C₂³×#3494 #72301667

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂³×#3494

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂² ▷ C₁

515957 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#62983, C₂³×#62818, C₂³×#62912
maximal normal subgroup of	C₂³×#62983, C₂³×#62818, C₂³×#62912
maximal quotient of	C₂³×#39148, C₂³×#59013, C₂³×#37662, …

GAP

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

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

C23×#3494 #72301667