C₂³×#34406 #72325490

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷×C₄ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂³×#34406

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂²	C₂⁷×C₄
Frattini, cosocle	C₂³	C₂⁸
prime radical	C₂⁴×#535	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

1480168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#7792761
maximal normal subgroup of	C₂³×#7792761

GAP

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

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

C23×#34406 #72325490