C₂×#8125618 #72324795

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂×#8125618

Subgroups

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

68093 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232068
maximal normal subgroup of	C₂×#72232068
maximal quotient of	C₂×#72232150

GAP

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

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

C2×#8125618 #72324795