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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

C1C2 — D4×C17
C1C2C34C2×C34 — D4×C17
C1C2 — D4×C17
C1C34 — D4×C17

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

2C2
2C2
2C34
2C34

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

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

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

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

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

85 conjugacy classes

class 1 2A2B2C 4 17A···17P34A···34P34Q···34AV68A···68P
order1222417···1734···3434···3468···68
size112221···11···12···22···2

85 irreducible representations

dim11111122
type++++
imageC1C2C2C17C34C34D4D4×C17
kernelD4×C17C68C2×C34D4C4C22C17C1
# reps112161632116

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

500
050
,
0136
10
,
10
0136
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

Export

Subgroup lattice of D4×C17 in TeX

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