C₅×Dic₆₈ #13062672

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

Aliases

direct product

C₅×Dic₆₈

Splittings

Hall	C₅×C₁₇ ⋊ Q₁₆
Sylow	C₁₇ ⋊ C₅×Q₁₆

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, prime radical	C₂×C₅×C₁₇	D₄
hypercenter	C₅×C₈	D₁₇
hypocenter	C₁₇	C₅×Q₁₆
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 1360

A minimal faithful permutation representation — 2 generators on 38 points:

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

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, 40);

C5×Dic68 #13062672