direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C32×C13⋊C4, C39⋊7C12, (C3×C39)⋊4C4, C13⋊3(C3×C12), D13.2(C3×C6), (C3×D13).7C6, (C32×D13).3C2, SmallGroup(468,36)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C32×C13⋊C4 |
Generators and relations for C32×C13⋊C4
G = < a,b,c,d | a3=b3=c13=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >
Smallest permutation representation of C32×C13⋊C4
►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)
(2 9 13 6)(3 4 12 11)(5 7 10 8)(15 22 26 19)(16 17 25 24)(18 20 23 21)(28 35 39 32)(29 30 38 37)(31 33 36 34)(41 48 52 45)(42 43 51 50)(44 46 49 47)(54 61 65 58)(55 56 64 63)(57 59 62 60)(67 74 78 71)(68 69 77 76)(70 72 75 73)(80 87 91 84)(81 82 90 89)(83 85 88 86)(93 100 104 97)(94 95 103 102)(96 98 101 99)(106 113 117 110)(107 108 116 115)(109 111 114 112)
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), (2,9,13,6)(3,4,12,11)(5,7,10,8)(15,22,26,19)(16,17,25,24)(18,20,23,21)(28,35,39,32)(29,30,38,37)(31,33,36,34)(41,48,52,45)(42,43,51,50)(44,46,49,47)(54,61,65,58)(55,56,64,63)(57,59,62,60)(67,74,78,71)(68,69,77,76)(70,72,75,73)(80,87,91,84)(81,82,90,89)(83,85,88,86)(93,100,104,97)(94,95,103,102)(96,98,101,99)(106,113,117,110)(107,108,116,115)(109,111,114,112)>;
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), (2,9,13,6)(3,4,12,11)(5,7,10,8)(15,22,26,19)(16,17,25,24)(18,20,23,21)(28,35,39,32)(29,30,38,37)(31,33,36,34)(41,48,52,45)(42,43,51,50)(44,46,49,47)(54,61,65,58)(55,56,64,63)(57,59,62,60)(67,74,78,71)(68,69,77,76)(70,72,75,73)(80,87,91,84)(81,82,90,89)(83,85,88,86)(93,100,104,97)(94,95,103,102)(96,98,101,99)(106,113,117,110)(107,108,116,115)(109,111,114,112) );
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)], [(2,9,13,6),(3,4,12,11),(5,7,10,8),(15,22,26,19),(16,17,25,24),(18,20,23,21),(28,35,39,32),(29,30,38,37),(31,33,36,34),(41,48,52,45),(42,43,51,50),(44,46,49,47),(54,61,65,58),(55,56,64,63),(57,59,62,60),(67,74,78,71),(68,69,77,76),(70,72,75,73),(80,87,91,84),(81,82,90,89),(83,85,88,86),(93,100,104,97),(94,95,103,102),(96,98,101,99),(106,113,117,110),(107,108,116,115),(109,111,114,112)]])
63 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 4A | 4B | 6A | ··· | 6H | 12A | ··· | 12P | 13A | 13B | 13C | 39A | ··· | 39X |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 13 | 13 | 13 | 39 | ··· | 39 |
| size | 1 | 13 | 1 | ··· | 1 | 13 | 13 | 13 | ··· | 13 | 13 | ··· | 13 | 4 | 4 | 4 | 4 | ··· | 4 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | |||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | C13⋊C4 | C3×C13⋊C4 |
| kernel | C32×C13⋊C4 | C32×D13 | C3×C13⋊C4 | C3×C39 | C3×D13 | C39 | C32 | C3 |
| # reps | 1 | 1 | 8 | 2 | 8 | 16 | 3 | 24 |
Matrix representation of C32×C13⋊C4 ►in GL5(𝔽157)
| 144 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 103 | 122 | 103 | 156 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 54 | 89 | 109 | 55 |
| 0 | 122 | 48 | 67 | 102 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(157))| [144,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,103,1,0,0,0,122,0,1,0,0,103,0,0,1,0,156,0,0,0],[28,0,0,0,0,0,1,54,122,0,0,0,89,48,0,0,0,109,67,1,0,0,55,102,0] >;
C32×C13⋊C4 in GAP, Magma, Sage, TeX
C_3^2\times C_{13}\rtimes C_4 % in TeX
G:=Group("C3^2xC13:C4"); // GroupNames label
G:=SmallGroup(468,36);
// by ID
G=gap.SmallGroup(468,36);
# by ID
G:=PCGroup([5,-2,-3,-3,-2,-13,90,7204,619]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^13=d^4=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^5>;
// generators/relations
Export