metacyclic, supersoluble, monomial
Aliases: C9⋊4F7, C63⋊11C6, D7⋊23- 1+2, C7⋊C9⋊2C6, C7⋊C18⋊2C3, (C9×D7)⋊3C3, C63⋊C3⋊3C2, C3.4(C3×F7), C21.3(C3×C6), (C3×F7).2C3, (C3×D7).3C32, C7⋊2(C2×3- 1+2), (C3×C7⋊C3).2C6, SmallGroup(378,9)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9⋊4F7
G = < a,b,c | a9=b7=c6=1, ab=ba, cac-1=a4, cbc-1=b5 >
Smallest permutation representation of C9⋊4F7
►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 23 59 34 14 44 46)(2 24 60 35 15 45 47)(3 25 61 36 16 37 48)(4 26 62 28 17 38 49)(5 27 63 29 18 39 50)(6 19 55 30 10 40 51)(7 20 56 31 11 41 52)(8 21 57 32 12 42 53)(9 22 58 33 13 43 54)
(2 8 5)(3 6 9)(10 43 25 30 58 48)(11 41 20 31 56 52)(12 39 24 32 63 47)(13 37 19 33 61 51)(14 44 23 34 59 46)(15 42 27 35 57 50)(16 40 22 36 55 54)(17 38 26 28 62 49)(18 45 21 29 60 53)
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), (1,23,59,34,14,44,46)(2,24,60,35,15,45,47)(3,25,61,36,16,37,48)(4,26,62,28,17,38,49)(5,27,63,29,18,39,50)(6,19,55,30,10,40,51)(7,20,56,31,11,41,52)(8,21,57,32,12,42,53)(9,22,58,33,13,43,54), (2,8,5)(3,6,9)(10,43,25,30,58,48)(11,41,20,31,56,52)(12,39,24,32,63,47)(13,37,19,33,61,51)(14,44,23,34,59,46)(15,42,27,35,57,50)(16,40,22,36,55,54)(17,38,26,28,62,49)(18,45,21,29,60,53)>;
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), (1,23,59,34,14,44,46)(2,24,60,35,15,45,47)(3,25,61,36,16,37,48)(4,26,62,28,17,38,49)(5,27,63,29,18,39,50)(6,19,55,30,10,40,51)(7,20,56,31,11,41,52)(8,21,57,32,12,42,53)(9,22,58,33,13,43,54), (2,8,5)(3,6,9)(10,43,25,30,58,48)(11,41,20,31,56,52)(12,39,24,32,63,47)(13,37,19,33,61,51)(14,44,23,34,59,46)(15,42,27,35,57,50)(16,40,22,36,55,54)(17,38,26,28,62,49)(18,45,21,29,60,53) );
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)], [(1,23,59,34,14,44,46),(2,24,60,35,15,45,47),(3,25,61,36,16,37,48),(4,26,62,28,17,38,49),(5,27,63,29,18,39,50),(6,19,55,30,10,40,51),(7,20,56,31,11,41,52),(8,21,57,32,12,42,53),(9,22,58,33,13,43,54)], [(2,8,5),(3,6,9),(10,43,25,30,58,48),(11,41,20,31,56,52),(12,39,24,32,63,47),(13,37,19,33,61,51),(14,44,23,34,59,46),(15,42,27,35,57,50),(16,40,22,36,55,54),(17,38,26,28,62,49),(18,45,21,29,60,53)]])
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⋊4F7 |
| kernel | C9⋊4F7 | C63⋊C3 | 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⋊4F7 ►in GL6(𝔽127)
| 85 | 0 | 121 | 7 | 7 | 121 |
| 6 | 91 | 6 | 0 | 13 | 13 |
| 114 | 120 | 78 | 120 | 114 | 0 |
| 0 | 114 | 120 | 78 | 120 | 114 |
| 13 | 13 | 0 | 6 | 91 | 6 |
| 121 | 7 | 7 | 121 | 0 | 85 |
| 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))| [85,6,114,0,13,121,0,91,120,114,13,7,121,6,78,120,0,7,7,0,120,78,6,121,7,13,114,120,91,0,121,13,0,114,6,85],[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⋊4F7 in GAP, Magma, Sage, TeX
C_9\rtimes_4F_7
% in TeX
G:=Group("C9:4F7"); // GroupNames label
G:=SmallGroup(378,9);
// by ID
G=gap.SmallGroup(378,9);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-7,187,102,8104,2709]);
// Polycyclic
G:=Group<a,b,c|a^9=b^7=c^6=1,a*b=b*a,c*a*c^-1=a^4,c*b*c^-1=b^5>;
// generators/relations
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