direct product, metabelian, supersoluble, monomial, A-group
Aliases: C7×C3⋊S3, C21⋊3S3, C32⋊2C14, C3⋊(S3×C7), (C3×C21)⋊5C2, SmallGroup(126,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C7×C3⋊S3 |
Generators and relations for C7×C3⋊S3
G = < a,b,c,d | a7=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Smallest permutation representation of C7×C3⋊S3
►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 58 27)(2 59 28)(3 60 22)(4 61 23)(5 62 24)(6 63 25)(7 57 26)(8 53 39)(9 54 40)(10 55 41)(11 56 42)(12 50 36)(13 51 37)(14 52 38)(15 29 43)(16 30 44)(17 31 45)(18 32 46)(19 33 47)(20 34 48)(21 35 49)
(1 44 9)(2 45 10)(3 46 11)(4 47 12)(5 48 13)(6 49 14)(7 43 8)(15 53 57)(16 54 58)(17 55 59)(18 56 60)(19 50 61)(20 51 62)(21 52 63)(22 32 42)(23 33 36)(24 34 37)(25 35 38)(26 29 39)(27 30 40)(28 31 41)
(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 39)(16 40)(17 41)(18 42)(19 36)(20 37)(21 38)(22 60)(23 61)(24 62)(25 63)(26 57)(27 58)(28 59)(29 53)(30 54)(31 55)(32 56)(33 50)(34 51)(35 52)
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), (1,58,27)(2,59,28)(3,60,22)(4,61,23)(5,62,24)(6,63,25)(7,57,26)(8,53,39)(9,54,40)(10,55,41)(11,56,42)(12,50,36)(13,51,37)(14,52,38)(15,29,43)(16,30,44)(17,31,45)(18,32,46)(19,33,47)(20,34,48)(21,35,49), (1,44,9)(2,45,10)(3,46,11)(4,47,12)(5,48,13)(6,49,14)(7,43,8)(15,53,57)(16,54,58)(17,55,59)(18,56,60)(19,50,61)(20,51,62)(21,52,63)(22,32,42)(23,33,36)(24,34,37)(25,35,38)(26,29,39)(27,30,40)(28,31,41), (8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,39)(16,40)(17,41)(18,42)(19,36)(20,37)(21,38)(22,60)(23,61)(24,62)(25,63)(26,57)(27,58)(28,59)(29,53)(30,54)(31,55)(32,56)(33,50)(34,51)(35,52)>;
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), (1,58,27)(2,59,28)(3,60,22)(4,61,23)(5,62,24)(6,63,25)(7,57,26)(8,53,39)(9,54,40)(10,55,41)(11,56,42)(12,50,36)(13,51,37)(14,52,38)(15,29,43)(16,30,44)(17,31,45)(18,32,46)(19,33,47)(20,34,48)(21,35,49), (1,44,9)(2,45,10)(3,46,11)(4,47,12)(5,48,13)(6,49,14)(7,43,8)(15,53,57)(16,54,58)(17,55,59)(18,56,60)(19,50,61)(20,51,62)(21,52,63)(22,32,42)(23,33,36)(24,34,37)(25,35,38)(26,29,39)(27,30,40)(28,31,41), (8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,39)(16,40)(17,41)(18,42)(19,36)(20,37)(21,38)(22,60)(23,61)(24,62)(25,63)(26,57)(27,58)(28,59)(29,53)(30,54)(31,55)(32,56)(33,50)(34,51)(35,52) );
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)], [(1,58,27),(2,59,28),(3,60,22),(4,61,23),(5,62,24),(6,63,25),(7,57,26),(8,53,39),(9,54,40),(10,55,41),(11,56,42),(12,50,36),(13,51,37),(14,52,38),(15,29,43),(16,30,44),(17,31,45),(18,32,46),(19,33,47),(20,34,48),(21,35,49)], [(1,44,9),(2,45,10),(3,46,11),(4,47,12),(5,48,13),(6,49,14),(7,43,8),(15,53,57),(16,54,58),(17,55,59),(18,56,60),(19,50,61),(20,51,62),(21,52,63),(22,32,42),(23,33,36),(24,34,37),(25,35,38),(26,29,39),(27,30,40),(28,31,41)], [(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,49),(15,39),(16,40),(17,41),(18,42),(19,36),(20,37),(21,38),(22,60),(23,61),(24,62),(25,63),(26,57),(27,58),(28,59),(29,53),(30,54),(31,55),(32,56),(33,50),(34,51),(35,52)]])
C7×C3⋊S3 is a maximal subgroup of
C32⋊Dic7 S32×C7 D21⋊S3 C7⋊He3⋊C2
42 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 7A | ··· | 7F | 14A | ··· | 14F | 21A | ··· | 21X |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 |
| size | 1 | 9 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C7 | C14 | S3 | S3×C7 |
| kernel | C7×C3⋊S3 | C3×C21 | C3⋊S3 | C32 | C21 | C3 |
| # reps | 1 | 1 | 6 | 6 | 4 | 24 |
Matrix representation of C7×C3⋊S3 ►in GL4(𝔽43) generated by
| 21 | 0 | 0 | 0 |
| 0 | 21 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 42 | 1 |
| 0 | 0 | 42 | 0 |
| 0 | 42 | 0 | 0 |
| 1 | 42 | 0 | 0 |
| 0 | 0 | 0 | 42 |
| 0 | 0 | 1 | 42 |
| 1 | 42 | 0 | 0 |
| 0 | 42 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 42 |
G:=sub<GL(4,GF(43))| [21,0,0,0,0,21,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,42,42,0,0,1,0],[0,1,0,0,42,42,0,0,0,0,0,1,0,0,42,42],[1,0,0,0,42,42,0,0,0,0,1,1,0,0,0,42] >;
C7×C3⋊S3 in GAP, Magma, Sage, TeX
C_7\times C_3\rtimes S_3
% in TeX
G:=Group("C7xC3:S3"); // GroupNames label
G:=SmallGroup(126,14);
// by ID
G=gap.SmallGroup(126,14);
# by ID
G:=PCGroup([4,-2,-7,-3,-3,338,1347]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
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