C₂²×#63017 #72301256

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

Aliases

direct product

C₂²×#63017

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

480505 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:

(5,6,9,12)(7,8,11,10)(13,14)(15,19)(16,20)(17,18)(21,26)(22,23)(24,25)(27,28);(5,6)(7,10)(8,11)(9,12)(14,18)(19,22)(20,24)(26,28);(5,7)(6,10)(8,12)(9,11);(5,8,9,10)(6,11,12,7)(13,15,17,23)(14,19,18,22)(16,21,25,27)(20,26,24,28);(13,16)(14,20)(15,21)(17,25)(18,24)(19,26)(22,28)(23,27);(5,6)(7,8)(9,12)(10,11)(13,14)(15,22)(16,24)(17,18)(19,23)(20,25)(21,26)(27,28);(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×#10586564
maximal normal subgroup of	C₂²×#10586564
maximal quotient of	C₂²×#7737955

GAP

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

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

C22×#63017 #72301256