direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×D17, C4⋊1D34, C68⋊C22, D68⋊3C2, C22⋊1D34, D34⋊2C22, C34.5C23, Dic17⋊1C22, C17⋊2(C2×D4), (C2×C34)⋊C22, (C4×D17)⋊1C2, (D4×C17)⋊2C2, C17⋊D4⋊1C2, (C22×D17)⋊2C2, C2.6(C22×D17), SmallGroup(272,40)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×D17
G = < a,b,c,d | a4=b2=c17=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 470 in 54 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, D4, D4, C23, C2×D4, C17, D17, D17, C34, C34, Dic17, C68, D34, D34, D34, C2×C34, C4×D17, D68, C17⋊D4, D4×C17, C22×D17, D4×D17
Quotients: C1, C2, C22, D4, C23, C2×D4, D17, D34, C22×D17, D4×D17
►On 68 points
Generators in S68
(1 37 25 52)(2 38 26 53)(3 39 27 54)(4 40 28 55)(5 41 29 56)(6 42 30 57)(7 43 31 58)(8 44 32 59)(9 45 33 60)(10 46 34 61)(11 47 18 62)(12 48 19 63)(13 49 20 64)(14 50 21 65)(15 51 22 66)(16 35 23 67)(17 36 24 68)
(35 67)(36 68)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(46 61)(47 62)(48 63)(49 64)(50 65)(51 66)
(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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 25)(35 53)(36 52)(37 68)(38 67)(39 66)(40 65)(41 64)(42 63)(43 62)(44 61)(45 60)(46 59)(47 58)(48 57)(49 56)(50 55)(51 54)
G:=sub<Sym(68)| (1,37,25,52)(2,38,26,53)(3,39,27,54)(4,40,28,55)(5,41,29,56)(6,42,30,57)(7,43,31,58)(8,44,32,59)(9,45,33,60)(10,46,34,61)(11,47,18,62)(12,48,19,63)(13,49,20,64)(14,50,21,65)(15,51,22,66)(16,35,23,67)(17,36,24,68), (35,67)(36,68)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66), (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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(35,53)(36,52)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)>;
G:=Group( (1,37,25,52)(2,38,26,53)(3,39,27,54)(4,40,28,55)(5,41,29,56)(6,42,30,57)(7,43,31,58)(8,44,32,59)(9,45,33,60)(10,46,34,61)(11,47,18,62)(12,48,19,63)(13,49,20,64)(14,50,21,65)(15,51,22,66)(16,35,23,67)(17,36,24,68), (35,67)(36,68)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66), (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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(35,53)(36,52)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54) );
G=PermutationGroup([[(1,37,25,52),(2,38,26,53),(3,39,27,54),(4,40,28,55),(5,41,29,56),(6,42,30,57),(7,43,31,58),(8,44,32,59),(9,45,33,60),(10,46,34,61),(11,47,18,62),(12,48,19,63),(13,49,20,64),(14,50,21,65),(15,51,22,66),(16,35,23,67),(17,36,24,68)], [(35,67),(36,68),(37,52),(38,53),(39,54),(40,55),(41,56),(42,57),(43,58),(44,59),(45,60),(46,61),(47,62),(48,63),(49,64),(50,65),(51,66)], [(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)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,34),(9,33),(10,32),(11,31),(12,30),(13,29),(14,28),(15,27),(16,26),(17,25),(35,53),(36,52),(37,68),(38,67),(39,66),(40,65),(41,64),(42,63),(43,62),(44,61),(45,60),(46,59),(47,58),(48,57),(49,56),(50,55),(51,54)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 17A | ··· | 17H | 34A | ··· | 34H | 34I | ··· | 34X | 68A | ··· | 68H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 17 | ··· | 17 | 34 | ··· | 34 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 2 | 2 | 17 | 17 | 34 | 34 | 2 | 34 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D17 | D34 | D34 | D4×D17 |
| kernel | D4×D17 | C4×D17 | D68 | C17⋊D4 | D4×C17 | C22×D17 | D17 | D4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 8 | 8 | 16 | 8 |
Matrix representation of D4×D17 ►in GL4(𝔽137) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 136 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 136 |
| 7 | 1 | 0 | 0 |
| 23 | 23 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 88 | 40 | 0 | 0 |
| 77 | 49 | 0 | 0 |
| 0 | 0 | 136 | 0 |
| 0 | 0 | 0 | 136 |
G:=sub<GL(4,GF(137))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,136,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,136],[7,23,0,0,1,23,0,0,0,0,1,0,0,0,0,1],[88,77,0,0,40,49,0,0,0,0,136,0,0,0,0,136] >;
D4×D17 in GAP, Magma, Sage, TeX
D_4\times D_{17} % in TeX
G:=Group("D4xD17"); // GroupNames label
G:=SmallGroup(272,40);
// by ID
G=gap.SmallGroup(272,40);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,97,6404]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^17=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations