C₂³×#14299 #72325794

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

Aliases

direct product

C₂³×#14299

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator	C₂³	C₂⁶×C₄
Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁴×#392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1278904 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)(12,14)(15,16,17,18)(19,20)(21,23)(22,24)(25,26);(8,10)(12,14)(20,21)(24,25);(3,4);(5,6);(11,12)(13,14)(19,21,23,20)(22,24,26,25);(1,2)(3,4)(5,6);(7,8)(9,10)(11,12)(13,14)(19,22)(20,24)(21,25)(23,26)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#1262353, C₂³×#1259281
maximal normal subgroup of	C₂³×#1262353, C₂³×#1259281

GAP

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

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

C23×#14299 #72325794