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## G = A4×C17order 204 = 22·3·17

### Direct product of C17 and A4

Aliases: A4×C17, C22⋊C51, (C2×C34)⋊C3, SmallGroup(204,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C17
 Chief series C1 — C22 — C2×C34 — A4×C17
 Lower central C22 — A4×C17
 Upper central C1 — C17

Generators and relations for A4×C17
G = < a,b,c,d | a17=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

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

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

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

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

A4×C17 is a maximal subgroup of   C17⋊S4

68 conjugacy classes

 class 1 2 3A 3B 17A ··· 17P 34A ··· 34P 51A ··· 51AF order 1 2 3 3 17 ··· 17 34 ··· 34 51 ··· 51 size 1 3 4 4 1 ··· 1 3 ··· 3 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 3 3 type + + image C1 C3 C17 C51 A4 A4×C17 kernel A4×C17 C2×C34 A4 C22 C17 C1 # reps 1 2 16 32 1 16

Matrix representation of A4×C17 in GL3(𝔽103) generated by

 61 0 0 0 61 0 0 0 61
,
 102 0 0 102 0 1 102 1 0
,
 0 1 102 1 0 102 0 0 102
,
 0 1 0 0 0 1 1 0 0
G:=sub<GL(3,GF(103))| [61,0,0,0,61,0,0,0,61],[102,102,102,0,0,1,0,1,0],[0,1,0,1,0,0,102,102,102],[0,0,1,1,0,0,0,1,0] >;

A4×C17 in GAP, Magma, Sage, TeX

A_4\times C_{17}
% in TeX

G:=Group("A4xC17");
// GroupNames label

G:=SmallGroup(204,8);
// by ID

G=gap.SmallGroup(204,8);
# by ID

G:=PCGroup([4,-3,-17,-2,2,1226,2451]);
// Polycyclic

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

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