direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C2xS3xC13:C3, C78:3C6, (S3xC26):C3, C13:3(S3xC6), C26:2(C3xS3), C39:4(C2xC6), (S3xC13):2C6, C6:(C2xC13:C3), C3:(C22xC13:C3), (C6xC13:C3):3C2, (C3xC13:C3):4C22, SmallGroup(468,34)
Series: Derived ►Chief ►Lower central ►Upper central
| C39 — C2xS3xC13:C3 |
Generators and relations for C2xS3xC13:C3
G = < a,b,c,d,e | a2=b3=c2=d13=e3=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d9 >
Quotients: C1, C2, C3, C22, S3, C6, D6, C2xC6, C3xS3, S3xC6, C13:C3, C2xC13:C3, C22xC13:C3, S3xC13:C3, C2xS3xC13:C3
Smallest permutation representation of C2xS3xC13:C3
►On 78 points
Generators in S78
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 56)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(33 72)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)
(1 14 27)(2 15 28)(3 16 29)(4 17 30)(5 18 31)(6 19 32)(7 20 33)(8 21 34)(9 22 35)(10 23 36)(11 24 37)(12 25 38)(13 26 39)(40 53 66)(41 54 67)(42 55 68)(43 56 69)(44 57 70)(45 58 71)(46 59 72)(47 60 73)(48 61 74)(49 62 75)(50 63 76)(51 64 77)(52 65 78)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 65)
(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)
(2 4 10)(3 7 6)(5 13 11)(8 9 12)(15 17 23)(16 20 19)(18 26 24)(21 22 25)(28 30 36)(29 33 32)(31 39 37)(34 35 38)(41 43 49)(42 46 45)(44 52 50)(47 48 51)(54 56 62)(55 59 58)(57 65 63)(60 61 64)(67 69 75)(68 72 71)(70 78 76)(73 74 77)
G:=sub<Sym(78)| (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78), (1,14,27)(2,15,28)(3,16,29)(4,17,30)(5,18,31)(6,19,32)(7,20,33)(8,21,34)(9,22,35)(10,23,36)(11,24,37)(12,25,38)(13,26,39)(40,53,66)(41,54,67)(42,55,68)(43,56,69)(44,57,70)(45,58,71)(46,59,72)(47,60,73)(48,61,74)(49,62,75)(50,63,76)(51,64,77)(52,65,78), (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,65), (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), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(15,17,23)(16,20,19)(18,26,24)(21,22,25)(28,30,36)(29,33,32)(31,39,37)(34,35,38)(41,43,49)(42,46,45)(44,52,50)(47,48,51)(54,56,62)(55,59,58)(57,65,63)(60,61,64)(67,69,75)(68,72,71)(70,78,76)(73,74,77)>;
G:=Group( (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78), (1,14,27)(2,15,28)(3,16,29)(4,17,30)(5,18,31)(6,19,32)(7,20,33)(8,21,34)(9,22,35)(10,23,36)(11,24,37)(12,25,38)(13,26,39)(40,53,66)(41,54,67)(42,55,68)(43,56,69)(44,57,70)(45,58,71)(46,59,72)(47,60,73)(48,61,74)(49,62,75)(50,63,76)(51,64,77)(52,65,78), (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,65), (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), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(15,17,23)(16,20,19)(18,26,24)(21,22,25)(28,30,36)(29,33,32)(31,39,37)(34,35,38)(41,43,49)(42,46,45)(44,52,50)(47,48,51)(54,56,62)(55,59,58)(57,65,63)(60,61,64)(67,69,75)(68,72,71)(70,78,76)(73,74,77) );
G=PermutationGroup([[(1,40),(2,41),(3,42),(4,43),(5,44),(6,45),(7,46),(8,47),(9,48),(10,49),(11,50),(12,51),(13,52),(14,53),(15,54),(16,55),(17,56),(18,57),(19,58),(20,59),(21,60),(22,61),(23,62),(24,63),(25,64),(26,65),(27,66),(28,67),(29,68),(30,69),(31,70),(32,71),(33,72),(34,73),(35,74),(36,75),(37,76),(38,77),(39,78)], [(1,14,27),(2,15,28),(3,16,29),(4,17,30),(5,18,31),(6,19,32),(7,20,33),(8,21,34),(9,22,35),(10,23,36),(11,24,37),(12,25,38),(13,26,39),(40,53,66),(41,54,67),(42,55,68),(43,56,69),(44,57,70),(45,58,71),(46,59,72),(47,60,73),(48,61,74),(49,62,75),(50,63,76),(51,64,77),(52,65,78)], [(1,40),(2,41),(3,42),(4,43),(5,44),(6,45),(7,46),(8,47),(9,48),(10,49),(11,50),(12,51),(13,52),(14,66),(15,67),(16,68),(17,69),(18,70),(19,71),(20,72),(21,73),(22,74),(23,75),(24,76),(25,77),(26,78),(27,53),(28,54),(29,55),(30,56),(31,57),(32,58),(33,59),(34,60),(35,61),(36,62),(37,63),(38,64),(39,65)], [(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)], [(2,4,10),(3,7,6),(5,13,11),(8,9,12),(15,17,23),(16,20,19),(18,26,24),(21,22,25),(28,30,36),(29,33,32),(31,39,37),(34,35,38),(41,43,49),(42,46,45),(44,52,50),(47,48,51),(54,56,62),(55,59,58),(57,65,63),(60,61,64),(67,69,75),(68,72,71),(70,78,76),(73,74,77)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 13A | 13B | 13C | 13D | 26A | 26B | 26C | 26D | 26E | ··· | 26L | 39A | 39B | 39C | 39D | 78A | 78B | 78C | 78D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 13 | 13 | 13 | 13 | 26 | 26 | 26 | 26 | 26 | ··· | 26 | 39 | 39 | 39 | 39 | 78 | 78 | 78 | 78 |
| size | 1 | 1 | 3 | 3 | 2 | 13 | 13 | 26 | 26 | 2 | 13 | 13 | 26 | 26 | 39 | 39 | 39 | 39 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 9 | ··· | 9 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 6 | 6 |
| type | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D6 | C3xS3 | S3xC6 | C13:C3 | C2xC13:C3 | C2xC13:C3 | S3xC13:C3 | C2xS3xC13:C3 |
| kernel | C2xS3xC13:C3 | S3xC13:C3 | C6xC13:C3 | S3xC26 | S3xC13 | C78 | C2xC13:C3 | C13:C3 | C26 | C13 | D6 | S3 | C6 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 2 | 4 | 8 | 4 | 4 | 4 |
Matrix representation of C2xS3xC13:C3 ►in GL5(F79)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 78 | 0 | 0 |
| 0 | 0 | 0 | 78 | 0 |
| 0 | 0 | 0 | 0 | 78 |
| 1 | 42 | 0 | 0 | 0 |
| 62 | 77 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 78 | 37 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 78 | 0 | 0 |
| 0 | 0 | 0 | 78 | 0 |
| 0 | 0 | 0 | 0 | 78 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 66 | 39 | 66 |
| 0 | 0 | 1 | 0 | 30 |
| 0 | 0 | 0 | 1 | 38 |
| 23 | 0 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 | 0 |
| 0 | 0 | 52 | 39 | 22 |
| 0 | 0 | 55 | 23 | 78 |
| 0 | 0 | 45 | 59 | 4 |
G:=sub<GL(5,GF(79))| [1,0,0,0,0,0,1,0,0,0,0,0,78,0,0,0,0,0,78,0,0,0,0,0,78],[1,62,0,0,0,42,77,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[78,0,0,0,0,37,1,0,0,0,0,0,78,0,0,0,0,0,78,0,0,0,0,0,78],[1,0,0,0,0,0,1,0,0,0,0,0,66,1,0,0,0,39,0,1,0,0,66,30,38],[23,0,0,0,0,0,23,0,0,0,0,0,52,55,45,0,0,39,23,59,0,0,22,78,4] >;
C2xS3xC13:C3 in GAP, Magma, Sage, TeX
C_2\times S_3\times C_{13}\rtimes C_3 % in TeX
G:=Group("C2xS3xC13:C3"); // GroupNames label
G:=SmallGroup(468,34);
// by ID
G=gap.SmallGroup(468,34);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-13,483,689]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^13=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^9>;
// generators/relations
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