C₂×#8181079 #72241024

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

Aliases

direct product

C₂×#8181079

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 10432 conjugacy classes of subgroups; only the 29621 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232071, C₂×#72232275
maximal normal subgroup of	C₂×#72232071, C₂×#72232275
maximal quotient of	C₂×#72232168, C₂×#72232172, C₂×#72232173, …

GAP

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

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

C2×#8181079 #72241024