C₅×Dic₃×C₃₇ #14555682

order	2220
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₄×C₅×C₃₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×Dic₃×C₃₇
unambiguous semidirect product	C₃⋊(C₄×C₅×C₃₇)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅×C₃₇	D₃
commutator, hypocenter	C₃	C₄×C₅×C₃₇
Frattini	C₂	C₅×D₃×C₃₇
Fitting, socle, prime radical	C₂×C₃×C₅×C₃₇	C₂
cosocle	C₂×C₃	C₂×C₅×C₃₇
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₅×C₃₇

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);(6,8,7,9)(11,12)

Conjugacy classes & character table

1110 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₃×C₂₅×C₃₇, C₅²×Dic₃×C₃₇, C₅×C₃₇×Dic₁₅, …
maximal normal subgroup of	Dic₃×C₂₅×C₃₇, C₅²×Dic₃×C₃₇, C₅×C₇×Dic₃×C₃₇, …
maximal quotient of	Dic₃×C₂₅×C₃₇, C₅²×Dic₃×C₃₇, C₅×C₃₇×Dic₁₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2220, 9);

C5×Dic3×C37 #14555682