C₅₉×#12062 #28424171

order	15104
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆×C₅₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₉×#12062

Subgroups

	subgroup	G/H
center	C₂×C₈×C₅₉	C₂×D₄
commutator	C₄	C₂²×C₁₆×C₅₉
Frattini, cosocle	C₄×C₈	C₂³×C₅₉
socle	C₂²×C₅₉	C₈×D₄
prime radical	C₂×D₈×C₅₉	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₅₉ ◁ C₄×C₁₆×C₅₉ ◁ G

252 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 7552

A minimal faithful permutation representation — 3 generators on 83 points:

(1,2,4,6)(3,5,8,7)(9,10,12,16,13,17,20,23,14,11,15,18,21,19,22,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83);(2,5)(3,8)(6,7);(2,6)(5,7)(9,11,12,18,13,19,20,24,14,10,15,16,21,17,22,23)

Conjugacy classes & character table

5192 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15104, 5050);