metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C17⋊2D4, C22⋊D17, D34⋊2C2, Dic17⋊C2, C2.5D34, C34.5C22, (C2×C34)⋊2C2, SmallGroup(136,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C17⋊D4
G = < a,b,c | a17=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C17⋊D4
►On 68 points
Generators in S68
(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 35 24 55)(2 51 25 54)(3 50 26 53)(4 49 27 52)(5 48 28 68)(6 47 29 67)(7 46 30 66)(8 45 31 65)(9 44 32 64)(10 43 33 63)(11 42 34 62)(12 41 18 61)(13 40 19 60)(14 39 20 59)(15 38 21 58)(16 37 22 57)(17 36 23 56)
(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(31 34)(32 33)(35 55)(36 54)(37 53)(38 52)(39 68)(40 67)(41 66)(42 65)(43 64)(44 63)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)
G:=sub<Sym(68)| (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), (1,35,24,55)(2,51,25,54)(3,50,26,53)(4,49,27,52)(5,48,28,68)(6,47,29,67)(7,46,30,66)(8,45,31,65)(9,44,32,64)(10,43,33,63)(11,42,34,62)(12,41,18,61)(13,40,19,60)(14,39,20,59)(15,38,21,58)(16,37,22,57)(17,36,23,56), (2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(31,34)(32,33)(35,55)(36,54)(37,53)(38,52)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)>;
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)(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68), (1,35,24,55)(2,51,25,54)(3,50,26,53)(4,49,27,52)(5,48,28,68)(6,47,29,67)(7,46,30,66)(8,45,31,65)(9,44,32,64)(10,43,33,63)(11,42,34,62)(12,41,18,61)(13,40,19,60)(14,39,20,59)(15,38,21,58)(16,37,22,57)(17,36,23,56), (2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(31,34)(32,33)(35,55)(36,54)(37,53)(38,52)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56) );
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),(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68)], [(1,35,24,55),(2,51,25,54),(3,50,26,53),(4,49,27,52),(5,48,28,68),(6,47,29,67),(7,46,30,66),(8,45,31,65),(9,44,32,64),(10,43,33,63),(11,42,34,62),(12,41,18,61),(13,40,19,60),(14,39,20,59),(15,38,21,58),(16,37,22,57),(17,36,23,56)], [(2,17),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10),(18,30),(19,29),(20,28),(21,27),(22,26),(23,25),(31,34),(32,33),(35,55),(36,54),(37,53),(38,52),(39,68),(40,67),(41,66),(42,65),(43,64),(44,63),(45,62),(46,61),(47,60),(48,59),(49,58),(50,57),(51,56)]])
C17⋊D4 is a maximal subgroup of
D68⋊5C2 D4×D17 D4⋊2D17 C51⋊D4 C17⋊D12 C51⋊7D4 C17⋊S4
C17⋊D4 is a maximal quotient of
C34.D4 D34⋊C4 D4⋊D17 D4.D17 Q8⋊D17 C17⋊Q16 C23.D17 C51⋊D4 C17⋊D12 C51⋊7D4
37 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 17A | ··· | 17H | 34A | ··· | 34X |
| order | 1 | 2 | 2 | 2 | 4 | 17 | ··· | 17 | 34 | ··· | 34 |
| size | 1 | 1 | 2 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 |
37 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D17 | D34 | C17⋊D4 |
| kernel | C17⋊D4 | Dic17 | D34 | C2×C34 | C17 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 8 | 8 | 16 |
Matrix representation of C17⋊D4 ►in GL2(𝔽137) generated by
| 0 | 1 |
| 136 | 58 |
| 10 | 28 |
| 60 | 127 |
| 1 | 0 |
| 58 | 136 |
G:=sub<GL(2,GF(137))| [0,136,1,58],[10,60,28,127],[1,58,0,136] >;
C17⋊D4 in GAP, Magma, Sage, TeX
C_{17}\rtimes D_4 % in TeX
G:=Group("C17:D4"); // GroupNames label
G:=SmallGroup(136,8);
// by ID
G=gap.SmallGroup(136,8);
# by ID
G:=PCGroup([4,-2,-2,-2,-17,49,2051]);
// Polycyclic
G:=Group<a,b,c|a^17=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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