C₂³×#3562 #72325000

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

Aliases

direct product

C₂³×#3562

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

480781 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:

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

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×#47930, C₂³×#43546, C₂³×#45002

GAP

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

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

C23×#3562 #72325000