C₂³×#14825 #72325806

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

Aliases

direct product

C₂³×#14825

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

1025304 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)(16,19)(17,21)(24,27)(25,29);(8,9)(12,13)(16,19)(17,21)(23,24,26,27)(25,28,29,30);(3,4);(5,6);(8,9)(11,12,14,13)(15,16,18,19)(17,22,21,20)(23,24)(25,30)(26,27)(28,29);(1,2)(3,4)(5,6);(7,9,10,8)(11,13,14,12)(15,17,18,21)(16,20,19,22)(23,25)(24,28)(26,29)(27,30)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#1420724
maximal normal subgroup of	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)(16,19)(17,21)(24,27)(25,29), (8,9)(12,13)(16,19)(17,21)(23,24,26,27)(25,28,29,30), (3,4), (5,6), (8,9)(11,12,14,13)(15,16,18,19)(17,22,21,20)(23,24)(25,30)(26,27)(28,29), (1,2)(3,4)(5,6), (7,9,10,8)(11,13,14,12)(15,17,18,21)(16,20,19,22)(23,25)(24,28)(26,29)(27,30));

C23×#14825 #72325806