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## G = C22×D17order 136 = 23·17

### Direct product of C22 and D17

Aliases: C22×D17, C17⋊C23, C34⋊C22, (C2×C34)⋊3C2, SmallGroup(136,14)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C17 — C22×D17
 Chief series C1 — C17 — D17 — D34 — C22×D17
 Lower central C17 — C22×D17
 Upper central C1 — C22

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   D685C2  D42D17  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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