C₅₉×#11478 #28423587

order	15104
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₅₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅₉×#11478

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅₉	#99
commutator	C₂×C₄	C₂²×C₈×C₅₉
Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₅₉
socle	C₂²×C₅₉	C₂×#99
prime radical	C₂³×C₅₉	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₉ ◁ C₂²×C₄×C₅₉ ◁ C₂×C₄×Q₈×C₅₉ ◁ G

428 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 3776

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

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

Conjugacy classes & character table

3068 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15104, 4466);