C₂×#3334777 #72294527

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#3334777

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

47687 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293937, C₂×#72293915
maximal normal subgroup of	C₂×#72293937, C₂×#72293915
maximal quotient of	C₂×#72293968, C₂×#72293936

GAP

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

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

C2×#3334777 #72294527