C₂³×#63041 #72325007

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

Aliases

direct product

C₂³×#63041

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂²	C₂⁹
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂² ▷ C₁

9171184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 9 generators on 30 points:

(15,16)(17,21)(18,20)(19,26)(22,27)(23,30)(24,25)(28,29);(8,11)(9,13)(16,18)(19,27)(21,25)(23,28);(3,4)(8,11)(12,14)(15,17)(16,21)(18,25)(19,28)(20,24)(22,30)(23,27)(26,29);(5,6);(8,11)(9,13)(19,27)(22,26)(23,28)(29,30);(1,2)(3,4)(5,6);(8,11)(12,14)(15,17)(16,21)(18,25)(19,28)(20,24)(22,30)(23,27)(26,29);(7,8,10,11)(9,12,13,14)(15,18,20,16)(17,21,24,25)(19,26,27,22)(23,29,28,30);(7,9)(8,12)(10,13)(11,14)(15,19,20,27)(16,22,18,26)(17,23,24,28)(21,29,25,30)

Conjugacy classes & character table

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

Relationships

maximal quotient of

C₂³×#7918389, C₂³×#7837029

GAP

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

gap> G := Group((15,16)(17,21)(18,20)(19,26)(22,27)(23,30)(24,25)(28,29), (8,11)(9,13)(16,18)(19,27)(21,25)(23,28), (3,4)(8,11)(12,14)(15,17)(16,21)(18,25)(19,28)(20,24)(22,30)(23,27)(26,29), (5,6), (8,11)(9,13)(19,27)(22,26)(23,28)(29,30), (1,2)(3,4)(5,6), (8,11)(12,14)(15,17)(16,21)(18,25)(19,28)(20,24)(22,30)(23,27)(26,29), (7,8,10,11)(9,12,13,14)(15,18,20,16)(17,21,24,25)(19,26,27,22)(23,29,28,30), (7,9)(8,12)(10,13)(11,14)(15,19,20,27)(16,22,18,26)(17,23,24,28)(21,29,25,30));

C23×#63041 #72325007