direct product, metacyclic, supersoluble, monomial, A-group, 3-hyperelementary
Aliases: C6xC13:C3, C78:C3, C26:C32, C39:4C6, C13:2(C3xC6), SmallGroup(234,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C6xC13:C3 |
Generators and relations for C6xC13:C3
G = < a,b,c | a6=b13=c3=1, ab=ba, ac=ca, cbc-1=b9 >
Quotients: C1, C2, C3, C6, C32, C3xC6, C13:C3, C2xC13:C3, C3xC13:C3, C6xC13:C3
Smallest permutation representation of C6xC13:C3
►On 78 points
Generators in S78
(1 53 27 40 14 66)(2 54 28 41 15 67)(3 55 29 42 16 68)(4 56 30 43 17 69)(5 57 31 44 18 70)(6 58 32 45 19 71)(7 59 33 46 20 72)(8 60 34 47 21 73)(9 61 35 48 22 74)(10 62 36 49 23 75)(11 63 37 50 24 76)(12 64 38 51 25 77)(13 65 39 52 26 78)
(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 27 14)(2 30 23)(3 33 19)(4 36 15)(5 39 24)(6 29 20)(7 32 16)(8 35 25)(9 38 21)(10 28 17)(11 31 26)(12 34 22)(13 37 18)(40 66 53)(41 69 62)(42 72 58)(43 75 54)(44 78 63)(45 68 59)(46 71 55)(47 74 64)(48 77 60)(49 67 56)(50 70 65)(51 73 61)(52 76 57)
G:=sub<Sym(78)| (1,53,27,40,14,66)(2,54,28,41,15,67)(3,55,29,42,16,68)(4,56,30,43,17,69)(5,57,31,44,18,70)(6,58,32,45,19,71)(7,59,33,46,20,72)(8,60,34,47,21,73)(9,61,35,48,22,74)(10,62,36,49,23,75)(11,63,37,50,24,76)(12,64,38,51,25,77)(13,65,39,52,26,78), (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), (1,27,14)(2,30,23)(3,33,19)(4,36,15)(5,39,24)(6,29,20)(7,32,16)(8,35,25)(9,38,21)(10,28,17)(11,31,26)(12,34,22)(13,37,18)(40,66,53)(41,69,62)(42,72,58)(43,75,54)(44,78,63)(45,68,59)(46,71,55)(47,74,64)(48,77,60)(49,67,56)(50,70,65)(51,73,61)(52,76,57)>;
G:=Group( (1,53,27,40,14,66)(2,54,28,41,15,67)(3,55,29,42,16,68)(4,56,30,43,17,69)(5,57,31,44,18,70)(6,58,32,45,19,71)(7,59,33,46,20,72)(8,60,34,47,21,73)(9,61,35,48,22,74)(10,62,36,49,23,75)(11,63,37,50,24,76)(12,64,38,51,25,77)(13,65,39,52,26,78), (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), (1,27,14)(2,30,23)(3,33,19)(4,36,15)(5,39,24)(6,29,20)(7,32,16)(8,35,25)(9,38,21)(10,28,17)(11,31,26)(12,34,22)(13,37,18)(40,66,53)(41,69,62)(42,72,58)(43,75,54)(44,78,63)(45,68,59)(46,71,55)(47,74,64)(48,77,60)(49,67,56)(50,70,65)(51,73,61)(52,76,57) );
G=PermutationGroup([[(1,53,27,40,14,66),(2,54,28,41,15,67),(3,55,29,42,16,68),(4,56,30,43,17,69),(5,57,31,44,18,70),(6,58,32,45,19,71),(7,59,33,46,20,72),(8,60,34,47,21,73),(9,61,35,48,22,74),(10,62,36,49,23,75),(11,63,37,50,24,76),(12,64,38,51,25,77),(13,65,39,52,26,78)], [(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)], [(1,27,14),(2,30,23),(3,33,19),(4,36,15),(5,39,24),(6,29,20),(7,32,16),(8,35,25),(9,38,21),(10,28,17),(11,31,26),(12,34,22),(13,37,18),(40,66,53),(41,69,62),(42,72,58),(43,75,54),(44,78,63),(45,68,59),(46,71,55),(47,74,64),(48,77,60),(49,67,56),(50,70,65),(51,73,61),(52,76,57)]])
C6xC13:C3 is a maximal subgroup of
C39:3C12
42 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3H | 6A | 6B | 6C | ··· | 6H | 13A | 13B | 13C | 13D | 26A | 26B | 26C | 26D | 39A | ··· | 39H | 78A | ··· | 78H |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 6 | 6 | 6 | ··· | 6 | 13 | 13 | 13 | 13 | 26 | 26 | 26 | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 1 | 1 | 13 | ··· | 13 | 1 | 1 | 13 | ··· | 13 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C13:C3 | C2xC13:C3 | C3xC13:C3 | C6xC13:C3 |
| kernel | C6xC13:C3 | C3xC13:C3 | C2xC13:C3 | C78 | C13:C3 | C39 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 4 | 4 | 8 | 8 |
Matrix representation of C6xC13:C3 ►in GL3(F79) generated by
| 24 | 0 | 0 |
| 0 | 24 | 0 |
| 0 | 0 | 24 |
| 68 | 4 | 1 |
| 4 | 42 | 50 |
| 56 | 74 | 38 |
| 15 | 12 | 3 |
| 64 | 8 | 49 |
| 64 | 43 | 56 |
G:=sub<GL(3,GF(79))| [24,0,0,0,24,0,0,0,24],[68,4,56,4,42,74,1,50,38],[15,64,64,12,8,43,3,49,56] >;
C6xC13:C3 in GAP, Magma, Sage, TeX
C_6\times C_{13}\rtimes C_3 % in TeX
G:=Group("C6xC13:C3"); // GroupNames label
G:=SmallGroup(234,10);
// by ID
G=gap.SmallGroup(234,10);
# by ID
G:=PCGroup([4,-2,-3,-3,-13,439]);
// Polycyclic
G:=Group<a,b,c|a^6=b^13=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations
Export