C₂²×#63066 #72301081

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

Aliases

direct product

C₂²×#63066

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

477689 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:

(13,14,18,22)(15,19,21,17)(16,24,27,20)(23,26,25,28);(7,11)(10,12)(13,15)(14,19)(16,23)(17,22)(18,21)(20,28)(24,26)(25,27);(5,6)(7,12)(8,9)(10,11)(13,15,18,21)(14,19,22,17)(16,25,27,23)(20,26,24,28);(6,9)(10,12)(13,15)(14,17)(16,23)(18,21)(19,22)(20,26)(24,28)(25,27);(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,23)(17,26)(18,27)(19,28)(21,25)(22,24);(5,7)(6,10)(8,11)(9,12)(13,17)(14,21)(15,22)(16,26)(18,19)(20,25)(23,24)(27,28);(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
maximal normal subgroup of	C₂²×#10586740
maximal quotient of	C₂²×#8063858

GAP

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

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

C22×#63066 #72301081