direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C7×C7⋊C9, C7⋊C63, C21.C21, C72⋊1C9, C21.5(C7⋊C3), (C7×C21).1C3, C3.(C7×C7⋊C3), SmallGroup(441,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C7×C7⋊C9 |
Generators and relations for C7×C7⋊C9
G = < a,b,c | a7=b7=c9=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C7×C7⋊C9
►On 63 points
Generators in S63
(1 21 17 39 32 51 56)(2 22 18 40 33 52 57)(3 23 10 41 34 53 58)(4 24 11 42 35 54 59)(5 25 12 43 36 46 60)(6 26 13 44 28 47 61)(7 27 14 45 29 48 62)(8 19 15 37 30 49 63)(9 20 16 38 31 50 55)
(1 21 17 39 32 51 56)(2 33 22 52 18 57 40)(3 10 34 58 23 41 53)(4 24 11 42 35 54 59)(5 36 25 46 12 60 43)(6 13 28 61 26 44 47)(7 27 14 45 29 48 62)(8 30 19 49 15 63 37)(9 16 31 55 20 38 50)
(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)
G:=sub<Sym(63)| (1,21,17,39,32,51,56)(2,22,18,40,33,52,57)(3,23,10,41,34,53,58)(4,24,11,42,35,54,59)(5,25,12,43,36,46,60)(6,26,13,44,28,47,61)(7,27,14,45,29,48,62)(8,19,15,37,30,49,63)(9,20,16,38,31,50,55), (1,21,17,39,32,51,56)(2,33,22,52,18,57,40)(3,10,34,58,23,41,53)(4,24,11,42,35,54,59)(5,36,25,46,12,60,43)(6,13,28,61,26,44,47)(7,27,14,45,29,48,62)(8,30,19,49,15,63,37)(9,16,31,55,20,38,50), (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)>;
G:=Group( (1,21,17,39,32,51,56)(2,22,18,40,33,52,57)(3,23,10,41,34,53,58)(4,24,11,42,35,54,59)(5,25,12,43,36,46,60)(6,26,13,44,28,47,61)(7,27,14,45,29,48,62)(8,19,15,37,30,49,63)(9,20,16,38,31,50,55), (1,21,17,39,32,51,56)(2,33,22,52,18,57,40)(3,10,34,58,23,41,53)(4,24,11,42,35,54,59)(5,36,25,46,12,60,43)(6,13,28,61,26,44,47)(7,27,14,45,29,48,62)(8,30,19,49,15,63,37)(9,16,31,55,20,38,50), (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) );
G=PermutationGroup([[(1,21,17,39,32,51,56),(2,22,18,40,33,52,57),(3,23,10,41,34,53,58),(4,24,11,42,35,54,59),(5,25,12,43,36,46,60),(6,26,13,44,28,47,61),(7,27,14,45,29,48,62),(8,19,15,37,30,49,63),(9,20,16,38,31,50,55)], [(1,21,17,39,32,51,56),(2,33,22,52,18,57,40),(3,10,34,58,23,41,53),(4,24,11,42,35,54,59),(5,36,25,46,12,60,43),(6,13,28,61,26,44,47),(7,27,14,45,29,48,62),(8,30,19,49,15,63,37),(9,16,31,55,20,38,50)], [(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)]])
105 conjugacy classes
| class | 1 | 3A | 3B | 7A | ··· | 7F | 7G | ··· | 7T | 9A | ··· | 9F | 21A | ··· | 21L | 21M | ··· | 21AN | 63A | ··· | 63AJ |
| order | 1 | 3 | 3 | 7 | ··· | 7 | 7 | ··· | 7 | 9 | ··· | 9 | 21 | ··· | 21 | 21 | ··· | 21 | 63 | ··· | 63 |
| size | 1 | 1 | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 7 | ··· | 7 | 1 | ··· | 1 | 3 | ··· | 3 | 7 | ··· | 7 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | |||||||||
| image | C1 | C3 | C7 | C9 | C21 | C63 | C7⋊C3 | C7⋊C9 | C7×C7⋊C3 | C7×C7⋊C9 |
| kernel | C7×C7⋊C9 | C7×C21 | C7⋊C9 | C72 | C21 | C7 | C21 | C7 | C3 | C1 |
| # reps | 1 | 2 | 6 | 6 | 12 | 36 | 2 | 4 | 12 | 24 |
Matrix representation of C7×C7⋊C9 ►in GL3(𝔽127) generated by
| 16 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 16 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 4 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 19 | 0 | 0 |
G:=sub<GL(3,GF(127))| [16,0,0,0,16,0,0,0,16],[16,0,0,0,2,0,0,0,4],[0,0,19,1,0,0,0,1,0] >;
C7×C7⋊C9 in GAP, Magma, Sage, TeX
C_7\times C_7\rtimes C_9
% in TeX
G:=Group("C7xC7:C9"); // GroupNames label
G:=SmallGroup(441,5);
// by ID
G=gap.SmallGroup(441,5);
# by ID
G:=PCGroup([4,-3,-7,-3,-7,84,2019]);
// Polycyclic
G:=Group<a,b,c|a^7=b^7=c^9=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
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