C₃×C₅×Dic₃₇ #14555679

order	2220
isoclinism	isoclinic class of D₃₇
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×C₅×Dic₃₇

Splittings

derived, Sylow	C₃₇ ⋊ C₃×C₄×C₅
Hall	C₃×C₅×C₃₇ ⋊ C₄
Hall	C₅×C₃₇ ⋊ C₃×C₄
Hall	C₃×C₃₇ ⋊ C₄×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅	D₃₇
commutator, hypocenter	C₃₇	C₃×C₄×C₅
Frattini	C₂	C₃×C₅×D₃₇
Fitting, socle, prime radical	C₂×C₃×C₅×C₃₇	C₂
cosocle	C₂×C₃₇	C₂×C₃×C₅
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₂×C₃×C₅

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 2220

A minimal faithful permutation representation — 2 generators on 49 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);(9,11,10,12)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×C₂₅×Dic₃₇, C₃×C₅²×Dic₃₇, C₃×C₅×Dic₁₈₅, …
maximal normal subgroup of	C₃×C₂₅×Dic₃₇, C₃×C₅²×Dic₃₇, C₃×C₅×C₇×Dic₃₇, …
maximal quotient of	C₃×C₂₅×Dic₃₇, C₃×C₅²×Dic₃₇, C₃×C₅×Dic₁₈₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2220, 6);

C3×C5×Dic37 #14555679