C₂³×#15412 #72325786

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹×C₄

Aliases

direct product

C₂³×#15412

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1420568 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#1418391, C₂³×#1082737
maximal normal subgroup of	C₂³×#1418391, C₂³×#1082737

GAP

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

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

C23×#15412 #72325786