C₂×#8837083 #72324809

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

Aliases

direct product

C₂×#8837083

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 9948 conjugacy classes of subgroups; only the 29469 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)(5,8)(6,7)(9,10)(19,20)(21,24)(22,25)(23,26);(11,12)(13,16)(14,17)(15,18)(19,21)(20,23)(22,26)(24,25);(4,7)(8,9)(11,13)(12,15)(14,18)(16,17)(19,21)(20,24)(22,26)(23,25);(13,18)(15,16)(19,20)(21,23)(22,25)(24,26);(1,2);(3,5)(4,8)(6,10)(7,9)(11,12)(13,16)(14,17)(15,18);(3,4)(5,9)(6,7)(8,10)(11,13)(12,16)(14,18)(15,17)

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232068, C₂×#72232275
maximal normal subgroup of	C₂×#72232068, C₂×#72232275
maximal quotient of	C₂×#72232103, C₂×#72232342

GAP

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

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

C2×#8837083 #72324809