C₅×D₁₅₃ #13095252

order	1530
isoclinism	isoclinic class of D₁₅₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×D₁₅₃
unambiguous semidirect product	C₉⋊(C₅×D₁₇)
unambiguous semidirect product	C₁₇⋊(C₅×D₉)

Splittings

Fitting, Hall	C₅×C₉×C₁₇ ⋊ C₂
derived, Hall	C₉×C₁₇ ⋊ C₂×C₅
Hall	C₅×C₁₇ ⋊ D₉
Hall	C₅×C₉ ⋊ D₁₇
Sylow	C₁₇ ⋊ C₅×D₉
Sylow	C₉ ⋊ C₅×D₁₇

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₁₅₃
commutator, cosocle, hypocenter	C₉×C₁₇	C₂×C₅
Frattini	C₃	C₅×D₅₁
Fitting	C₅×C₉×C₁₇	C₂
socle	C₃×C₅×C₁₇	D₃
LCS	G ▷ C₉×C₁₇
UCS	C₁ ◁ C₅

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 765

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);(7,14)(8,13)(9,12)(10,11)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×D₉×D₁₇, D₅×D₁₅₃, D₄₅⋊D₁₇, …
maximal normal subgroup of	C₅×D₉×D₁₇, D₅×D₁₅₃, D₄₅⋊D₁₇, …
maximal quotient of	C₅×Dic₁₅₃, C₂×C₅×D₁₅₃, C₅×C₁₁×D₁₅₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1530, 5);

C5×D153 #13095252