direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary
Aliases: C7×3- 1+2, C9⋊C21, C63⋊4C3, C32.C21, C21.8C32, (C3×C21).1C3, C3.2(C3×C21), SmallGroup(189,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×3- 1+2
G = < a,b,c | a7=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C7×3- 1+2
►On 63 points
Generators in S63
(1 39 56 52 20 10 30)(2 40 57 53 21 11 31)(3 41 58 54 22 12 32)(4 42 59 46 23 13 33)(5 43 60 47 24 14 34)(6 44 61 48 25 15 35)(7 45 62 49 26 16 36)(8 37 63 50 27 17 28)(9 38 55 51 19 18 29)
(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 8 5)(3 6 9)(11 17 14)(12 15 18)(19 22 25)(21 27 24)(28 34 31)(29 32 35)(37 43 40)(38 41 44)(47 53 50)(48 51 54)(55 58 61)(57 63 60)
G:=sub<Sym(63)| (1,39,56,52,20,10,30)(2,40,57,53,21,11,31)(3,41,58,54,22,12,32)(4,42,59,46,23,13,33)(5,43,60,47,24,14,34)(6,44,61,48,25,15,35)(7,45,62,49,26,16,36)(8,37,63,50,27,17,28)(9,38,55,51,19,18,29), (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), (2,8,5)(3,6,9)(11,17,14)(12,15,18)(19,22,25)(21,27,24)(28,34,31)(29,32,35)(37,43,40)(38,41,44)(47,53,50)(48,51,54)(55,58,61)(57,63,60)>;
G:=Group( (1,39,56,52,20,10,30)(2,40,57,53,21,11,31)(3,41,58,54,22,12,32)(4,42,59,46,23,13,33)(5,43,60,47,24,14,34)(6,44,61,48,25,15,35)(7,45,62,49,26,16,36)(8,37,63,50,27,17,28)(9,38,55,51,19,18,29), (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), (2,8,5)(3,6,9)(11,17,14)(12,15,18)(19,22,25)(21,27,24)(28,34,31)(29,32,35)(37,43,40)(38,41,44)(47,53,50)(48,51,54)(55,58,61)(57,63,60) );
G=PermutationGroup([[(1,39,56,52,20,10,30),(2,40,57,53,21,11,31),(3,41,58,54,22,12,32),(4,42,59,46,23,13,33),(5,43,60,47,24,14,34),(6,44,61,48,25,15,35),(7,45,62,49,26,16,36),(8,37,63,50,27,17,28),(9,38,55,51,19,18,29)], [(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)], [(2,8,5),(3,6,9),(11,17,14),(12,15,18),(19,22,25),(21,27,24),(28,34,31),(29,32,35),(37,43,40),(38,41,44),(47,53,50),(48,51,54),(55,58,61),(57,63,60)]])
C7×3- 1+2 is a maximal subgroup of
D63⋊C3
77 conjugacy classes
| class | 1 | 3A | 3B | 3C | 3D | 7A | ··· | 7F | 9A | ··· | 9F | 21A | ··· | 21L | 21M | ··· | 21X | 63A | ··· | 63AJ |
| order | 1 | 3 | 3 | 3 | 3 | 7 | ··· | 7 | 9 | ··· | 9 | 21 | ··· | 21 | 21 | ··· | 21 | 63 | ··· | 63 |
| size | 1 | 1 | 1 | 3 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 |
77 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||||
| image | C1 | C3 | C3 | C7 | C21 | C21 | 3- 1+2 | C7×3- 1+2 |
| kernel | C7×3- 1+2 | C63 | C3×C21 | 3- 1+2 | C9 | C32 | C7 | C1 |
| # reps | 1 | 6 | 2 | 6 | 36 | 12 | 2 | 12 |
Matrix representation of C7×3- 1+2 ►in GL3(𝔽127) generated by
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 0 | 1 | 0 |
| 0 | 0 | 107 |
| 1 | 0 | 0 |
| 1 | 0 | 0 |
| 0 | 107 | 0 |
| 0 | 0 | 19 |
G:=sub<GL(3,GF(127))| [4,0,0,0,4,0,0,0,4],[0,0,1,1,0,0,0,107,0],[1,0,0,0,107,0,0,0,19] >;
C7×3- 1+2 in GAP, Magma, Sage, TeX
C_7\times 3_-^{1+2} % in TeX
G:=Group("C7xES-(3,1)"); // GroupNames label
G:=SmallGroup(189,11);
// by ID
G=gap.SmallGroup(189,11);
# by ID
G:=PCGroup([4,-3,-3,-7,-3,252,529]);
// Polycyclic
G:=Group<a,b,c|a^7=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
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