metabelian, soluble, monomial, A-group
Aliases: C32⋊Dic13, (C3×C39)⋊6C4, C3⋊S3.D13, C13⋊2(C32⋊C4), (C13×C3⋊S3).2C2, SmallGroup(468,40)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C39 — C32⋊Dic13 |
Generators and relations for C32⋊Dic13
G = < a,b,c,d | a3=b3=c26=1, d2=c13, ab=ba, cac-1=a-1, dad-1=ab-1, cbc-1=b-1, dbd-1=a-1b-1, dcd-1=c-1 >
Smallest permutation representation of C32⋊Dic13
►On 78 points
Generators in S78
(1 76 63)(2 64 77)(3 78 65)(4 66 53)(5 54 67)(6 68 55)(7 56 69)(8 70 57)(9 58 71)(10 72 59)(11 60 73)(12 74 61)(13 62 75)(14 38 51)(15 52 39)(16 40 27)(17 28 41)(18 42 29)(19 30 43)(20 44 31)(21 32 45)(22 46 33)(23 34 47)(24 48 35)(25 36 49)(26 50 37)
(1 63 76)(2 77 64)(3 65 78)(4 53 66)(5 67 54)(6 55 68)(7 69 56)(8 57 70)(9 71 58)(10 59 72)(11 73 60)(12 61 74)(13 75 62)
(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 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 26)(8 25)(9 24)(10 23)(11 22)(12 21)(13 20)(27 53 40 66)(28 78 41 65)(29 77 42 64)(30 76 43 63)(31 75 44 62)(32 74 45 61)(33 73 46 60)(34 72 47 59)(35 71 48 58)(36 70 49 57)(37 69 50 56)(38 68 51 55)(39 67 52 54)
G:=sub<Sym(78)| (1,76,63)(2,64,77)(3,78,65)(4,66,53)(5,54,67)(6,68,55)(7,56,69)(8,70,57)(9,58,71)(10,72,59)(11,60,73)(12,74,61)(13,62,75)(14,38,51)(15,52,39)(16,40,27)(17,28,41)(18,42,29)(19,30,43)(20,44,31)(21,32,45)(22,46,33)(23,34,47)(24,48,35)(25,36,49)(26,50,37), (1,63,76)(2,77,64)(3,65,78)(4,53,66)(5,67,54)(6,55,68)(7,69,56)(8,57,70)(9,71,58)(10,59,72)(11,73,60)(12,61,74)(13,75,62), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(27,53,40,66)(28,78,41,65)(29,77,42,64)(30,76,43,63)(31,75,44,62)(32,74,45,61)(33,73,46,60)(34,72,47,59)(35,71,48,58)(36,70,49,57)(37,69,50,56)(38,68,51,55)(39,67,52,54)>;
G:=Group( (1,76,63)(2,64,77)(3,78,65)(4,66,53)(5,54,67)(6,68,55)(7,56,69)(8,70,57)(9,58,71)(10,72,59)(11,60,73)(12,74,61)(13,62,75)(14,38,51)(15,52,39)(16,40,27)(17,28,41)(18,42,29)(19,30,43)(20,44,31)(21,32,45)(22,46,33)(23,34,47)(24,48,35)(25,36,49)(26,50,37), (1,63,76)(2,77,64)(3,65,78)(4,53,66)(5,67,54)(6,55,68)(7,69,56)(8,57,70)(9,71,58)(10,59,72)(11,73,60)(12,61,74)(13,75,62), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(27,53,40,66)(28,78,41,65)(29,77,42,64)(30,76,43,63)(31,75,44,62)(32,74,45,61)(33,73,46,60)(34,72,47,59)(35,71,48,58)(36,70,49,57)(37,69,50,56)(38,68,51,55)(39,67,52,54) );
G=PermutationGroup([[(1,76,63),(2,64,77),(3,78,65),(4,66,53),(5,54,67),(6,68,55),(7,56,69),(8,70,57),(9,58,71),(10,72,59),(11,60,73),(12,74,61),(13,62,75),(14,38,51),(15,52,39),(16,40,27),(17,28,41),(18,42,29),(19,30,43),(20,44,31),(21,32,45),(22,46,33),(23,34,47),(24,48,35),(25,36,49),(26,50,37)], [(1,63,76),(2,77,64),(3,65,78),(4,53,66),(5,67,54),(6,55,68),(7,69,56),(8,57,70),(9,71,58),(10,59,72),(11,73,60),(12,61,74),(13,75,62)], [(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)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,26),(8,25),(9,24),(10,23),(11,22),(12,21),(13,20),(27,53,40,66),(28,78,41,65),(29,77,42,64),(30,76,43,63),(31,75,44,62),(32,74,45,61),(33,73,46,60),(34,72,47,59),(35,71,48,58),(36,70,49,57),(37,69,50,56),(38,68,51,55),(39,67,52,54)]])
42 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 13A | ··· | 13F | 26A | ··· | 26F | 39A | ··· | 39X |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 |
| size | 1 | 9 | 4 | 4 | 117 | 117 | 2 | ··· | 2 | 18 | ··· | 18 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | - | + | ||
| image | C1 | C2 | C4 | D13 | Dic13 | C32⋊C4 | C32⋊Dic13 |
| kernel | C32⋊Dic13 | C13×C3⋊S3 | C3×C39 | C3⋊S3 | C32 | C13 | C1 |
| # reps | 1 | 1 | 2 | 6 | 6 | 2 | 24 |
Matrix representation of C32⋊Dic13 ►in GL6(𝔽157)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 155 | 112 | 0 | 0 |
| 0 | 0 | 21 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 45 |
| 0 | 0 | 0 | 0 | 136 | 155 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 155 | 112 |
| 0 | 0 | 0 | 0 | 21 | 1 |
| 0 | 156 | 0 | 0 | 0 | 0 |
| 1 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 156 | 112 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 156 | 112 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 62 | 62 | 0 | 0 | 0 | 0 |
| 57 | 95 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 156 | 112 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(157))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,155,21,0,0,0,0,112,1,0,0,0,0,0,0,1,136,0,0,0,0,45,155],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,155,21,0,0,0,0,112,1],[0,1,0,0,0,0,156,33,0,0,0,0,0,0,156,0,0,0,0,0,112,1,0,0,0,0,0,0,156,0,0,0,0,0,112,1],[62,57,0,0,0,0,62,95,0,0,0,0,0,0,0,0,156,0,0,0,0,0,112,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
C32⋊Dic13 in GAP, Magma, Sage, TeX
C_3^2\rtimes {\rm Dic}_{13} % in TeX
G:=Group("C3^2:Dic13"); // GroupNames label
G:=SmallGroup(468,40);
// by ID
G=gap.SmallGroup(468,40);
# by ID
G:=PCGroup([5,-2,-2,-3,3,-13,10,302,67,323,248,10804]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^26=1,d^2=c^13,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^-1,c*b*c^-1=b^-1,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^-1>;
// generators/relations
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