direct product, metabelian, soluble, monomial, A-group
Aliases: A4×D17, C17⋊(C2×A4), (C2×C34)⋊C6, (C22×D17)⋊C3, C22⋊(C3×D17), (A4×C17)⋊2C2, SmallGroup(408,38)
Series: Derived ►Chief ►Lower central ►Upper central
| C2×C34 — A4×D17 |
Generators and relations for A4×D17
G = < a,b,c,d,e | a2=b2=c3=d17=e2=1, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Smallest permutation representation of A4×D17
►On 68 points
Generators in S68
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(35 66)(36 67)(37 68)(38 52)(39 53)(40 54)(41 55)(42 56)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(49 63)(50 64)(51 65)
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 51)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 41)(18 56)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(31 52)(32 53)(33 54)(34 55)
(18 42 56)(19 43 57)(20 44 58)(21 45 59)(22 46 60)(23 47 61)(24 48 62)(25 49 63)(26 50 64)(27 51 65)(28 35 66)(29 36 67)(30 37 68)(31 38 52)(32 39 53)(33 40 54)(34 41 55)
(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 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)(25 27)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)(49 51)(52 59)(53 58)(54 57)(55 56)(60 68)(61 67)(62 66)(63 65)
G:=sub<Sym(68)| (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(35,66)(36,67)(37,68)(38,52)(39,53)(40,54)(41,55)(42,56)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,51)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,52)(32,53)(33,54)(34,55), (18,42,56)(19,43,57)(20,44,58)(21,45,59)(22,46,60)(23,47,61)(24,48,62)(25,49,63)(26,50,64)(27,51,65)(28,35,66)(29,36,67)(30,37,68)(31,38,52)(32,39,53)(33,40,54)(34,41,55), (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)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28)(25,27)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(49,51)(52,59)(53,58)(54,57)(55,56)(60,68)(61,67)(62,66)(63,65)>;
G:=Group( (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(35,66)(36,67)(37,68)(38,52)(39,53)(40,54)(41,55)(42,56)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,51)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,52)(32,53)(33,54)(34,55), (18,42,56)(19,43,57)(20,44,58)(21,45,59)(22,46,60)(23,47,61)(24,48,62)(25,49,63)(26,50,64)(27,51,65)(28,35,66)(29,36,67)(30,37,68)(31,38,52)(32,39,53)(33,40,54)(34,41,55), (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)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28)(25,27)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(49,51)(52,59)(53,58)(54,57)(55,56)(60,68)(61,67)(62,66)(63,65) );
G=PermutationGroup([[(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,33),(17,34),(35,66),(36,67),(37,68),(38,52),(39,53),(40,54),(41,55),(42,56),(43,57),(44,58),(45,59),(46,60),(47,61),(48,62),(49,63),(50,64),(51,65)], [(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,49),(9,50),(10,51),(11,35),(12,36),(13,37),(14,38),(15,39),(16,40),(17,41),(18,56),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62),(25,63),(26,64),(27,65),(28,66),(29,67),(30,68),(31,52),(32,53),(33,54),(34,55)], [(18,42,56),(19,43,57),(20,44,58),(21,45,59),(22,46,60),(23,47,61),(24,48,62),(25,49,63),(26,50,64),(27,51,65),(28,35,66),(29,36,67),(30,37,68),(31,38,52),(32,39,53),(33,40,54),(34,41,55)], [(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),(18,34),(19,33),(20,32),(21,31),(22,30),(23,29),(24,28),(25,27),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43),(41,42),(49,51),(52,59),(53,58),(54,57),(55,56),(60,68),(61,67),(62,66),(63,65)]])
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 17A | ··· | 17H | 34A | ··· | 34H | 51A | ··· | 51P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 |
| size | 1 | 3 | 17 | 51 | 4 | 4 | 68 | 68 | 2 | ··· | 2 | 6 | ··· | 6 | 8 | ··· | 8 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 6 |
| type | + | + | + | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D17 | C3×D17 | A4 | C2×A4 | A4×D17 |
| kernel | A4×D17 | A4×C17 | C22×D17 | C2×C34 | A4 | C22 | D17 | C17 | C1 |
| # reps | 1 | 1 | 2 | 2 | 8 | 16 | 1 | 1 | 8 |
Matrix representation of A4×D17 ►in GL5(𝔽103)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 102 | 102 | 102 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 102 | 102 | 102 |
| 0 | 0 | 1 | 0 | 0 |
| 46 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 102 | 102 | 102 |
| 0 | 0 | 0 | 1 | 0 |
| 34 | 1 | 0 | 0 | 0 |
| 102 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(103))| [1,0,0,0,0,0,1,0,0,0,0,0,0,1,102,0,0,1,0,102,0,0,0,0,102],[1,0,0,0,0,0,1,0,0,0,0,0,0,102,1,0,0,0,102,0,0,0,1,102,0],[46,0,0,0,0,0,46,0,0,0,0,0,1,102,0,0,0,0,102,1,0,0,0,102,0],[34,102,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;
A4×D17 in GAP, Magma, Sage, TeX
A_4\times D_{17} % in TeX
G:=Group("A4xD17"); // GroupNames label
G:=SmallGroup(408,38);
// by ID
G=gap.SmallGroup(408,38);
# by ID
G:=PCGroup([5,-2,-3,-2,2,-17,142,68,9604]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^17=e^2=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
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