direct product, metacyclic, supersoluble, monomial
Aliases: C7×C9⋊C6, C9⋊C42, D9⋊C21, C63⋊7C6, 3- 1+2⋊C14, (C7×D9)⋊3C3, C32.(S3×C7), C3.3(S3×C21), (C3×C21).2S3, C21.17(C3×S3), (C7×3- 1+2)⋊2C2, SmallGroup(378,35)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C7×C9⋊C6 |
Generators and relations for C7×C9⋊C6
G = < a,b,c | a7=b9=c6=1, ab=ba, ac=ca, cbc-1=b2 >
Smallest permutation representation of C7×C9⋊C6
►On 63 points
Generators in S63
(1 62 53 44 35 26 17)(2 63 54 45 36 27 18)(3 55 46 37 28 19 10)(4 56 47 38 29 20 11)(5 57 48 39 30 21 12)(6 58 49 40 31 22 13)(7 59 50 41 32 23 14)(8 60 51 42 33 24 15)(9 61 52 43 34 25 16)
(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)
(2 6 8 9 5 3)(4 7)(10 18 13 15 16 12)(11 14)(19 27 22 24 25 21)(20 23)(28 36 31 33 34 30)(29 32)(37 45 40 42 43 39)(38 41)(46 54 49 51 52 48)(47 50)(55 63 58 60 61 57)(56 59)
G:=sub<Sym(63)| (1,62,53,44,35,26,17)(2,63,54,45,36,27,18)(3,55,46,37,28,19,10)(4,56,47,38,29,20,11)(5,57,48,39,30,21,12)(6,58,49,40,31,22,13)(7,59,50,41,32,23,14)(8,60,51,42,33,24,15)(9,61,52,43,34,25,16), (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), (2,6,8,9,5,3)(4,7)(10,18,13,15,16,12)(11,14)(19,27,22,24,25,21)(20,23)(28,36,31,33,34,30)(29,32)(37,45,40,42,43,39)(38,41)(46,54,49,51,52,48)(47,50)(55,63,58,60,61,57)(56,59)>;
G:=Group( (1,62,53,44,35,26,17)(2,63,54,45,36,27,18)(3,55,46,37,28,19,10)(4,56,47,38,29,20,11)(5,57,48,39,30,21,12)(6,58,49,40,31,22,13)(7,59,50,41,32,23,14)(8,60,51,42,33,24,15)(9,61,52,43,34,25,16), (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), (2,6,8,9,5,3)(4,7)(10,18,13,15,16,12)(11,14)(19,27,22,24,25,21)(20,23)(28,36,31,33,34,30)(29,32)(37,45,40,42,43,39)(38,41)(46,54,49,51,52,48)(47,50)(55,63,58,60,61,57)(56,59) );
G=PermutationGroup([[(1,62,53,44,35,26,17),(2,63,54,45,36,27,18),(3,55,46,37,28,19,10),(4,56,47,38,29,20,11),(5,57,48,39,30,21,12),(6,58,49,40,31,22,13),(7,59,50,41,32,23,14),(8,60,51,42,33,24,15),(9,61,52,43,34,25,16)], [(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)], [(2,6,8,9,5,3),(4,7),(10,18,13,15,16,12),(11,14),(19,27,22,24,25,21),(20,23),(28,36,31,33,34,30),(29,32),(37,45,40,42,43,39),(38,41),(46,54,49,51,52,48),(47,50),(55,63,58,60,61,57),(56,59)]])
70 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 6A | 6B | 7A | ··· | 7F | 9A | 9B | 9C | 14A | ··· | 14F | 21A | ··· | 21F | 21G | ··· | 21R | 42A | ··· | 42L | 63A | ··· | 63R |
| order | 1 | 2 | 3 | 3 | 3 | 6 | 6 | 7 | ··· | 7 | 9 | 9 | 9 | 14 | ··· | 14 | 21 | ··· | 21 | 21 | ··· | 21 | 42 | ··· | 42 | 63 | ··· | 63 |
| size | 1 | 9 | 2 | 3 | 3 | 9 | 9 | 1 | ··· | 1 | 6 | 6 | 6 | 9 | ··· | 9 | 2 | ··· | 2 | 3 | ··· | 3 | 9 | ··· | 9 | 6 | ··· | 6 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 |
| type | + | + | + | + | ||||||||||
| image | C1 | C2 | C3 | C6 | C7 | C14 | C21 | C42 | S3 | C3×S3 | S3×C7 | S3×C21 | C9⋊C6 | C7×C9⋊C6 |
| kernel | C7×C9⋊C6 | C7×3- 1+2 | C7×D9 | C63 | C9⋊C6 | 3- 1+2 | D9 | C9 | C3×C21 | C21 | C32 | C3 | C7 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 | 1 | 2 | 6 | 12 | 1 | 6 |
Matrix representation of C7×C9⋊C6 ►in GL6(𝔽127)
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 126 | 0 | 0 |
| 0 | 0 | 1 | 126 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 126 |
| 0 | 0 | 0 | 0 | 1 | 126 |
| 126 | 1 | 0 | 0 | 0 | 0 |
| 126 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 126 | 0 | 0 |
| 0 | 0 | 0 | 126 | 0 | 0 |
G:=sub<GL(6,GF(127))| [8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[0,0,0,0,126,126,0,0,0,0,1,0,0,1,0,0,0,0,126,126,0,0,0,0,0,0,0,1,0,0,0,0,126,126,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,126,126,0,0,0,1,0,0,0,0,1,0,0,0] >;
C7×C9⋊C6 in GAP, Magma, Sage, TeX
C_7\times C_9\rtimes C_6
% in TeX
G:=Group("C7xC9:C6"); // GroupNames label
G:=SmallGroup(378,35);
// by ID
G=gap.SmallGroup(378,35);
# by ID
G:=PCGroup([5,-2,-3,-7,-3,-3,4203,1688,138,6304]);
// Polycyclic
G:=Group<a,b,c|a^7=b^9=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;
// generators/relations
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