C₂×#7659329 #72298908

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

Aliases

direct product

C₂×#7659329

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

At least 6162 conjugacy classes of subgroups; only the 29973 normal subgroups are shown.

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297796
maximal normal subgroup of	C₂×#72297796
maximal quotient of	C₂×#72261010

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

C2×#7659329 #72298908