C₅₉×#12773 #28424882

order	15104
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅₉×#12773

Subgroups

	subgroup	G/H
center	C₄×C₅₉	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₅₉
Frattini, cosocle	C₂×#41	C₂³×C₅₉
socle	C₂×C₅₉	#1790
prime radical	C₅₉×#3483	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₉ ◁ C₂×C₄×C₅₉ ◁ C₂×#41×C₅₉ ◁ G

832 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 832 conjugacy classes of subgroups — too many to draw. Only its 138 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 — 3 generators on 75 points:

(1,2)(3,7,4,9)(5,10)(6,11)(8,12,16,13)(14,15)(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);(2,7)(3,4)(8,10)(9,11)(12,13)(15,16);(1,3)(2,8)(4,6)(5,13)(7,15)(9,10)(11,16)(12,14)

Conjugacy classes & character table

2360 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15104, 5761);