C₂×#8181165 #72337758

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

Aliases

direct product

C₂×#8181165

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

73305 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232275
maximal normal subgroup of	C₂×#72232275
maximal quotient of	C₂×#72232360

GAP

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

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

C2×#8181165 #72337758