C₂×#7804412 #72337803

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

Aliases

direct product

C₂×#7804412

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

67681 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232275
maximal normal subgroup of	C₂×#72232275
maximal quotient of	C₂×#72232287

GAP

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

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

C2×#7804412 #72337803