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## G = D4×C17order 136 = 23·17

### Direct product of C17 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C17, C4⋊C34, C683C2, C22⋊C34, C34.6C22, (C2×C34)⋊1C2, C2.1(C2×C34), SmallGroup(136,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C17
 Chief series C1 — C2 — C34 — C2×C34 — D4×C17
 Lower central C1 — C2 — D4×C17
 Upper central C1 — C34 — D4×C17

Generators and relations for D4×C17
G = < a,b,c | a17=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D4×C17
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 62 36 24)(2 63 37 25)(3 64 38 26)(4 65 39 27)(5 66 40 28)(6 67 41 29)(7 68 42 30)(8 52 43 31)(9 53 44 32)(10 54 45 33)(11 55 46 34)(12 56 47 18)(13 57 48 19)(14 58 49 20)(15 59 50 21)(16 60 51 22)(17 61 35 23)
(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)

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), (1,62,36,24)(2,63,37,25)(3,64,38,26)(4,65,39,27)(5,66,40,28)(6,67,41,29)(7,68,42,30)(8,52,43,31)(9,53,44,32)(10,54,45,33)(11,55,46,34)(12,56,47,18)(13,57,48,19)(14,58,49,20)(15,59,50,21)(16,60,51,22)(17,61,35,23), (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)>;

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), (1,62,36,24)(2,63,37,25)(3,64,38,26)(4,65,39,27)(5,66,40,28)(6,67,41,29)(7,68,42,30)(8,52,43,31)(9,53,44,32)(10,54,45,33)(11,55,46,34)(12,56,47,18)(13,57,48,19)(14,58,49,20)(15,59,50,21)(16,60,51,22)(17,61,35,23), (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) );

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)], [(1,62,36,24),(2,63,37,25),(3,64,38,26),(4,65,39,27),(5,66,40,28),(6,67,41,29),(7,68,42,30),(8,52,43,31),(9,53,44,32),(10,54,45,33),(11,55,46,34),(12,56,47,18),(13,57,48,19),(14,58,49,20),(15,59,50,21),(16,60,51,22),(17,61,35,23)], [(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)]])

D4×C17 is a maximal subgroup of   D4⋊D17  D4.D17  D42D17

85 conjugacy classes

 class 1 2A 2B 2C 4 17A ··· 17P 34A ··· 34P 34Q ··· 34AV 68A ··· 68P order 1 2 2 2 4 17 ··· 17 34 ··· 34 34 ··· 34 68 ··· 68 size 1 1 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

85 irreducible representations

 dim 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C17 C34 C34 D4 D4×C17 kernel D4×C17 C68 C2×C34 D4 C4 C22 C17 C1 # reps 1 1 2 16 16 32 1 16

Matrix representation of D4×C17 in GL2(𝔽137) generated by

 50 0 0 50
,
 0 136 1 0
,
 1 0 0 136
G:=sub<GL(2,GF(137))| [50,0,0,50],[0,1,136,0],[1,0,0,136] >;

D4×C17 in GAP, Magma, Sage, TeX

D_4\times C_{17}
% in TeX

G:=Group("D4xC17");
// GroupNames label

G:=SmallGroup(136,10);
// by ID

G=gap.SmallGroup(136,10);
# by ID

G:=PCGroup([4,-2,-2,-17,-2,561]);
// Polycyclic

G:=Group<a,b,c|a^17=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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