C₂³×#1765 #72308159

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄² · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂³×#1765

Subgroups

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₄
commutator	C₄	C₂⁴×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁶
socle	C₂⁵	C₄×D₄
prime radical	C₂⁴×#41	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁵×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

49173 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#21546
maximal normal subgroup of	C₂³×#21546

GAP

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

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

C23×#1765 #72308159