direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3xC72:3C3, C72:4C32, C21:2(C7:C3), (C7xC21):3C3, C7:3(C3xC7:C3), SmallGroup(441,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C72 — C3xC72:3C3 |
Generators and relations for C3xC72:3C3
G = < a,b,c,d | a3=b7=c7=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b4, dcd-1=c2 >
Quotients: C1, C3, C32, C7:C3, C3xC7:C3, C72:3C3, C3xC72:3C3
Smallest permutation representation of C3xC72:3C3
►On 63 points
Generators in S63
(1 19 12)(2 20 13)(3 21 14)(4 15 8)(5 16 9)(6 17 10)(7 18 11)(22 36 29)(23 37 30)(24 38 31)(25 39 32)(26 40 33)(27 41 34)(28 42 35)(43 57 50)(44 58 51)(45 59 52)(46 60 53)(47 61 54)(48 62 55)(49 63 56)
(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 4 7 3 6 2 5)(8 11 14 10 13 9 12)(15 18 21 17 20 16 19)(22 28 27 26 25 24 23)(29 35 34 33 32 31 30)(36 42 41 40 39 38 37)(43 48 46 44 49 47 45)(50 55 53 51 56 54 52)(57 62 60 58 63 61 59)
(1 45 27)(2 47 24)(3 49 28)(4 44 25)(5 46 22)(6 48 26)(7 43 23)(8 51 32)(9 53 29)(10 55 33)(11 50 30)(12 52 34)(13 54 31)(14 56 35)(15 58 39)(16 60 36)(17 62 40)(18 57 37)(19 59 41)(20 61 38)(21 63 42)
G:=sub<Sym(63)| (1,19,12)(2,20,13)(3,21,14)(4,15,8)(5,16,9)(6,17,10)(7,18,11)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35)(43,57,50)(44,58,51)(45,59,52)(46,60,53)(47,61,54)(48,62,55)(49,63,56), (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), (1,4,7,3,6,2,5)(8,11,14,10,13,9,12)(15,18,21,17,20,16,19)(22,28,27,26,25,24,23)(29,35,34,33,32,31,30)(36,42,41,40,39,38,37)(43,48,46,44,49,47,45)(50,55,53,51,56,54,52)(57,62,60,58,63,61,59), (1,45,27)(2,47,24)(3,49,28)(4,44,25)(5,46,22)(6,48,26)(7,43,23)(8,51,32)(9,53,29)(10,55,33)(11,50,30)(12,52,34)(13,54,31)(14,56,35)(15,58,39)(16,60,36)(17,62,40)(18,57,37)(19,59,41)(20,61,38)(21,63,42)>;
G:=Group( (1,19,12)(2,20,13)(3,21,14)(4,15,8)(5,16,9)(6,17,10)(7,18,11)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35)(43,57,50)(44,58,51)(45,59,52)(46,60,53)(47,61,54)(48,62,55)(49,63,56), (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), (1,4,7,3,6,2,5)(8,11,14,10,13,9,12)(15,18,21,17,20,16,19)(22,28,27,26,25,24,23)(29,35,34,33,32,31,30)(36,42,41,40,39,38,37)(43,48,46,44,49,47,45)(50,55,53,51,56,54,52)(57,62,60,58,63,61,59), (1,45,27)(2,47,24)(3,49,28)(4,44,25)(5,46,22)(6,48,26)(7,43,23)(8,51,32)(9,53,29)(10,55,33)(11,50,30)(12,52,34)(13,54,31)(14,56,35)(15,58,39)(16,60,36)(17,62,40)(18,57,37)(19,59,41)(20,61,38)(21,63,42) );
G=PermutationGroup([[(1,19,12),(2,20,13),(3,21,14),(4,15,8),(5,16,9),(6,17,10),(7,18,11),(22,36,29),(23,37,30),(24,38,31),(25,39,32),(26,40,33),(27,41,34),(28,42,35),(43,57,50),(44,58,51),(45,59,52),(46,60,53),(47,61,54),(48,62,55),(49,63,56)], [(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)], [(1,4,7,3,6,2,5),(8,11,14,10,13,9,12),(15,18,21,17,20,16,19),(22,28,27,26,25,24,23),(29,35,34,33,32,31,30),(36,42,41,40,39,38,37),(43,48,46,44,49,47,45),(50,55,53,51,56,54,52),(57,62,60,58,63,61,59)], [(1,45,27),(2,47,24),(3,49,28),(4,44,25),(5,46,22),(6,48,26),(7,43,23),(8,51,32),(9,53,29),(10,55,33),(11,50,30),(12,52,34),(13,54,31),(14,56,35),(15,58,39),(16,60,36),(17,62,40),(18,57,37),(19,59,41),(20,61,38),(21,63,42)]])
57 conjugacy classes
| class | 1 | 3A | 3B | 3C | ··· | 3H | 7A | ··· | 7P | 21A | ··· | 21AF |
| order | 1 | 3 | 3 | 3 | ··· | 3 | 7 | ··· | 7 | 21 | ··· | 21 |
| size | 1 | 1 | 1 | 49 | ··· | 49 | 3 | ··· | 3 | 3 | ··· | 3 |
57 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | ||||||
| image | C1 | C3 | C3 | C7:C3 | C3xC7:C3 | C72:3C3 | C3xC72:3C3 |
| kernel | C3xC72:3C3 | C72:3C3 | C7xC21 | C21 | C7 | C3 | C1 |
| # reps | 1 | 6 | 2 | 4 | 8 | 12 | 24 |
Matrix representation of C3xC72:3C3 ►in GL3(F43) generated by
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 11 | 0 | 0 |
| 0 | 35 | 0 |
| 0 | 0 | 21 |
| 41 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 4 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(43))| [6,0,0,0,6,0,0,0,6],[11,0,0,0,35,0,0,0,21],[41,0,0,0,16,0,0,0,4],[0,0,1,1,0,0,0,1,0] >;
C3xC72:3C3 in GAP, Magma, Sage, TeX
C_3\times C_7^2\rtimes_3C_3
% in TeX
G:=Group("C3xC7^2:3C3"); // GroupNames label
G:=SmallGroup(441,12);
// by ID
G=gap.SmallGroup(441,12);
# by ID
G:=PCGroup([4,-3,-3,-7,-7,434,2019]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^7=c^7=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^4,d*c*d^-1=c^2>;
// generators/relations
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