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## G = C15×S4order 360 = 23·32·5

### Direct product of C15 and S4

Aliases: C15×S4, A4⋊C30, (C5×A4)⋊3C6, (C2×C30)⋊1S3, C22⋊(S3×C15), (A4×C15)⋊1C2, (C3×A4)⋊1C10, (C2×C6)⋊1(C5×S3), (C2×C10)⋊1(C3×S3), SmallGroup(360,138)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C15×S4
 Chief series C1 — C22 — A4 — C5×A4 — A4×C15 — C15×S4
 Lower central A4 — C15×S4
 Upper central C1 — C15

Generators and relations for C15×S4
G = < a,b,c,d,e | a15=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

Smallest permutation representation of C15×S4
On 60 points
Generators in S60
(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)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 16)(13 17)(14 18)(15 19)(31 47)(32 48)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 46)
(1 57)(2 58)(3 59)(4 60)(5 46)(6 47)(7 48)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(16 53 37)(17 54 38)(18 55 39)(19 56 40)(20 57 41)(21 58 42)(22 59 43)(23 60 44)(24 46 45)(25 47 31)(26 48 32)(27 49 33)(28 50 34)(29 51 35)(30 52 36)
(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)

G:=sub<Sym(60)| (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), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,16)(13,17)(14,18)(15,19)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,46), (1,57)(2,58)(3,59)(4,60)(5,46)(6,47)(7,48)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (16,53,37)(17,54,38)(18,55,39)(19,56,40)(20,57,41)(21,58,42)(22,59,43)(23,60,44)(24,46,45)(25,47,31)(26,48,32)(27,49,33)(28,50,34)(29,51,35)(30,52,36), (16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)>;

G:=Group( (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), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,16)(13,17)(14,18)(15,19)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,46), (1,57)(2,58)(3,59)(4,60)(5,46)(6,47)(7,48)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (16,53,37)(17,54,38)(18,55,39)(19,56,40)(20,57,41)(21,58,42)(22,59,43)(23,60,44)(24,46,45)(25,47,31)(26,48,32)(27,49,33)(28,50,34)(29,51,35)(30,52,36), (16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36) );

G=PermutationGroup([[(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)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,16),(13,17),(14,18),(15,19),(31,47),(32,48),(33,49),(34,50),(35,51),(36,52),(37,53),(38,54),(39,55),(40,56),(41,57),(42,58),(43,59),(44,60),(45,46)], [(1,57),(2,58),(3,59),(4,60),(5,46),(6,47),(7,48),(8,49),(9,50),(10,51),(11,52),(12,53),(13,54),(14,55),(15,56),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36)], [(16,53,37),(17,54,38),(18,55,39),(19,56,40),(20,57,41),(21,58,42),(22,59,43),(23,60,44),(24,46,45),(25,47,31),(26,48,32),(27,49,33),(28,50,34),(29,51,35),(30,52,36)], [(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36)]])

75 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 4 5A 5B 5C 5D 6A 6B 6C 6D 10A 10B 10C 10D 10E 10F 10G 10H 12A 12B 15A ··· 15H 15I ··· 15T 20A 20B 20C 20D 30A ··· 30H 30I ··· 30P 60A ··· 60H order 1 2 2 3 3 3 3 3 4 5 5 5 5 6 6 6 6 10 10 10 10 10 10 10 10 12 12 15 ··· 15 15 ··· 15 20 20 20 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 3 6 1 1 8 8 8 6 1 1 1 1 3 3 6 6 3 3 3 3 6 6 6 6 6 6 1 ··· 1 8 ··· 8 6 6 6 6 3 ··· 3 6 ··· 6 6 ··· 6

75 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 type + + + + image C1 C2 C3 C5 C6 C10 C15 C30 S3 C3×S3 C5×S3 S3×C15 S4 C3×S4 C5×S4 C15×S4 kernel C15×S4 A4×C15 C5×S4 C3×S4 C5×A4 C3×A4 S4 A4 C2×C30 C2×C10 C2×C6 C22 C15 C5 C3 C1 # reps 1 1 2 4 2 4 8 8 1 2 4 8 2 4 8 16

Matrix representation of C15×S4 in GL5(𝔽61)

 13 0 0 0 0 0 13 0 0 0 0 0 34 0 0 0 0 0 34 0 0 0 0 0 34
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 60 60 60 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 60 60 60 0 0 0 0 1 0 0 0 1 0
,
 0 60 0 0 0 1 60 0 0 0 0 0 1 0 0 0 0 60 60 60 0 0 0 1 0
,
 60 0 0 0 0 60 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(61))| [13,0,0,0,0,0,13,0,0,0,0,0,34,0,0,0,0,0,34,0,0,0,0,0,34],[1,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,60,0,0,0,1,60,0],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,60,0,1,0,0,60,1,0],[0,1,0,0,0,60,60,0,0,0,0,0,1,60,0,0,0,0,60,1,0,0,0,60,0],[60,60,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C15×S4 in GAP, Magma, Sage, TeX

C_{15}\times S_4
% in TeX

G:=Group("C15xS4");
// GroupNames label

G:=SmallGroup(360,138);
// by ID

G=gap.SmallGroup(360,138);
# by ID

G:=PCGroup([6,-2,-3,-5,-3,-2,2,1443,5404,202,3245,347]);
// Polycyclic

G:=Group<a,b,c,d,e|a^15=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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