C₂³×#59220 #72301707

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂³×#59220

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

807640 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 1024

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#8195808, C₂³×#7667013
maximal normal subgroup of	C₂³×#8195808, C₂³×#7667013

GAP

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

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

C23×#59220 #72301707