C₂×#7675577 #72331687

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

Aliases

direct product

C₂×#7675577

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

68217 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:

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72241227, C₂×#72241247
maximal normal subgroup of	C₂×#72241227, C₂×#72241247
maximal quotient of	C₂×#72327602

GAP

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

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

C2×#7675577 #72331687