C₅₉×#7316 #28419425

order	15104
nilpotency class	4
isoclinism	isoclinic class of #7316
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅₉×#7316

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₉	#1304
commutator	C₂⁴	C₄²×C₅₉
Frattini, cosocle	C₂³×D₄	C₂²×C₅₉
prime radical	C₂³×D₄×C₅₉	C₂²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₉ ◁ C₂³×C₅₉ ◁ C₂³×D₄×C₅₉ ◁ G

620 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 75
transitive degree	μ_t(G) = 944

A minimal faithful permutation representation — 2 generators on 75 points:

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

Conjugacy classes & character table

1652 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15104, 304);