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C₅×#3675 #10623594

order	720
isoclinism	isoclinic class of #3672
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Fitting	C₅×Q₈×C₉ ⋊ C₂
Hall	C₅×C₉ ⋊ SD₁₆
Sylow	C₉ ⋊ C₅×SD₁₆

Subgroups

	subgroup	G/H
center	C₂×C₅	#611
commutator, cosocle	C₄×C₉	C₂²×C₅
Frattini	C₃×C₄	C₂×C₅×D₃
Fitting	C₅×Q₈×C₉	C₂
socle	C₂×C₃×C₅	#67
prime radical	C₅×D₃₆	C₂
hypercenter	C₅×Q₈	D₉
hypocenter	C₉	C₅×SD₁₆
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×Q₈

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 360

A minimal faithful permutation representation — 2 generators on 22 points:

(1,2,3,4,5)(6,7,10,12,14,13,8,11,9)(15,16,18,20)(17,22,21,19);(7,9)(8,12)(10,11)(13,14)(16,19)(17,21)(20,22)

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×#3675, #13084263, #13084264, …
maximal normal subgroup of	D₅×#3675, #13084263, #13084264, …
maximal quotient of	C₅×#63367, C₅×#63369, C₅×#63395, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 63);