C₅×Dic₂₃₇ #18011607

order	4740
automorphism	?
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×Dic₂₃₇
unambiguous semidirect product	C₃⋊(C₅×Dic₇₉)
unambiguous semidirect product	C₇₉⋊(C₅×Dic₃)

Splittings

derived, Hall	C₃×C₇₉ ⋊ C₄×C₅
Hall	C₃×C₅×C₇₉ ⋊ C₄
Hall	C₅×C₇₉ ⋊ Dic₃
Sylow	C₇₉ ⋊ C₅×Dic₃
Hall	C₃×C₅ ⋊ Dic₇₉
Sylow	C₃ ⋊ C₅×Dic₇₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	D₂₃₇
commutator, hypocenter	C₃×C₇₉	C₄×C₅
Frattini	C₂	C₅×D₂₃₇
Fitting, socle, prime radical	C₂×C₃×C₅×C₇₉	C₂
cosocle	C₂×C₃×C₇₉	C₂×C₅
LCS	G ▷ C₃×C₇₉
UCS	C₁ ◁ C₂×C₅

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 4740

A minimal faithful permutation representation — 2 generators on 91 points:

(1,2,3,4,5)(6,7)(8,9)(10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91);(6,8,7,9)(11,12)(14,91)(15,90)(16,89)(17,88)(18,87)(19,86)(20,85)(21,84)(22,83)(23,82)(24,81)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)

Conjugacy classes & character table

1200 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅×Dic₇₁₁, C₅×C₃⋊((C₇₉⋊C₃)⋊C₄), C₃×C₅×Dic₂₃₇, …
maximal normal subgroup of	C₅×C₃⋊((C₇₉⋊C₃)⋊C₄), C₃×C₅×Dic₂₃₇, C₂₅×Dic₂₃₇, …
maximal quotient of	C₅×Dic₇₁₁, C₃×C₅×Dic₂₃₇, C₅×C₃⋊Dic₂₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4740, 7);

C5×Dic237 #18011607