C₂×#720184 #72325094

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#720184

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

49129 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232077, C₂×#72232131, C₂×#72232159
maximal normal subgroup of	C₂×#72232077, C₂×#72232131, C₂×#72232159

GAP

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

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

C2×#720184 #72325094