C₂³×#35921 #72289387

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶×C₄

Aliases

direct product

C₂³×#35921

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

2867520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#7684106, C₂²×#72320249
maximal normal subgroup of	C₂³×#7684106, C₂²×#72320249
maximal quotient of	C₂³×D₄×#392, C₂²×#72320249

GAP

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

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

C23×#35921 #72289387