C₂×#7682192 #72331008

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

Aliases

direct product

C₂×#7682192

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

56887 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)(5,9)(19,20)(21,25)(22,26)(23,24);(11,12,15,13)(14,16,18,17)(19,20,23,24)(21,25,26,22);(3,4)(5,10)(6,7)(8,9)(11,12,15,13)(14,17,18,16)(19,21,23,26)(20,22,24,25);(3,4)(5,8)(6,7)(9,10)(11,12,15,13)(14,16,18,17)(19,22,23,25)(20,21,24,26);(1,2);(3,4)(5,10)(6,7)(8,9)(11,13,15,12)(14,16,18,17)(19,21)(20,22)(23,26)(24,25);(3,5)(4,8)(6,9)(7,10)(11,14,15,18)(12,16,13,17)

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232290
maximal normal subgroup of	C₂×#72232290

GAP

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

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

C2×#7682192 #72331008