direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4xD17, C68:2C2, C2.1D34, D34.2C2, Dic17:2C2, C34.2C22, C17:2(C2xC4), SmallGroup(136,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C4xD17 |
Generators and relations for C4xD17
G = < a,b,c | a4=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >
Quotients: C1, C2, C4, C22, C2xC4, D17, D34, C4xD17
Smallest permutation representation of C4xD17
►On 68 points
Generators in S68
(1 63 20 38)(2 64 21 39)(3 65 22 40)(4 66 23 41)(5 67 24 42)(6 68 25 43)(7 52 26 44)(8 53 27 45)(9 54 28 46)(10 55 29 47)(11 56 30 48)(12 57 31 49)(13 58 32 50)(14 59 33 51)(15 60 34 35)(16 61 18 36)(17 62 19 37)
(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 64 65 66 67 68)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(18 21)(19 20)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(35 40)(36 39)(37 38)(41 51)(42 50)(43 49)(44 48)(45 47)(52 56)(53 55)(57 68)(58 67)(59 66)(60 65)(61 64)(62 63)
G:=sub<Sym(68)| (1,63,20,38)(2,64,21,39)(3,65,22,40)(4,66,23,41)(5,67,24,42)(6,68,25,43)(7,52,26,44)(8,53,27,45)(9,54,28,46)(10,55,29,47)(11,56,30,48)(12,57,31,49)(13,58,32,50)(14,59,33,51)(15,60,34,35)(16,61,18,36)(17,62,19,37), (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)(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,21)(19,20)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(35,40)(36,39)(37,38)(41,51)(42,50)(43,49)(44,48)(45,47)(52,56)(53,55)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)>;
G:=Group( (1,63,20,38)(2,64,21,39)(3,65,22,40)(4,66,23,41)(5,67,24,42)(6,68,25,43)(7,52,26,44)(8,53,27,45)(9,54,28,46)(10,55,29,47)(11,56,30,48)(12,57,31,49)(13,58,32,50)(14,59,33,51)(15,60,34,35)(16,61,18,36)(17,62,19,37), (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)(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,21)(19,20)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(35,40)(36,39)(37,38)(41,51)(42,50)(43,49)(44,48)(45,47)(52,56)(53,55)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63) );
G=PermutationGroup([[(1,63,20,38),(2,64,21,39),(3,65,22,40),(4,66,23,41),(5,67,24,42),(6,68,25,43),(7,52,26,44),(8,53,27,45),(9,54,28,46),(10,55,29,47),(11,56,30,48),(12,57,31,49),(13,58,32,50),(14,59,33,51),(15,60,34,35),(16,61,18,36),(17,62,19,37)], [(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),(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(18,21),(19,20),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(35,40),(36,39),(37,38),(41,51),(42,50),(43,49),(44,48),(45,47),(52,56),(53,55),(57,68),(58,67),(59,66),(60,65),(61,64),(62,63)]])
C4xD17 is a maximal subgroup of
C8:D17 C68.C4 D34.4C4 C68:C4 D68:5C2 D4:2D17 D68:C2 D51:2C4
C4xD17 is a maximal quotient of
C8:D17 C34.D4 D34:C4 D51:2C4
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 17A | ··· | 17H | 34A | ··· | 34H | 68A | ··· | 68P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 17 | 17 | 1 | 1 | 17 | 17 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | D17 | D34 | C4xD17 |
| kernel | C4xD17 | Dic17 | C68 | D34 | D17 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 16 |
Matrix representation of C4xD17 ►in GL3(F137) generated by
| 37 | 0 | 0 |
| 0 | 136 | 0 |
| 0 | 0 | 136 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 136 | 76 |
| 136 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(137))| [37,0,0,0,136,0,0,0,136],[1,0,0,0,0,136,0,1,76],[136,0,0,0,0,1,0,1,0] >;
C4xD17 in GAP, Magma, Sage, TeX
C_4\times D_{17} % in TeX
G:=Group("C4xD17"); // GroupNames label
G:=SmallGroup(136,5);
// by ID
G=gap.SmallGroup(136,5);
# by ID
G:=PCGroup([4,-2,-2,-2,-17,21,2051]);
// Polycyclic
G:=Group<a,b,c|a^4=b^17=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export