direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C6×C13⋊C6, C13⋊C62, C78⋊2C6, D26⋊C32, C26⋊(C3×C6), D13⋊(C3×C6), (C6×D13)⋊C3, C39⋊3(C2×C6), (C3×D13)⋊2C6, (C2×C13⋊C3)⋊C6, C13⋊C3⋊(C2×C6), (C6×C13⋊C3)⋊2C2, (C3×C13⋊C3)⋊3C22, SmallGroup(468,33)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C6×C13⋊C6 |
Generators and relations for C6×C13⋊C6
G = < a,b,c | a6=b13=c6=1, ab=ba, ac=ca, cbc-1=b10 >
Subgroups: 372 in 60 conjugacy classes, 34 normal (13 characteristic)
C1, C2, C2, C3, C3, C22, C6, C6, C32, C2×C6, C13, C3×C6, D13, C26, C62, C13⋊C3, C39, D26, C13⋊C6, C2×C13⋊C3, C3×D13, C78, C3×C13⋊C3, C2×C13⋊C6, C6×D13, C3×C13⋊C6, C6×C13⋊C3, C6×C13⋊C6
Quotients: C1, C2, C3, C22, C6, C32, C2×C6, C3×C6, C62, C13⋊C6, C2×C13⋊C6, C3×C13⋊C6, C6×C13⋊C6
►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 66 14 40 27 53)(2 70 17 52 36 63)(3 74 20 51 32 60)(4 78 23 50 28 57)(5 69 26 49 37 54)(6 73 16 48 33 64)(7 77 19 47 29 61)(8 68 22 46 38 58)(9 72 25 45 34 55)(10 76 15 44 30 65)(11 67 18 43 39 62)(12 71 21 42 35 59)(13 75 24 41 31 56)
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), (1,66,14,40,27,53)(2,70,17,52,36,63)(3,74,20,51,32,60)(4,78,23,50,28,57)(5,69,26,49,37,54)(6,73,16,48,33,64)(7,77,19,47,29,61)(8,68,22,46,38,58)(9,72,25,45,34,55)(10,76,15,44,30,65)(11,67,18,43,39,62)(12,71,21,42,35,59)(13,75,24,41,31,56)>;
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), (1,66,14,40,27,53)(2,70,17,52,36,63)(3,74,20,51,32,60)(4,78,23,50,28,57)(5,69,26,49,37,54)(6,73,16,48,33,64)(7,77,19,47,29,61)(8,68,22,46,38,58)(9,72,25,45,34,55)(10,76,15,44,30,65)(11,67,18,43,39,62)(12,71,21,42,35,59)(13,75,24,41,31,56) );
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)], [(1,66,14,40,27,53),(2,70,17,52,36,63),(3,74,20,51,32,60),(4,78,23,50,28,57),(5,69,26,49,37,54),(6,73,16,48,33,64),(7,77,19,47,29,61),(8,68,22,46,38,58),(9,72,25,45,34,55),(10,76,15,44,30,65),(11,67,18,43,39,62),(12,71,21,42,35,59),(13,75,24,41,31,56)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3H | 6A | 6B | 6C | ··· | 6X | 13A | 13B | 26A | 26B | 39A | 39B | 39C | 39D | 78A | 78B | 78C | 78D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 6 | 6 | 6 | ··· | 6 | 13 | 13 | 26 | 26 | 39 | 39 | 39 | 39 | 78 | 78 | 78 | 78 |
| size | 1 | 1 | 13 | 13 | 1 | 1 | 13 | ··· | 13 | 1 | 1 | 13 | ··· | 13 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C3 | C3 | C6 | C6 | C6 | C6 | C13⋊C6 | C2×C13⋊C6 | C3×C13⋊C6 | C6×C13⋊C6 |
| kernel | C6×C13⋊C6 | C3×C13⋊C6 | C6×C13⋊C3 | C2×C13⋊C6 | C6×D13 | C13⋊C6 | C2×C13⋊C3 | C3×D13 | C78 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 6 | 2 | 12 | 6 | 4 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C6×C13⋊C6 ►in GL7(𝔽79)
| 23 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 78 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 78 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 78 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 78 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 78 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 78 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 77 | 16 | 77 | 15 | 78 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 56 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 78 | 0 | 0 | 0 | 0 | 0 |
| 0 | 66 | 50 | 2 | 64 | 67 | 65 |
| 0 | 15 | 13 | 15 | 78 | 14 | 14 |
| 0 | 0 | 0 | 0 | 0 | 78 | 0 |
| 0 | 0 | 78 | 0 | 0 | 0 | 0 |
| 0 | 14 | 14 | 78 | 15 | 13 | 15 |
G:=sub<GL(7,GF(79))| [23,0,0,0,0,0,0,0,78,0,0,0,0,0,0,0,78,0,0,0,0,0,0,0,78,0,0,0,0,0,0,0,78,0,0,0,0,0,0,0,78,0,0,0,0,0,0,0,78],[1,0,0,0,0,0,0,0,15,1,0,0,0,0,0,77,0,1,0,0,0,0,16,0,0,1,0,0,0,77,0,0,0,1,0,0,15,0,0,0,0,1,0,78,0,0,0,0,0],[56,0,0,0,0,0,0,0,78,66,15,0,0,14,0,0,50,13,0,78,14,0,0,2,15,0,0,78,0,0,64,78,0,0,15,0,0,67,14,78,0,13,0,0,65,14,0,0,15] >;
C6×C13⋊C6 in GAP, Magma, Sage, TeX
C_6\times C_{13}\rtimes C_6 % in TeX
G:=Group("C6xC13:C6"); // GroupNames label
G:=SmallGroup(468,33);
// by ID
G=gap.SmallGroup(468,33);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-13,10804,689]);
// Polycyclic
G:=Group<a,b,c|a^6=b^13=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^10>;
// generators/relations