C₂³×#56384 #72301709

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

Aliases

direct product

C₂³×#56384

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

1068952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#8698894, C₂³×#8195808
maximal normal subgroup of	C₂³×#8698894, C₂³×#8195808

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

C23×#56384 #72301709