C₄×C₅₉×#354 #28424735

order	15104
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₅₉ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₄×C₅₉×#354

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅₉	#99
commutator	C₂×C₄	C₂×C₄²×C₅₉
Frattini, cosocle	C₂²×D₄	C₂³×C₅₉
socle	C₂²×C₅₉	C₂×#99
prime radical	C₂²×D₄×C₅₉	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₉ ◁ C₂²×C₄×C₅₉ ◁ C₂×C₄×D₄×C₅₉ ◁ G

584 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 79
transitive degree	μ_t(G) = 3776

A minimal faithful permutation representation — 3 generators on 79 points:

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

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, 5614);