24)(17,18)(20,23)(21,25)(22,28)(26,27);(5,8)(6,11)(7,12)(9,10)(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,27)(24,28);(5,8)(6,11)(7,12)(9,10)(13,16)(14,20)(15,22)(17,23)(18,24)(19,26)(21,27)(25,28);(3,4);(1,2)(3,4)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#10586564
maximal normal subgroup of	C₂²×#10586564
maximal quotient of	C₂²×#7677211

GAP

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

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

C22×#62918 #72301255