C₂×#9122465 #72294563

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

Aliases

direct product

C₂×#9122465

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 10076 conjugacy classes of subgroups; only the 29525 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)(11,12,15,18)(13,16,17,14)(19,20,23,26)(21,25,24,22);(5,9)(8,10)(11,13,15,17)(12,16,18,14)(19,20)(21,22)(23,26)(24,25);(4,7)(5,9)(11,13,15,17)(12,14,18,16)(19,21)(20,22)(23,24)(25,26);(11,14)(12,17)(13,18)(15,16)(19,22)(20,24)(21,26)(23,25);(3,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,25)(24,26);(3,5)(4,8)(6,9)(7,10)(11,13)(12,16)(14,18)(15,17)(19,21)(20,25)(22,26)(23,24);(1,2)

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232277
maximal normal subgroup of	C₂×#72232277
maximal quotient of	C₂×#72293940, C₂×#72293942, C₂×#72232681, …

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

C2×#9122465 #72294563