direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22×D17, C17⋊C23, C34⋊C22, (C2×C34)⋊3C2, SmallGroup(136,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C22×D17 |
Generators and relations for C22×D17
G = < a,b,c,d | a2=b2=c17=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of C22×D17
►On 68 points
Generators in S68
(1 65)(2 66)(3 67)(4 68)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 64)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 35)(27 36)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 18)(35 55)(36 56)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 52)(50 53)(51 54)
(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 64)(2 63)(3 62)(4 61)(5 60)(6 59)(7 58)(8 57)(9 56)(10 55)(11 54)(12 53)(13 52)(14 68)(15 67)(16 66)(17 65)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 51)(30 50)(31 49)(32 48)(33 47)(34 46)
G:=sub<Sym(68)| (1,65)(2,66)(3,67)(4,68)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,64)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,35)(27,36)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,18)(35,55)(36,56)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,52)(50,53)(51,54), (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), (1,64)(2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,68)(15,67)(16,66)(17,65)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)>;
G:=Group( (1,65)(2,66)(3,67)(4,68)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,64)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,35)(27,36)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,18)(35,55)(36,56)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,52)(50,53)(51,54), (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), (1,64)(2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,68)(15,67)(16,66)(17,65)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46) );
G=PermutationGroup([[(1,65),(2,66),(3,67),(4,68),(5,52),(6,53),(7,54),(8,55),(9,56),(10,57),(11,58),(12,59),(13,60),(14,61),(15,62),(16,63),(17,64),(18,44),(19,45),(20,46),(21,47),(22,48),(23,49),(24,50),(25,51),(26,35),(27,36),(28,37),(29,38),(30,39),(31,40),(32,41),(33,42),(34,43)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,18),(35,55),(36,56),(37,57),(38,58),(39,59),(40,60),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,52),(50,53),(51,54)], [(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)], [(1,64),(2,63),(3,62),(4,61),(5,60),(6,59),(7,58),(8,57),(9,56),(10,55),(11,54),(12,53),(13,52),(14,68),(15,67),(16,66),(17,65),(18,45),(19,44),(20,43),(21,42),(22,41),(23,40),(24,39),(25,38),(26,37),(27,36),(28,35),(29,51),(30,50),(31,49),(32,48),(33,47),(34,46)]])
C22×D17 is a maximal subgroup of
D34⋊C4 D17.D4
C22×D17 is a maximal quotient of D68⋊5C2 D4⋊2D17 D68⋊C2
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 17A | ··· | 17H | 34A | ··· | 34X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 17 | ··· | 17 | 34 | ··· | 34 |
| size | 1 | 1 | 1 | 1 | 17 | 17 | 17 | 17 | 2 | ··· | 2 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | C2 | D17 | D34 |
| kernel | C22×D17 | D34 | C2×C34 | C22 | C2 |
| # reps | 1 | 6 | 1 | 8 | 24 |
Matrix representation of C22×D17 ►in GL3(𝔽103) generated by
| 1 | 0 | 0 |
| 0 | 102 | 0 |
| 0 | 0 | 102 |
| 102 | 0 | 0 |
| 0 | 102 | 0 |
| 0 | 0 | 102 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 102 | 21 |
| 102 | 0 | 0 |
| 0 | 0 | 102 |
| 0 | 102 | 0 |
G:=sub<GL(3,GF(103))| [1,0,0,0,102,0,0,0,102],[102,0,0,0,102,0,0,0,102],[1,0,0,0,0,102,0,1,21],[102,0,0,0,0,102,0,102,0] >;
C22×D17 in GAP, Magma, Sage, TeX
C_2^2\times D_{17} % in TeX
G:=Group("C2^2xD17"); // GroupNames label
G:=SmallGroup(136,14);
// by ID
G=gap.SmallGroup(136,14);
# by ID
G:=PCGroup([4,-2,-2,-2,-17,2051]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^17=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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