C₂×#9223150 #72324568

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

Aliases

direct product

C₂×#9223150

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

67627 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)(5,8,9,10)(11,12,15,17)(13,14,18,16)(20,23)(24,25);(3,4)(5,8)(6,7)(9,10)(11,13)(12,16)(14,17)(15,18)(19,20)(21,24)(22,23)(25,26);(4,7)(5,9)(12,17)(14,16)(20,23)(24,25);(4,7)(8,10)(12,17)(14,16)(19,20,22,23)(21,25,26,24);(1,2);(3,4)(5,10)(6,7)(8,9)(11,13)(12,14)(15,18)(16,17)(19,20)(21,24)(22,23)(25,26);(3,5)(4,8)(6,9)(7,10)(11,14)(12,18)(13,17)(15,16)(19,21)(20,24)(22,26)(23,25)

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232069
maximal normal subgroup of	C₂×#72232069
maximal quotient of	C₂×#72232089, C₂×#72232126

GAP

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

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

C2×#9223150 #72324568