C₅×D₁₃₃ #13050596

order	1330
isoclinism	isoclinic class of D₁₃₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×D₁₃₃
unambiguous semidirect product	C₇⋊(C₅×D₁₉)
unambiguous semidirect product	C₁₉⋊(C₅×D₇)

Splittings

Fitting, socle, Hall	C₅×C₇×C₁₉ ⋊ C₂
derived, Hall	C₇×C₁₉ ⋊ C₂×C₅
Hall	C₅×C₁₉ ⋊ D₇
Hall	C₅×C₇ ⋊ D₁₉
Sylow	C₁₉ ⋊ C₅×D₇
Sylow	C₇ ⋊ C₅×D₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₁₃₃
commutator, cosocle, hypocenter	C₇×C₁₉	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₁₉	C₂
LCS	G ▷ C₇×C₁₉
UCS	C₁ ◁ C₅

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 665

A minimal faithful permutation representation — 2 generators on 31 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);(7,12)(8,11)(9,10)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×D₇×D₁₉, D₅×D₁₃₃, D₃₅⋊D₁₉, …
maximal normal subgroup of	C₅×D₇×D₁₉, D₅×D₁₃₃, D₃₅⋊D₁₉, …
maximal quotient of	C₅×Dic₁₃₃, C₂×C₅×D₁₃₃, C₃×C₅×D₁₃₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1330, 3);

C5×D133 #13050596