direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D17, C68⋊2C2, C2.1D34, D34.2C2, Dic17⋊2C2, C34.2C22, C17⋊2(C2×C4), SmallGroup(136,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C4×D17 |
Generators and relations for C4×D17
G = < a,b,c | a4=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D17
►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)]])
C4×D17 is a maximal subgroup of
C8⋊D17 C68.C4 D34.4C4 C68⋊C4 D68⋊5C2 D4⋊2D17 D68⋊C2 D51⋊2C4
C4×D17 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 | C4×D17 |
| kernel | C4×D17 | Dic17 | C68 | D34 | D17 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 16 |
Matrix representation of C4×D17 ►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] >;
C4×D17 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