C₅×D₄₇₇ #18012007

order	4770
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×D₄₇₇
unambiguous semidirect product	C₉⋊(C₅×D₅₃)
unambiguous semidirect product	C₅₃⋊(C₅×D₉)

Splittings

Fitting, Hall	C₅×C₉×C₅₃ ⋊ C₂
derived, Hall	C₉×C₅₃ ⋊ C₂×C₅
Hall	C₅×C₅₃ ⋊ D₉
Sylow	C₅₃ ⋊ C₅×D₉
Hall	C₅×C₉ ⋊ D₅₃
Sylow	C₉ ⋊ C₅×D₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₄₇₇
commutator, cosocle, hypocenter	C₉×C₅₃	C₂×C₅
Frattini	C₃	C₅×D₁₅₉
Fitting	C₅×C₉×C₅₃	C₂
socle	C₃×C₅×C₅₃	D₃
LCS	G ▷ C₉×C₅₃
UCS	C₁ ◁ C₅

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 2385

A minimal faithful permutation representation — 2 generators on 67 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);(7,14)(8,13)(9,12)(10,11)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂₅×D₄₇₇, C₅²×D₄₇₇, C₅×D₂₃₈₅, …
maximal normal subgroup of	C₂₅×D₄₇₇, C₅²×D₄₇₇, C₅×C₇×D₄₇₇
maximal quotient of	C₂₅×D₄₇₇, C₅²×D₄₇₇, C₅×D₂₃₈₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4770, 6);

C5×D477 #18012007