C₅₉×Dic₁₅ #16812995

order	3540
isoclinism	isoclinic class of D₁₅
automorphism	?
homology	Abelianization (H₁) = C₄×C₅₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅₉×Dic₁₅
unambiguous semidirect product	C₃⋊(Dic₅×C₅₉)

Splittings

derived, Hall	C₃×C₅ ⋊ C₄×C₅₉
Hall	C₃×C₅×C₅₉ ⋊ C₄
Hall	C₅×C₅₉ ⋊ Dic₃
Hall	C₃×C₅₉ ⋊ Dic₅
Sylow	C₅ ⋊ Dic₃×C₅₉
Sylow	C₃ ⋊ Dic₅×C₅₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅₉	D₁₅
commutator, hypocenter	C₃×C₅	C₄×C₅₉
Frattini	C₂	D₁₅×C₅₉
Fitting, socle, prime radical	C₂×C₃×C₅×C₅₉	C₂
cosocle	C₂×C₃×C₅	C₂×C₅₉
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₅₉

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 3540

A minimal faithful permutation representation — 2 generators on 71 points:

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

Conjugacy classes & character table

1062 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅₉×Dic₇₅, C₅×C₅₉×Dic₁₅, C₅₉×C₃⋊#1039, …
maximal normal subgroup of	C₅×C₅₉×Dic₁₅, C₇×C₅₉×Dic₁₅
maximal quotient of	C₅₉×Dic₇₅, C₅×C₅₉×Dic₁₅, C₅₉×C₃⋊#1039, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3540, 7);

C59×Dic15 #16812995