C₂³×#14823 #72325678

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂³×#14823

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁶×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1009496 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#850465, C₂³×#1420724
maximal normal subgroup of	C₂³×#850465, C₂³×#1420724

GAP

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

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

C23×#14823 #72325678