metabelian, supersoluble, monomial, A-group
Aliases: C3⋊D21, C21⋊1S3, C32⋊2D7, C7⋊(C3⋊S3), (C3×C21)⋊1C2, SmallGroup(126,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C21 — C3⋊D21 |
Generators and relations for C3⋊D21
G = < a,b,c | a3=b21=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Smallest permutation representation of C3⋊D21
►On 63 points
Generators in S63
(1 50 28)(2 51 29)(3 52 30)(4 53 31)(5 54 32)(6 55 33)(7 56 34)(8 57 35)(9 58 36)(10 59 37)(11 60 38)(12 61 39)(13 62 40)(14 63 41)(15 43 42)(16 44 22)(17 45 23)(18 46 24)(19 47 25)(20 48 26)(21 49 27)
(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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 55)(23 54)(24 53)(25 52)(26 51)(27 50)(28 49)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 63)(36 62)(37 61)(38 60)(39 59)(40 58)(41 57)(42 56)
G:=sub<Sym(63)| (1,50,28)(2,51,29)(3,52,30)(4,53,31)(5,54,32)(6,55,33)(7,56,34)(8,57,35)(9,58,36)(10,59,37)(11,60,38)(12,61,39)(13,62,40)(14,63,41)(15,43,42)(16,44,22)(17,45,23)(18,46,24)(19,47,25)(20,48,26)(21,49,27), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,63)(36,62)(37,61)(38,60)(39,59)(40,58)(41,57)(42,56)>;
G:=Group( (1,50,28)(2,51,29)(3,52,30)(4,53,31)(5,54,32)(6,55,33)(7,56,34)(8,57,35)(9,58,36)(10,59,37)(11,60,38)(12,61,39)(13,62,40)(14,63,41)(15,43,42)(16,44,22)(17,45,23)(18,46,24)(19,47,25)(20,48,26)(21,49,27), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,63)(36,62)(37,61)(38,60)(39,59)(40,58)(41,57)(42,56) );
G=PermutationGroup([[(1,50,28),(2,51,29),(3,52,30),(4,53,31),(5,54,32),(6,55,33),(7,56,34),(8,57,35),(9,58,36),(10,59,37),(11,60,38),(12,61,39),(13,62,40),(14,63,41),(15,43,42),(16,44,22),(17,45,23),(18,46,24),(19,47,25),(20,48,26),(21,49,27)], [(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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(22,55),(23,54),(24,53),(25,52),(26,51),(27,50),(28,49),(29,48),(30,47),(31,46),(32,45),(33,44),(34,43),(35,63),(36,62),(37,61),(38,60),(39,59),(40,58),(41,57),(42,56)]])
C3⋊D21 is a maximal subgroup of
D7×C3⋊S3 S3×D21 C32⋊F7 He3⋊D7 C3⋊D63 C32⋊4F7 C33⋊D7
C3⋊D21 is a maximal quotient of C3⋊Dic21 C3⋊D63 C32⋊D21 C33⋊D7
33 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 7A | 7B | 7C | 21A | ··· | 21X |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 7 | 7 | 7 | 21 | ··· | 21 |
| size | 1 | 63 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
33 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | S3 | D7 | D21 |
| kernel | C3⋊D21 | C3×C21 | C21 | C32 | C3 |
| # reps | 1 | 1 | 4 | 3 | 24 |
Matrix representation of C3⋊D21 ►in GL4(𝔽43) generated by
| 4 | 8 | 0 | 0 |
| 35 | 38 | 0 | 0 |
| 0 | 0 | 20 | 6 |
| 0 | 0 | 23 | 22 |
| 27 | 18 | 0 | 0 |
| 25 | 39 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 27 | 18 | 0 | 0 |
| 36 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 29 | 42 |
G:=sub<GL(4,GF(43))| [4,35,0,0,8,38,0,0,0,0,20,23,0,0,6,22],[27,25,0,0,18,39,0,0,0,0,1,0,0,0,0,1],[27,36,0,0,18,16,0,0,0,0,1,29,0,0,0,42] >;
C3⋊D21 in GAP, Magma, Sage, TeX
C_3\rtimes D_{21} % in TeX
G:=Group("C3:D21"); // GroupNames label
G:=SmallGroup(126,15);
// by ID
G=gap.SmallGroup(126,15);
# by ID
G:=PCGroup([4,-2,-3,-3,-7,33,146,1731]);
// Polycyclic
G:=Group<a,b,c|a^3=b^21=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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