C₂×#7642253 #72342112

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

Aliases

direct product

C₂×#7642253

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

62713 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72342077
maximal normal subgroup of	C₂×#72342077

GAP

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

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

C2×#7642253 #72342112