C₂³×#14363 #72325805

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²¹×C₄

Aliases

direct product

C₂³×#14363

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator	C₂³	C₂⁶×C₄
Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁵×#121	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1206200 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:

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

Conjugacy classes & character table

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

Relationships

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

GAP

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

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

C23×#14363 #72325805