C₅×#63219 #13062670

order	1360
isoclinism	isoclinic class of #63219
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₅×#63219

Splittings

Fitting	C₅×C₈×C₁₇ ⋊ C₂
Hall	C₅×C₁₇ ⋊ SD₁₆
Sylow	C₁₇ ⋊ C₅×SD₁₆

Subgroups

	subgroup	G/H
center	C₂×C₅	D₆₈
commutator, cosocle	C₄×C₁₇	C₂²×C₅
Frattini	C₄	C₂×C₅×D₁₇
Fitting	C₅×C₈×C₁₇	C₂
socle	C₂×C₅×C₁₇	D₄
prime radical	C₅×D₆₈	C₂
hypercenter	C₅×C₈	D₁₇
hypocenter	C₁₇	C₅×SD₁₆
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×C₈

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 680

A minimal faithful permutation representation — 2 generators on 30 points:

(1,2,3,4,5)(6,7,9,11,13,15,17,19,21,22,20,18,16,14,12,10,8)(23,24,25,27,26,28,29,30);(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(24,27)(25,29)(28,30)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1360, 38);

C5×#63219 #13062670