direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C32×D13, C39⋊6C6, (C3×C39)⋊3C2, C13⋊3(C3×C6), SmallGroup(234,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C32×D13 |
Generators and relations for C32×D13
G = < a,b,c,d | a3=b3=c13=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of C32×D13
►On 117 points
Generators in S117
(1 113 54)(2 114 55)(3 115 56)(4 116 57)(5 117 58)(6 105 59)(7 106 60)(8 107 61)(9 108 62)(10 109 63)(11 110 64)(12 111 65)(13 112 53)(14 80 66)(15 81 67)(16 82 68)(17 83 69)(18 84 70)(19 85 71)(20 86 72)(21 87 73)(22 88 74)(23 89 75)(24 90 76)(25 91 77)(26 79 78)(27 99 41)(28 100 42)(29 101 43)(30 102 44)(31 103 45)(32 104 46)(33 92 47)(34 93 48)(35 94 49)(36 95 50)(37 96 51)(38 97 52)(39 98 40)
(1 33 20)(2 34 21)(3 35 22)(4 36 23)(5 37 24)(6 38 25)(7 39 26)(8 27 14)(9 28 15)(10 29 16)(11 30 17)(12 31 18)(13 32 19)(40 78 60)(41 66 61)(42 67 62)(43 68 63)(44 69 64)(45 70 65)(46 71 53)(47 72 54)(48 73 55)(49 74 56)(50 75 57)(51 76 58)(52 77 59)(79 106 98)(80 107 99)(81 108 100)(82 109 101)(83 110 102)(84 111 103)(85 112 104)(86 113 92)(87 114 93)(88 115 94)(89 116 95)(90 117 96)(91 105 97)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 25)(15 24)(16 23)(17 22)(18 21)(19 20)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 54)(55 65)(56 64)(57 63)(58 62)(59 61)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(80 91)(81 90)(82 89)(83 88)(84 87)(85 86)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)(105 107)(108 117)(109 116)(110 115)(111 114)(112 113)
G:=sub<Sym(117)| (1,113,54)(2,114,55)(3,115,56)(4,116,57)(5,117,58)(6,105,59)(7,106,60)(8,107,61)(9,108,62)(10,109,63)(11,110,64)(12,111,65)(13,112,53)(14,80,66)(15,81,67)(16,82,68)(17,83,69)(18,84,70)(19,85,71)(20,86,72)(21,87,73)(22,88,74)(23,89,75)(24,90,76)(25,91,77)(26,79,78)(27,99,41)(28,100,42)(29,101,43)(30,102,44)(31,103,45)(32,104,46)(33,92,47)(34,93,48)(35,94,49)(36,95,50)(37,96,51)(38,97,52)(39,98,40), (1,33,20)(2,34,21)(3,35,22)(4,36,23)(5,37,24)(6,38,25)(7,39,26)(8,27,14)(9,28,15)(10,29,16)(11,30,17)(12,31,18)(13,32,19)(40,78,60)(41,66,61)(42,67,62)(43,68,63)(44,69,64)(45,70,65)(46,71,53)(47,72,54)(48,73,55)(49,74,56)(50,75,57)(51,76,58)(52,77,59)(79,106,98)(80,107,99)(81,108,100)(82,109,101)(83,110,102)(84,111,103)(85,112,104)(86,113,92)(87,114,93)(88,115,94)(89,116,95)(90,117,96)(91,105,97), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,54)(55,65)(56,64)(57,63)(58,62)(59,61)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86)(92,104)(93,103)(94,102)(95,101)(96,100)(97,99)(105,107)(108,117)(109,116)(110,115)(111,114)(112,113)>;
G:=Group( (1,113,54)(2,114,55)(3,115,56)(4,116,57)(5,117,58)(6,105,59)(7,106,60)(8,107,61)(9,108,62)(10,109,63)(11,110,64)(12,111,65)(13,112,53)(14,80,66)(15,81,67)(16,82,68)(17,83,69)(18,84,70)(19,85,71)(20,86,72)(21,87,73)(22,88,74)(23,89,75)(24,90,76)(25,91,77)(26,79,78)(27,99,41)(28,100,42)(29,101,43)(30,102,44)(31,103,45)(32,104,46)(33,92,47)(34,93,48)(35,94,49)(36,95,50)(37,96,51)(38,97,52)(39,98,40), (1,33,20)(2,34,21)(3,35,22)(4,36,23)(5,37,24)(6,38,25)(7,39,26)(8,27,14)(9,28,15)(10,29,16)(11,30,17)(12,31,18)(13,32,19)(40,78,60)(41,66,61)(42,67,62)(43,68,63)(44,69,64)(45,70,65)(46,71,53)(47,72,54)(48,73,55)(49,74,56)(50,75,57)(51,76,58)(52,77,59)(79,106,98)(80,107,99)(81,108,100)(82,109,101)(83,110,102)(84,111,103)(85,112,104)(86,113,92)(87,114,93)(88,115,94)(89,116,95)(90,117,96)(91,105,97), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,54)(55,65)(56,64)(57,63)(58,62)(59,61)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86)(92,104)(93,103)(94,102)(95,101)(96,100)(97,99)(105,107)(108,117)(109,116)(110,115)(111,114)(112,113) );
G=PermutationGroup([[(1,113,54),(2,114,55),(3,115,56),(4,116,57),(5,117,58),(6,105,59),(7,106,60),(8,107,61),(9,108,62),(10,109,63),(11,110,64),(12,111,65),(13,112,53),(14,80,66),(15,81,67),(16,82,68),(17,83,69),(18,84,70),(19,85,71),(20,86,72),(21,87,73),(22,88,74),(23,89,75),(24,90,76),(25,91,77),(26,79,78),(27,99,41),(28,100,42),(29,101,43),(30,102,44),(31,103,45),(32,104,46),(33,92,47),(34,93,48),(35,94,49),(36,95,50),(37,96,51),(38,97,52),(39,98,40)], [(1,33,20),(2,34,21),(3,35,22),(4,36,23),(5,37,24),(6,38,25),(7,39,26),(8,27,14),(9,28,15),(10,29,16),(11,30,17),(12,31,18),(13,32,19),(40,78,60),(41,66,61),(42,67,62),(43,68,63),(44,69,64),(45,70,65),(46,71,53),(47,72,54),(48,73,55),(49,74,56),(50,75,57),(51,76,58),(52,77,59),(79,106,98),(80,107,99),(81,108,100),(82,109,101),(83,110,102),(84,111,103),(85,112,104),(86,113,92),(87,114,93),(88,115,94),(89,116,95),(90,117,96),(91,105,97)], [(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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,25),(15,24),(16,23),(17,22),(18,21),(19,20),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(53,54),(55,65),(56,64),(57,63),(58,62),(59,61),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(80,91),(81,90),(82,89),(83,88),(84,87),(85,86),(92,104),(93,103),(94,102),(95,101),(96,100),(97,99),(105,107),(108,117),(109,116),(110,115),(111,114),(112,113)]])
C32×D13 is a maximal subgroup of
C39⋊Dic3
72 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 6A | ··· | 6H | 13A | ··· | 13F | 39A | ··· | 39AV |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 13 | ··· | 13 | 39 | ··· | 39 |
| size | 1 | 13 | 1 | ··· | 1 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D13 | C3×D13 |
| kernel | C32×D13 | C3×C39 | C3×D13 | C39 | C32 | C3 |
| # reps | 1 | 1 | 8 | 8 | 6 | 48 |
Matrix representation of C32×D13 ►in GL4(𝔽79) generated by
| 1 | 0 | 0 | 0 |
| 0 | 55 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 23 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 78 | 18 |
| 78 | 0 | 0 | 0 |
| 0 | 78 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(79))| [1,0,0,0,0,55,0,0,0,0,1,0,0,0,0,1],[23,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,78,0,0,1,18],[78,0,0,0,0,78,0,0,0,0,0,1,0,0,1,0] >;
C32×D13 in GAP, Magma, Sage, TeX
C_3^2\times D_{13} % in TeX
G:=Group("C3^2xD13"); // GroupNames label
G:=SmallGroup(234,11);
// by ID
G=gap.SmallGroup(234,11);
# by ID
G:=PCGroup([4,-2,-3,-3,-13,3459]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^13=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export