C₂³×#1446 #72325576

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂³×#1446

Subgroups

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

281877 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#16632, C₂³×#15670, C₂³×#14885, …
maximal normal subgroup of	C₂³×#16632, C₂³×#15670, C₂³×#14885, …

GAP

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

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

C23×#1446 #72325576