metabelian, soluble, monomial, A-group
Aliases: (C3×C39)⋊1C4, C13⋊(C32⋊C4), C32⋊(C13⋊C4), C3⋊D39.C2, SmallGroup(468,41)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C39 — (C3×C39)⋊C4 |
Generators and relations for (C3×C39)⋊C4
G = < a,b,c | a3=b39=c4=1, ab=ba, cac-1=a-1b13, cbc-1=ab31 >
Smallest permutation representation of (C3×C39)⋊C4
►On 78 points
Generators in S78
(1 37 23)(2 38 24)(3 39 25)(4 27 26)(5 28 14)(6 29 15)(7 30 16)(8 31 17)(9 32 18)(10 33 19)(11 34 20)(12 35 21)(13 36 22)(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)
(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)
(1 40 12 76)(2 61 11 55)(3 43 10 73)(4 64 9 52)(5 46 8 70)(6 67 7 49)(13 58)(14 72 31 44)(15 54 30 62)(16 75 29 41)(17 57 28 59)(18 78 27 77)(19 60 39 56)(20 42 38 74)(21 63 37 53)(22 45 36 71)(23 66 35 50)(24 48 34 68)(25 69 33 47)(26 51 32 65)
G:=sub<Sym(78)| (1,37,23)(2,38,24)(3,39,25)(4,27,26)(5,28,14)(6,29,15)(7,30,16)(8,31,17)(9,32,18)(10,33,19)(11,34,20)(12,35,21)(13,36,22)(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), (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), (1,40,12,76)(2,61,11,55)(3,43,10,73)(4,64,9,52)(5,46,8,70)(6,67,7,49)(13,58)(14,72,31,44)(15,54,30,62)(16,75,29,41)(17,57,28,59)(18,78,27,77)(19,60,39,56)(20,42,38,74)(21,63,37,53)(22,45,36,71)(23,66,35,50)(24,48,34,68)(25,69,33,47)(26,51,32,65)>;
G:=Group( (1,37,23)(2,38,24)(3,39,25)(4,27,26)(5,28,14)(6,29,15)(7,30,16)(8,31,17)(9,32,18)(10,33,19)(11,34,20)(12,35,21)(13,36,22)(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), (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), (1,40,12,76)(2,61,11,55)(3,43,10,73)(4,64,9,52)(5,46,8,70)(6,67,7,49)(13,58)(14,72,31,44)(15,54,30,62)(16,75,29,41)(17,57,28,59)(18,78,27,77)(19,60,39,56)(20,42,38,74)(21,63,37,53)(22,45,36,71)(23,66,35,50)(24,48,34,68)(25,69,33,47)(26,51,32,65) );
G=PermutationGroup([[(1,37,23),(2,38,24),(3,39,25),(4,27,26),(5,28,14),(6,29,15),(7,30,16),(8,31,17),(9,32,18),(10,33,19),(11,34,20),(12,35,21),(13,36,22),(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)], [(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)], [(1,40,12,76),(2,61,11,55),(3,43,10,73),(4,64,9,52),(5,46,8,70),(6,67,7,49),(13,58),(14,72,31,44),(15,54,30,62),(16,75,29,41),(17,57,28,59),(18,78,27,77),(19,60,39,56),(20,42,38,74),(21,63,37,53),(22,45,36,71),(23,66,35,50),(24,48,34,68),(25,69,33,47),(26,51,32,65)]])
33 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 13A | 13B | 13C | 39A | ··· | 39X |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 13 | 13 | 13 | 39 | ··· | 39 |
| size | 1 | 117 | 4 | 4 | 117 | 117 | 4 | 4 | 4 | 4 | ··· | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 4 | 4 | 4 |
| type | + | + | + | + | + | |
| image | C1 | C2 | C4 | C32⋊C4 | C13⋊C4 | (C3×C39)⋊C4 |
| kernel | (C3×C39)⋊C4 | C3⋊D39 | C3×C39 | C13 | C32 | C1 |
| # reps | 1 | 1 | 2 | 2 | 3 | 24 |
Matrix representation of (C3×C39)⋊C4 ►in GL4(𝔽157) generated by
| 4 | 111 | 0 | 0 |
| 38 | 152 | 0 | 0 |
| 44 | 4 | 121 | 111 |
| 117 | 130 | 24 | 35 |
| 132 | 103 | 0 | 0 |
| 147 | 60 | 0 | 0 |
| 131 | 51 | 48 | 58 |
| 8 | 63 | 38 | 95 |
| 133 | 33 | 151 | 31 |
| 101 | 8 | 81 | 154 |
| 3 | 31 | 1 | 94 |
| 56 | 77 | 41 | 15 |
G:=sub<GL(4,GF(157))| [4,38,44,117,111,152,4,130,0,0,121,24,0,0,111,35],[132,147,131,8,103,60,51,63,0,0,48,38,0,0,58,95],[133,101,3,56,33,8,31,77,151,81,1,41,31,154,94,15] >;
(C3×C39)⋊C4 in GAP, Magma, Sage, TeX
(C_3\times C_{39})\rtimes C_4 % in TeX
G:=Group("(C3xC39):C4"); // GroupNames label
G:=SmallGroup(468,41);
// by ID
G=gap.SmallGroup(468,41);
# by ID
G:=PCGroup([5,-2,-2,-3,3,-13,10,422,67,643,248,7204,5409]);
// Polycyclic
G:=Group<a,b,c|a^3=b^39=c^4=1,a*b=b*a,c*a*c^-1=a^-1*b^13,c*b*c^-1=a*b^31>;
// generators/relations
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