C₅₉×#7347 #28419456

order	15104
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₅₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₉×#7347

Subgroups

	subgroup	G/H
center	C₄×C₅₉	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂×C₈×C₅₉
Frattini, cosocle	#408	C₂²×C₅₉
socle	C₂×C₅₉	#1316
prime radical	C₅₉×#585	C₄
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₉ ◁ C₂×C₄×C₅₉ ◁ C₂²×C₄×C₅₉ ◁ C₅₉×#408 ◁ G

318 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 123
transitive degree	μ_t(G) = 3776

A minimal faithful permutation representation — 2 generators on 123 points:

(1,2,5,11,7,12,26,44,8,13,27,45,6,28,31,43)(3,14,16,46,17,47,51,63,18,38,52,60,29,59,24,56)(4,19,21,53,22,10,37,39,23,40,25,57,30,41,55,64)(9,32,33,15,35,48,42,50,36,58,61,20,34,54,62,49)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(1,3)(2,9)(4,20)(5,24)(6,29)(7,17)(8,18)(10,38)(11,42)(12,35)(13,36)(14,41)(15,23)(16,26)(19,47)(21,32)(22,49)(25,58)(27,51)(28,34)(30,50)(31,52)(33,43)(37,48)(39,63)(40,59)(44,61)(45,62)(46,53)(54,55)(56,64)(57,60)

Conjugacy classes & character table

2006 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15104, 335);