metacyclic, supersoluble, monomial
Aliases: C9⋊3F7, C63⋊10C6, D7⋊13- 1+2, C7⋊C9⋊1C6, C7⋊C18⋊1C3, (C9×D7)⋊2C3, C63⋊3C3⋊3C2, C3.3(C3×F7), C21.2(C3×C6), (C3×F7).1C3, (C3×D7).2C32, C7⋊1(C2×3- 1+2), (C3×C7⋊C3).1C6, SmallGroup(378,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9⋊3F7
G = < a,b,c | a9=b7=c6=1, ab=ba, cac-1=a7, cbc-1=b5 >
Smallest permutation representation of C9⋊3F7
►On 63 points
Generators in S63
(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)
(1 48 45 17 27 35 56)(2 49 37 18 19 36 57)(3 50 38 10 20 28 58)(4 51 39 11 21 29 59)(5 52 40 12 22 30 60)(6 53 41 13 23 31 61)(7 54 42 14 24 32 62)(8 46 43 15 25 33 63)(9 47 44 16 26 34 55)
(2 5 8)(3 9 6)(10 44 61 20 34 53)(11 39 59 21 29 51)(12 43 57 22 33 49)(13 38 55 23 28 47)(14 42 62 24 32 54)(15 37 60 25 36 52)(16 41 58 26 31 50)(17 45 56 27 35 48)(18 40 63 19 30 46)
G:=sub<Sym(63)| (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), (1,48,45,17,27,35,56)(2,49,37,18,19,36,57)(3,50,38,10,20,28,58)(4,51,39,11,21,29,59)(5,52,40,12,22,30,60)(6,53,41,13,23,31,61)(7,54,42,14,24,32,62)(8,46,43,15,25,33,63)(9,47,44,16,26,34,55), (2,5,8)(3,9,6)(10,44,61,20,34,53)(11,39,59,21,29,51)(12,43,57,22,33,49)(13,38,55,23,28,47)(14,42,62,24,32,54)(15,37,60,25,36,52)(16,41,58,26,31,50)(17,45,56,27,35,48)(18,40,63,19,30,46)>;
G:=Group( (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), (1,48,45,17,27,35,56)(2,49,37,18,19,36,57)(3,50,38,10,20,28,58)(4,51,39,11,21,29,59)(5,52,40,12,22,30,60)(6,53,41,13,23,31,61)(7,54,42,14,24,32,62)(8,46,43,15,25,33,63)(9,47,44,16,26,34,55), (2,5,8)(3,9,6)(10,44,61,20,34,53)(11,39,59,21,29,51)(12,43,57,22,33,49)(13,38,55,23,28,47)(14,42,62,24,32,54)(15,37,60,25,36,52)(16,41,58,26,31,50)(17,45,56,27,35,48)(18,40,63,19,30,46) );
G=PermutationGroup([[(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)], [(1,48,45,17,27,35,56),(2,49,37,18,19,36,57),(3,50,38,10,20,28,58),(4,51,39,11,21,29,59),(5,52,40,12,22,30,60),(6,53,41,13,23,31,61),(7,54,42,14,24,32,62),(8,46,43,15,25,33,63),(9,47,44,16,26,34,55)], [(2,5,8),(3,9,6),(10,44,61,20,34,53),(11,39,59,21,29,51),(12,43,57,22,33,49),(13,38,55,23,28,47),(14,42,62,24,32,54),(15,37,60,25,36,52),(16,41,58,26,31,50),(17,45,56,27,35,48),(18,40,63,19,30,46)]])
31 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 6A | 6B | 6C | 6D | 7 | 9A | 9B | 9C | 9D | 9E | 9F | 18A | ··· | 18F | 21A | 21B | 63A | ··· | 63F |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 7 | 9 | 9 | 9 | 9 | 9 | 9 | 18 | ··· | 18 | 21 | 21 | 63 | ··· | 63 |
| size | 1 | 7 | 1 | 1 | 21 | 21 | 7 | 7 | 21 | 21 | 6 | 3 | 3 | 21 | 21 | 21 | 21 | 21 | ··· | 21 | 6 | 6 | 6 | ··· | 6 |
31 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 6 | 6 | 6 |
| type | + | + | + | ||||||||||
| image | C1 | C2 | C3 | C3 | C3 | C6 | C6 | C6 | 3- 1+2 | C2×3- 1+2 | F7 | C3×F7 | C9⋊3F7 |
| kernel | C9⋊3F7 | C63⋊3C3 | C7⋊C18 | C9×D7 | C3×F7 | C7⋊C9 | C63 | C3×C7⋊C3 | D7 | C7 | C9 | C3 | C1 |
| # reps | 1 | 1 | 4 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 1 | 2 | 6 |
Matrix representation of C9⋊3F7 ►in GL6(𝔽127)
| 26 | 0 | 38 | 40 | 40 | 38 |
| 89 | 115 | 89 | 0 | 2 | 2 |
| 125 | 87 | 113 | 87 | 125 | 0 |
| 0 | 125 | 87 | 113 | 87 | 125 |
| 2 | 2 | 0 | 89 | 115 | 89 |
| 38 | 40 | 40 | 38 | 0 | 26 |
| 126 | 126 | 126 | 126 | 126 | 126 |
| 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 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 126 | 126 | 126 | 126 | 126 | 126 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(127))| [26,89,125,0,2,38,0,115,87,125,2,40,38,89,113,87,0,40,40,0,87,113,89,38,40,2,125,87,115,0,38,2,0,125,89,26],[126,1,0,0,0,0,126,0,1,0,0,0,126,0,0,1,0,0,126,0,0,0,1,0,126,0,0,0,0,1,126,0,0,0,0,0],[1,0,0,0,126,0,0,0,0,1,126,0,0,0,0,0,126,0,0,0,1,0,126,0,0,0,0,0,126,1,0,1,0,0,126,0] >;
C9⋊3F7 in GAP, Magma, Sage, TeX
C_9\rtimes_3F_7
% in TeX
G:=Group("C9:3F7"); // GroupNames label
G:=SmallGroup(378,8);
// by ID
G=gap.SmallGroup(378,8);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-7,187,57,8104,2709]);
// Polycyclic
G:=Group<a,b,c|a^9=b^7=c^6=1,a*b=b*a,c*a*c^-1=a^7,c*b*c^-1=b^5>;
// generators/relations
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