C₁₇×C₅²⋊C₉ #16838554

order	3825
isoclinism	isoclinic class of C₅²⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₉×C₁₇ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₁₇×C₅²⋊C₉
unambiguous semidirect product	C₅²⋊(C₉×C₁₇)
unambiguous semidirect product	(C₅²×C₁₇)⋊C₉

Splittings

derived, Sylow	C₅² ⋊ C₉×C₁₇
Hall	C₅²×C₁₇ ⋊ C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₁₇	C₅²⋊C₃
commutator, hypocenter	C₅²	C₉×C₁₇
Frattini	C₃	C₁₇×C₅²⋊C₃
Fitting, socle, prime radical	C₃×C₅²×C₁₇	C₃
cosocle	C₃×C₅²	C₃×C₁₇
LCS	G ▷ C₅²
UCS	C₁ ◁ C₃×C₁₇

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 41
transitive degree	μ_t(G) = 765

A minimal faithful permutation representation — 2 generators on 41 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,21)(20,23,25)(22,27,30)(24,29,31)(26,32,28)(33,34,36,35,37,39,38,40,41);(18,20,24,30,32)(19,23,29,22,28)(21,27,25,26,31)

Conjugacy classes & character table

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

Relationships

derived subgroup of	#29505665, #90646⋊D₁₇, ((C₅⋊D₅)⋊C₉)⋊D₁₇, …
maximal subgroup of	C₁₇×C₂²⋊(C₅²⋊C₉), C₁₇×C₇⋊(C₅²⋊C₉), C₇×C₁₇×C₅²⋊C₉
maximal normal subgroup of	C₇×C₁₇×C₅²⋊C₉
maximal quotient of	C₁₇×C₂²⋊(C₅²⋊C₉), C₁₇×C₇⋊(C₅²⋊C₉), C₇×C₁₇×C₅²⋊C₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3825, 4);

C17×C52⋊C9 #16838554