direct product, metacyclic, supersoluble, monomial, A-group, 3-hyperelementary
Aliases: C9×C7⋊C3, C63⋊1C3, C21.1C32, C7⋊C9⋊4C3, C7⋊1(C3×C9), C3.1(C3×C7⋊C3), (C3×C7⋊C3).3C3, SmallGroup(189,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C9×C7⋊C3 |
Generators and relations for C9×C7⋊C3
G = < a,b,c | a9=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C9×C7⋊C3
►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 46 29 21 39 16 56)(2 47 30 22 40 17 57)(3 48 31 23 41 18 58)(4 49 32 24 42 10 59)(5 50 33 25 43 11 60)(6 51 34 26 44 12 61)(7 52 35 27 45 13 62)(8 53 36 19 37 14 63)(9 54 28 20 38 15 55)
(1 7 4)(2 8 5)(3 9 6)(10 21 62)(11 22 63)(12 23 55)(13 24 56)(14 25 57)(15 26 58)(16 27 59)(17 19 60)(18 20 61)(28 44 48)(29 45 49)(30 37 50)(31 38 51)(32 39 52)(33 40 53)(34 41 54)(35 42 46)(36 43 47)
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), (1,46,29,21,39,16,56)(2,47,30,22,40,17,57)(3,48,31,23,41,18,58)(4,49,32,24,42,10,59)(5,50,33,25,43,11,60)(6,51,34,26,44,12,61)(7,52,35,27,45,13,62)(8,53,36,19,37,14,63)(9,54,28,20,38,15,55), (1,7,4)(2,8,5)(3,9,6)(10,21,62)(11,22,63)(12,23,55)(13,24,56)(14,25,57)(15,26,58)(16,27,59)(17,19,60)(18,20,61)(28,44,48)(29,45,49)(30,37,50)(31,38,51)(32,39,52)(33,40,53)(34,41,54)(35,42,46)(36,43,47)>;
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), (1,46,29,21,39,16,56)(2,47,30,22,40,17,57)(3,48,31,23,41,18,58)(4,49,32,24,42,10,59)(5,50,33,25,43,11,60)(6,51,34,26,44,12,61)(7,52,35,27,45,13,62)(8,53,36,19,37,14,63)(9,54,28,20,38,15,55), (1,7,4)(2,8,5)(3,9,6)(10,21,62)(11,22,63)(12,23,55)(13,24,56)(14,25,57)(15,26,58)(16,27,59)(17,19,60)(18,20,61)(28,44,48)(29,45,49)(30,37,50)(31,38,51)(32,39,52)(33,40,53)(34,41,54)(35,42,46)(36,43,47) );
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)], [(1,46,29,21,39,16,56),(2,47,30,22,40,17,57),(3,48,31,23,41,18,58),(4,49,32,24,42,10,59),(5,50,33,25,43,11,60),(6,51,34,26,44,12,61),(7,52,35,27,45,13,62),(8,53,36,19,37,14,63),(9,54,28,20,38,15,55)], [(1,7,4),(2,8,5),(3,9,6),(10,21,62),(11,22,63),(12,23,55),(13,24,56),(14,25,57),(15,26,58),(16,27,59),(17,19,60),(18,20,61),(28,44,48),(29,45,49),(30,37,50),(31,38,51),(32,39,52),(33,40,53),(34,41,54),(35,42,46),(36,43,47)]])
C9×C7⋊C3 is a maximal subgroup of
C9⋊5F7
45 conjugacy classes
| class | 1 | 3A | 3B | 3C | ··· | 3H | 7A | 7B | 9A | ··· | 9F | 9G | ··· | 9R | 21A | 21B | 21C | 21D | 63A | ··· | 63L |
| order | 1 | 3 | 3 | 3 | ··· | 3 | 7 | 7 | 9 | ··· | 9 | 9 | ··· | 9 | 21 | 21 | 21 | 21 | 63 | ··· | 63 |
| size | 1 | 1 | 1 | 7 | ··· | 7 | 3 | 3 | 1 | ··· | 1 | 7 | ··· | 7 | 3 | 3 | 3 | 3 | 3 | ··· | 3 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | |||||||
| image | C1 | C3 | C3 | C3 | C9 | C7⋊C3 | C3×C7⋊C3 | C9×C7⋊C3 |
| kernel | C9×C7⋊C3 | C7⋊C9 | C63 | C3×C7⋊C3 | C7⋊C3 | C9 | C3 | C1 |
| # reps | 1 | 4 | 2 | 2 | 18 | 2 | 4 | 12 |
Matrix representation of C9×C7⋊C3 ►in GL3(𝔽127) generated by
| 22 | 0 | 0 |
| 0 | 22 | 0 |
| 0 | 0 | 22 |
| 104 | 105 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 19 | 0 | 0 |
| 37 | 108 | 108 |
| 0 | 19 | 0 |
G:=sub<GL(3,GF(127))| [22,0,0,0,22,0,0,0,22],[104,1,0,105,0,1,1,0,0],[19,37,0,0,108,19,0,108,0] >;
C9×C7⋊C3 in GAP, Magma, Sage, TeX
C_9\times C_7\rtimes C_3
% in TeX
G:=Group("C9xC7:C3"); // GroupNames label
G:=SmallGroup(189,3);
// by ID
G=gap.SmallGroup(189,3);
# by ID
G:=PCGroup([4,-3,-3,-3,-7,29,867]);
// Polycyclic
G:=Group<a,b,c|a^9=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
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