C₂×#8161174 #72331700

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

Aliases

direct product

C₂×#8161174

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

68317 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72241227
maximal normal subgroup of	C₂×#72241227
maximal quotient of	C₂×#72297705

GAP

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

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

C2×#8161174 #72331700