C₂²×#10586740 #72301088

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁹ · Schur multiplier (H₂) = C₂³⁴

Aliases

direct product

C₂²×#10586740

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁷
commutator, Frattini, cosocle	C₂²	C₂⁹
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂² ▷ C₁

9091360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 9 generators on 28 points:

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

Conjugacy classes & character table

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

Relationships

maximal quotient of

C₂²×#72297477

GAP

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

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

C22×#10586740 #72301088