C₂²×#62846 #72299717

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

Aliases

direct product

C₂²×#62846

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

481237 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)(7,10)(8,12)(9,11)(13,14)(15,18)(16,21)(17,22)(19,24)(20,26)(23,27)(25,28);(5,7,9,8)(6,10,11,12)(13,14,18,15)(16,19,24,21)(17,20,26,22)(23,27,25,28);(5,6)(7,10)(8,12)(9,11)(14,15)(19,21)(20,22)(27,28);(5,8,9,7)(6,10,11,12)(13,15,18,14)(16,19,24,21)(17,22,26,20)(23,27,25,28);(13,16)(14,19)(15,21)(17,25)(18,24)(20,28)(22,27)(23,26);(13,17)(14,20)(15,22)(16,23)(18,26)(19,27)(21,28)(24,25);(3,4);(1,2)(3,4)

Conjugacy classes & character table

328 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×#10586570, C₂²×#10586564
maximal normal subgroup of	C₂²×#10586570, C₂²×#10586564
maximal quotient of	C₂²×#7696044, C₂²×#7677129, C₂²×#7694192

GAP

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

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

C22×#62846 #72299717