C₂×#3334799 #72325214

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂×#3334799

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

47749 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)(13,17)(14,18)(15,16)(19,20)(21,22)(23,25)(24,26);(3,4,6,5)(12,16)(14,17)(19,21,24,23)(20,25,26,22);(4,5)(12,16)(13,18)(21,23)(22,25);(7,8,9,10)(11,13)(12,17)(14,16)(15,18)(19,22,24,25)(20,21,26,23);(1,2);(3,5,6,4)(7,8)(9,10)(11,14,15,17)(12,13,16,18)(19,23,24,21)(20,25,26,22)

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232091, C₂×#72232095, C₂×#72232188, …
maximal normal subgroup of	C₂×#72232091, C₂×#72232095, C₂×#72232188, …

GAP

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

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

C2×#3334799 #72325214