C₅×Dic₃×C₆₇ #16860834

order	4020
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₄×C₅×C₆₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×Dic₃×C₆₇
unambiguous semidirect product	C₃⋊(C₄×C₅×C₆₇)

Splittings

derived, Sylow	C₃ ⋊ C₄×C₅×C₆₇
Hall	C₃×C₅×C₆₇ ⋊ C₄
Hall	C₃×C₆₇ ⋊ C₄×C₅
Hall	C₃×C₅ ⋊ C₄×C₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅×C₆₇	D₃
commutator, hypocenter	C₃	C₄×C₅×C₆₇
Frattini	C₂	C₅×D₃×C₆₇
Fitting, socle, prime radical	C₂×C₃×C₅×C₆₇	C₂
cosocle	C₂×C₃	C₂×C₅×C₆₇
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₅×C₆₇

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 4020

A minimal faithful permutation representation — 2 generators on 79 points:

(1,2,3,4,5)(6,7)(8,9)(10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79);(6,8,7,9)(11,12)

Conjugacy classes & character table

2010 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅×Dic₉×C₆₇, C₅×Dic₃×C₆₇⋊C₃, C₃×C₅×Dic₃×C₆₇, …
maximal normal subgroup of	C₅×Dic₃×C₆₇⋊C₃, C₃×C₅×Dic₃×C₆₇, Dic₃×C₂₅×C₆₇, …
maximal quotient of	C₅×Dic₉×C₆₇, C₃×C₅×Dic₃×C₆₇, C₅×C₃⋊Dic₃×C₆₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4020, 6);

C5×Dic3×C67 #16860834