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## G = C4×D17order 136 = 23·17

### Direct product of C4 and D17

Aliases: C4×D17, C682C2, C2.1D34, D34.2C2, Dic172C2, C34.2C22, C172(C2×C4), SmallGroup(136,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C17 — C4×D17
 Chief series C1 — C17 — C34 — D34 — C4×D17
 Lower central C17 — C4×D17
 Upper central C1 — C4

Generators and relations for C4×D17
G = < a,b,c | a4=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C4×D17
On 68 points
Generators in S68
(1 60 32 38)(2 61 33 39)(3 62 34 40)(4 63 18 41)(5 64 19 42)(6 65 20 43)(7 66 21 44)(8 67 22 45)(9 68 23 46)(10 52 24 47)(11 53 25 48)(12 54 26 49)(13 55 27 50)(14 56 28 51)(15 57 29 35)(16 58 30 36)(17 59 31 37)
(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 28)(19 27)(20 26)(21 25)(22 24)(29 34)(30 33)(31 32)(35 40)(36 39)(37 38)(41 51)(42 50)(43 49)(44 48)(45 47)(52 67)(53 66)(54 65)(55 64)(56 63)(57 62)(58 61)(59 60)

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

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

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

C4×D17 is a maximal subgroup of
C8⋊D17  C68.C4  D34.4C4  C68⋊C4  D685C2  D42D17  D68⋊C2  D512C4
C4×D17 is a maximal quotient of
C8⋊D17  C34.D4  D34⋊C4  D512C4

40 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 17A ··· 17H 34A ··· 34H 68A ··· 68P order 1 2 2 2 4 4 4 4 17 ··· 17 34 ··· 34 68 ··· 68 size 1 1 17 17 1 1 17 17 2 ··· 2 2 ··· 2 2 ··· 2

40 irreducible representations

 dim 1 1 1 1 1 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 D17 D34 C4×D17 kernel C4×D17 Dic17 C68 D34 D17 C4 C2 C1 # reps 1 1 1 1 4 8 8 16

Matrix representation of C4×D17 in GL3(𝔽137) generated by

 37 0 0 0 136 0 0 0 136
,
 1 0 0 0 0 1 0 136 76
,
 136 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(137))| [37,0,0,0,136,0,0,0,136],[1,0,0,0,0,136,0,1,76],[136,0,0,0,0,1,0,1,0] >;

C4×D17 in GAP, Magma, Sage, TeX

C_4\times D_{17}
% in TeX

G:=Group("C4xD17");
// GroupNames label

G:=SmallGroup(136,5);
// by ID

G=gap.SmallGroup(136,5);
# by ID

G:=PCGroup([4,-2,-2,-2,-17,21,2051]);
// Polycyclic

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

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