C₂²×#62923 #72301080

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

Aliases

direct product

C₂²×#62923

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

476401 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:

(13,14,19,22)(15,18,24,28)(16,20,17,21)(23,27,25,26);(7,11)(9,12)(13,15)(14,18)(16,25)(17,23)(19,24)(20,26)(21,27)(22,28);(5,6)(7,12)(8,10)(9,11)(13,16,19,17)(14,20,22,21)(15,23,24,25)(18,27,28,26);(5,6,8,10)(7,12,11,9)(13,17,19,16)(14,21,22,20)(15,23,24,25)(18,27,28,26);(5,6)(7,9)(8,10)(11,12)(13,18)(14,15)(16,26)(17,27)(19,28)(20,25)(21,23)(22,24);(5,7)(6,9)(8,11)(10,12)(13,14)(15,18)(16,21)(17,20)(19,22)(23,26)(24,28)(25,27);(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×#10586740
maximal normal subgroup of	C₂²×#10586740
maximal quotient of	C₂²×#7779660, C₂²×#7677129

GAP

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

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

C22×#62923 #72301080