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## G = A4×D17order 408 = 23·3·17

### Direct product of A4 and D17

Aliases: A4×D17, C17⋊(C2×A4), (C2×C34)⋊C6, (C22×D17)⋊C3, C22⋊(C3×D17), (A4×C17)⋊2C2, SmallGroup(408,38)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C34 — A4×D17
 Chief series C1 — C17 — C2×C34 — A4×C17 — A4×D17
 Lower central C2×C34 — A4×D17
 Upper central C1

Generators and relations for A4×D17
G = < a,b,c,d,e | a2=b2=c3=d17=e2=1, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

3C2
17C2
51C2
4C3
51C22
51C22
68C6
3C34
3D17
4C51
17C23
3D34
3D34
17C2×A4

Smallest permutation representation of A4×D17
On 68 points
Generators in S68
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(35 66)(36 67)(37 68)(38 52)(39 53)(40 54)(41 55)(42 56)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(49 63)(50 64)(51 65)
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 51)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 41)(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)
(18 42 56)(19 43 57)(20 44 58)(21 45 59)(22 46 60)(23 47 61)(24 48 62)(25 49 63)(26 50 64)(27 51 65)(28 35 66)(29 36 67)(30 37 68)(31 38 52)(32 39 53)(33 40 54)(34 41 55)
(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 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)(25 27)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)(49 51)(52 59)(53 58)(54 57)(55 56)(60 68)(61 67)(62 66)(63 65)

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

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

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

40 conjugacy classes

 class 1 2A 2B 2C 3A 3B 6A 6B 17A ··· 17H 34A ··· 34H 51A ··· 51P order 1 2 2 2 3 3 6 6 17 ··· 17 34 ··· 34 51 ··· 51 size 1 3 17 51 4 4 68 68 2 ··· 2 6 ··· 6 8 ··· 8

40 irreducible representations

 dim 1 1 1 1 2 2 3 3 6 type + + + + + + image C1 C2 C3 C6 D17 C3×D17 A4 C2×A4 A4×D17 kernel A4×D17 A4×C17 C22×D17 C2×C34 A4 C22 D17 C17 C1 # reps 1 1 2 2 8 16 1 1 8

Matrix representation of A4×D17 in GL5(𝔽103)

 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 102 102 102
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 102 102 102 0 0 1 0 0
,
 46 0 0 0 0 0 46 0 0 0 0 0 1 0 0 0 0 102 102 102 0 0 0 1 0
,
 34 1 0 0 0 102 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(103))| [1,0,0,0,0,0,1,0,0,0,0,0,0,1,102,0,0,1,0,102,0,0,0,0,102],[1,0,0,0,0,0,1,0,0,0,0,0,0,102,1,0,0,0,102,0,0,0,1,102,0],[46,0,0,0,0,0,46,0,0,0,0,0,1,102,0,0,0,0,102,1,0,0,0,102,0],[34,102,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

A4×D17 in GAP, Magma, Sage, TeX

A_4\times D_{17}
% in TeX

G:=Group("A4xD17");
// GroupNames label

G:=SmallGroup(408,38);
// by ID

G=gap.SmallGroup(408,38);
# by ID

G:=PCGroup([5,-2,-3,-2,2,-17,142,68,9604]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^17=e^2=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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