C₂×C₇×C₇₉⋊C₃ #15693285

order	3318
isoclinism	isoclinic class of C₇₉⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂×C₇×C₇₉⋊C₃
unambiguous semidirect product	(C₂×C₇₉)⋊(C₃×C₇)

Splittings

Fitting, socle, Hall	C₂×C₇×C₇₉ ⋊ C₃
derived, Sylow	C₇₉ ⋊ C₂×C₃×C₇
Hall	C₇×C₇₉ ⋊ C₂×C₃
Hall	C₂×C₇₉ ⋊ C₃×C₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇	C₇₉⋊C₃
commutator, cosocle, hypocenter	C₇₉	C₂×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₇₉	C₃
LCS	G ▷ C₇₉
UCS	C₁ ◁ C₂×C₇

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 88
transitive degree	μ_t(G) = 1106

A minimal faithful permutation representation — 2 generators on 88 points:

(1,2)(3,4,5,6,7,8,9)(11,12,14)(13,16,21)(15,19,18)(17,23,31)(20,26,28)(22,29,25)(24,33,44)(27,36,41)(30,39,38)(32,42,35)(34,46,59)(37,49,57)(40,52,54)(43,55,51)(45,47,48)(50,62,72)(53,65,70)(56,68,67)(58,63,64)(60,73,61)(66,74,81)(69,77,79)(71,75,76)(78,82,86)(80,83,84)(85,87,88);(10,11,13,17,24,34,47,31,41,54,67,76,62,33,45,58,71,80,85,87,77,52,26,12,15,20,27,37,50,63,42,21,28,38,51,64,46,60,59,72,81,86,88,82,65,36,16,22,30,40,53,66,75,55,29,14,18,25,35,48,61,73,44,57,70,79,84,74,49,23,32,43,56,69,78,83,68,39,19)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₅×C₇×C₇₉⋊C₃, C₇×D₅×C₇₉⋊C₃, C₂×C₄₉×C₇₉⋊C₃, …
maximal normal subgroup of	C₂×C₅×C₇×C₇₉⋊C₃, C₂×C₄₉×C₇₉⋊C₃, C₂×C₇²×C₇₉⋊C₃, …
maximal quotient of	C₂×C₅×C₇×C₇₉⋊C₃, C₇×D₅×C₇₉⋊C₃, C₂×C₄₉×C₇₉⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3318, 2);

C2×C7×C79⋊C3 #15693285