C₂²×#63022 #72301086

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

Aliases

direct product

C₂²×#63022

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

475593 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:

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

Conjugacy classes & character table

304 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×#7699591, C₂²×#7985575

GAP

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

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

C22×#63022 #72301086