C₂²×#62994 #72301082

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

Aliases

direct product

C₂²×#62994

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

475285 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:

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

Conjugacy classes & character table

340 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×#7687886

GAP

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

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

C22×#62994 #72301082