direct product, non-abelian, soluble, monomial
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
| A4 — C15×S4 |
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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