direct product, metabelian, supersoluble, monomial, A-group, 3-hyperelementary
Aliases: C32×C13⋊C3, C13⋊C33, C39⋊C32, (C3×C39)⋊3C3, SmallGroup(351,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C32×C13⋊C3 |
Generators and relations for C32×C13⋊C3
G = < a,b,c,d | a3=b3=c13=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c9 >
Subgroups: 320 in 56 conjugacy classes, 34 normal (5 characteristic)
C1, C3, C3, C32, C32, C13, C33, C13⋊C3, C39, C3×C13⋊C3, C3×C39, C32×C13⋊C3
Quotients: C1, C3, C32, C33, C13⋊C3, C3×C13⋊C3, C32×C13⋊C3
►On 117 points
Generators in S117
(1 105 53)(2 106 54)(3 107 55)(4 108 56)(5 109 57)(6 110 58)(7 111 59)(8 112 60)(9 113 61)(10 114 62)(11 115 63)(12 116 64)(13 117 65)(14 79 66)(15 80 67)(16 81 68)(17 82 69)(18 83 70)(19 84 71)(20 85 72)(21 86 73)(22 87 74)(23 88 75)(24 89 76)(25 90 77)(26 91 78)(27 92 40)(28 93 41)(29 94 42)(30 95 43)(31 96 44)(32 97 45)(33 98 46)(34 99 47)(35 100 48)(36 101 49)(37 102 50)(38 103 51)(39 104 52)
(1 27 14)(2 28 15)(3 29 16)(4 30 17)(5 31 18)(6 32 19)(7 33 20)(8 34 21)(9 35 22)(10 36 23)(11 37 24)(12 38 25)(13 39 26)(40 66 53)(41 67 54)(42 68 55)(43 69 56)(44 70 57)(45 71 58)(46 72 59)(47 73 60)(48 74 61)(49 75 62)(50 76 63)(51 77 64)(52 78 65)(79 105 92)(80 106 93)(81 107 94)(82 108 95)(83 109 96)(84 110 97)(85 111 98)(86 112 99)(87 113 100)(88 114 101)(89 115 102)(90 116 103)(91 117 104)
(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 80 81 82 83 84 85 86 87 88 89 90 91)(92 93 94 95 96 97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112 113 114 115 116 117)
(1 14 27)(2 17 36)(3 20 32)(4 23 28)(5 26 37)(6 16 33)(7 19 29)(8 22 38)(9 25 34)(10 15 30)(11 18 39)(12 21 35)(13 24 31)(40 53 66)(41 56 75)(42 59 71)(43 62 67)(44 65 76)(45 55 72)(46 58 68)(47 61 77)(48 64 73)(49 54 69)(50 57 78)(51 60 74)(52 63 70)(79 92 105)(80 95 114)(81 98 110)(82 101 106)(83 104 115)(84 94 111)(85 97 107)(86 100 116)(87 103 112)(88 93 108)(89 96 117)(90 99 113)(91 102 109)
G:=sub<Sym(117)| (1,105,53)(2,106,54)(3,107,55)(4,108,56)(5,109,57)(6,110,58)(7,111,59)(8,112,60)(9,113,61)(10,114,62)(11,115,63)(12,116,64)(13,117,65)(14,79,66)(15,80,67)(16,81,68)(17,82,69)(18,83,70)(19,84,71)(20,85,72)(21,86,73)(22,87,74)(23,88,75)(24,89,76)(25,90,77)(26,91,78)(27,92,40)(28,93,41)(29,94,42)(30,95,43)(31,96,44)(32,97,45)(33,98,46)(34,99,47)(35,100,48)(36,101,49)(37,102,50)(38,103,51)(39,104,52), (1,27,14)(2,28,15)(3,29,16)(4,30,17)(5,31,18)(6,32,19)(7,33,20)(8,34,21)(9,35,22)(10,36,23)(11,37,24)(12,38,25)(13,39,26)(40,66,53)(41,67,54)(42,68,55)(43,69,56)(44,70,57)(45,71,58)(46,72,59)(47,73,60)(48,74,61)(49,75,62)(50,76,63)(51,77,64)(52,78,65)(79,105,92)(80,106,93)(81,107,94)(82,108,95)(83,109,96)(84,110,97)(85,111,98)(86,112,99)(87,113,100)(88,114,101)(89,115,102)(90,116,103)(91,117,104), (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)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117), (1,14,27)(2,17,36)(3,20,32)(4,23,28)(5,26,37)(6,16,33)(7,19,29)(8,22,38)(9,25,34)(10,15,30)(11,18,39)(12,21,35)(13,24,31)(40,53,66)(41,56,75)(42,59,71)(43,62,67)(44,65,76)(45,55,72)(46,58,68)(47,61,77)(48,64,73)(49,54,69)(50,57,78)(51,60,74)(52,63,70)(79,92,105)(80,95,114)(81,98,110)(82,101,106)(83,104,115)(84,94,111)(85,97,107)(86,100,116)(87,103,112)(88,93,108)(89,96,117)(90,99,113)(91,102,109)>;
G:=Group( (1,105,53)(2,106,54)(3,107,55)(4,108,56)(5,109,57)(6,110,58)(7,111,59)(8,112,60)(9,113,61)(10,114,62)(11,115,63)(12,116,64)(13,117,65)(14,79,66)(15,80,67)(16,81,68)(17,82,69)(18,83,70)(19,84,71)(20,85,72)(21,86,73)(22,87,74)(23,88,75)(24,89,76)(25,90,77)(26,91,78)(27,92,40)(28,93,41)(29,94,42)(30,95,43)(31,96,44)(32,97,45)(33,98,46)(34,99,47)(35,100,48)(36,101,49)(37,102,50)(38,103,51)(39,104,52), (1,27,14)(2,28,15)(3,29,16)(4,30,17)(5,31,18)(6,32,19)(7,33,20)(8,34,21)(9,35,22)(10,36,23)(11,37,24)(12,38,25)(13,39,26)(40,66,53)(41,67,54)(42,68,55)(43,69,56)(44,70,57)(45,71,58)(46,72,59)(47,73,60)(48,74,61)(49,75,62)(50,76,63)(51,77,64)(52,78,65)(79,105,92)(80,106,93)(81,107,94)(82,108,95)(83,109,96)(84,110,97)(85,111,98)(86,112,99)(87,113,100)(88,114,101)(89,115,102)(90,116,103)(91,117,104), (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)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117), (1,14,27)(2,17,36)(3,20,32)(4,23,28)(5,26,37)(6,16,33)(7,19,29)(8,22,38)(9,25,34)(10,15,30)(11,18,39)(12,21,35)(13,24,31)(40,53,66)(41,56,75)(42,59,71)(43,62,67)(44,65,76)(45,55,72)(46,58,68)(47,61,77)(48,64,73)(49,54,69)(50,57,78)(51,60,74)(52,63,70)(79,92,105)(80,95,114)(81,98,110)(82,101,106)(83,104,115)(84,94,111)(85,97,107)(86,100,116)(87,103,112)(88,93,108)(89,96,117)(90,99,113)(91,102,109) );
G=PermutationGroup([[(1,105,53),(2,106,54),(3,107,55),(4,108,56),(5,109,57),(6,110,58),(7,111,59),(8,112,60),(9,113,61),(10,114,62),(11,115,63),(12,116,64),(13,117,65),(14,79,66),(15,80,67),(16,81,68),(17,82,69),(18,83,70),(19,84,71),(20,85,72),(21,86,73),(22,87,74),(23,88,75),(24,89,76),(25,90,77),(26,91,78),(27,92,40),(28,93,41),(29,94,42),(30,95,43),(31,96,44),(32,97,45),(33,98,46),(34,99,47),(35,100,48),(36,101,49),(37,102,50),(38,103,51),(39,104,52)], [(1,27,14),(2,28,15),(3,29,16),(4,30,17),(5,31,18),(6,32,19),(7,33,20),(8,34,21),(9,35,22),(10,36,23),(11,37,24),(12,38,25),(13,39,26),(40,66,53),(41,67,54),(42,68,55),(43,69,56),(44,70,57),(45,71,58),(46,72,59),(47,73,60),(48,74,61),(49,75,62),(50,76,63),(51,77,64),(52,78,65),(79,105,92),(80,106,93),(81,107,94),(82,108,95),(83,109,96),(84,110,97),(85,111,98),(86,112,99),(87,113,100),(88,114,101),(89,115,102),(90,116,103),(91,117,104)], [(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),(79,80,81,82,83,84,85,86,87,88,89,90,91),(92,93,94,95,96,97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112,113,114,115,116,117)], [(1,14,27),(2,17,36),(3,20,32),(4,23,28),(5,26,37),(6,16,33),(7,19,29),(8,22,38),(9,25,34),(10,15,30),(11,18,39),(12,21,35),(13,24,31),(40,53,66),(41,56,75),(42,59,71),(43,62,67),(44,65,76),(45,55,72),(46,58,68),(47,61,77),(48,64,73),(49,54,69),(50,57,78),(51,60,74),(52,63,70),(79,92,105),(80,95,114),(81,98,110),(82,101,106),(83,104,115),(84,94,111),(85,97,107),(86,100,116),(87,103,112),(88,93,108),(89,96,117),(90,99,113),(91,102,109)]])
63 conjugacy classes
| class | 1 | 3A | ··· | 3H | 3I | ··· | 3Z | 13A | 13B | 13C | 13D | 39A | ··· | 39AF |
| order | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 13 | 13 | 13 | 13 | 39 | ··· | 39 |
| size | 1 | 1 | ··· | 1 | 13 | ··· | 13 | 3 | 3 | 3 | 3 | 3 | ··· | 3 |
63 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 |
| type | + | ||||
| image | C1 | C3 | C3 | C13⋊C3 | C3×C13⋊C3 |
| kernel | C32×C13⋊C3 | C3×C13⋊C3 | C3×C39 | C32 | C3 |
| # reps | 1 | 24 | 2 | 4 | 32 |
Matrix representation of C32×C13⋊C3 ►in GL4(𝔽79) generated by
| 1 | 0 | 0 | 0 |
| 0 | 55 | 0 | 0 |
| 0 | 0 | 55 | 0 |
| 0 | 0 | 0 | 55 |
| 23 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 |
| 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 23 |
| 1 | 0 | 0 | 0 |
| 0 | 29 | 41 | 29 |
| 0 | 1 | 0 | 67 |
| 0 | 0 | 1 | 40 |
| 23 | 0 | 0 | 0 |
| 0 | 52 | 15 | 73 |
| 0 | 61 | 25 | 31 |
| 0 | 7 | 6 | 2 |
G:=sub<GL(4,GF(79))| [1,0,0,0,0,55,0,0,0,0,55,0,0,0,0,55],[23,0,0,0,0,23,0,0,0,0,23,0,0,0,0,23],[1,0,0,0,0,29,1,0,0,41,0,1,0,29,67,40],[23,0,0,0,0,52,61,7,0,15,25,6,0,73,31,2] >;
C32×C13⋊C3 in GAP, Magma, Sage, TeX
C_3^2\times C_{13}\rtimes C_3 % in TeX
G:=Group("C3^2xC13:C3"); // GroupNames label
G:=SmallGroup(351,13);
// by ID
G=gap.SmallGroup(351,13);
# by ID
G:=PCGroup([4,-3,-3,-3,-13,1299]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^13=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^9>;
// generators/relations