C₂³×#2826 #72301694

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂³×#2826

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×#392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

91765 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 30 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂³×#42566, C₂³×#56384, C₂³×#45599, …
maximal normal subgroup of	C₂³×#42566, C₂³×#56384, C₂³×#45599, …

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

C23×#2826 #72301694