C₂³×#3540 #72324997

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

Aliases

direct product

C₂³×#3540

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂² ▷ C₁

484877 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 8 generators on 30 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#63041
maximal normal subgroup of	C₂³×#63041
maximal quotient of	C₂³×#37800, C₂³×#37269

GAP

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

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

C23×#3540 #72324997