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## G = C3×C8.17D4order 192 = 26·3

### Direct product of C3 and C8.17D4

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C3×C8.17D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C24 — C3×C8.C4 — C3×C8.17D4
 Lower central C1 — C2 — C4 — C8 — C3×C8.17D4
 Upper central C1 — C6 — C2×C12 — C2×C24 — C3×C8.17D4

Generators and relations for C3×C8.17D4
G = < a,b,c,d | a3=b8=1, c4=b4, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >

Smallest permutation representation of C3×C8.17D4
On 96 points
Generators in S96
(1 96 45)(2 81 46)(3 82 47)(4 83 48)(5 84 33)(6 85 34)(7 86 35)(8 87 36)(9 88 37)(10 89 38)(11 90 39)(12 91 40)(13 92 41)(14 93 42)(15 94 43)(16 95 44)(17 61 66)(18 62 67)(19 63 68)(20 64 69)(21 49 70)(22 50 71)(23 51 72)(24 52 73)(25 53 74)(26 54 75)(27 55 76)(28 56 77)(29 57 78)(30 58 79)(31 59 80)(32 60 65)
(1 15 13 11 9 7 5 3)(2 16 14 12 10 8 6 4)(17 31 29 27 25 23 21 19)(18 32 30 28 26 24 22 20)(33 47 45 43 41 39 37 35)(34 48 46 44 42 40 38 36)(49 63 61 59 57 55 53 51)(50 64 62 60 58 56 54 52)(65 79 77 75 73 71 69 67)(66 80 78 76 74 72 70 68)(81 95 93 91 89 87 85 83)(82 96 94 92 90 88 86 84)
(1 63 13 51 9 55 5 59)(2 50 6 62 10 58 14 54)(3 61 15 49 11 53 7 57)(4 64 8 60 12 56 16 52)(17 43 21 39 25 35 29 47)(18 38 30 42 26 46 22 34)(19 41 23 37 27 33 31 45)(20 36 32 40 28 44 24 48)(65 91 77 95 73 83 69 87)(66 94 70 90 74 86 78 82)(67 89 79 93 75 81 71 85)(68 92 72 88 76 84 80 96)
(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 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)

G:=sub<Sym(96)| (1,96,45)(2,81,46)(3,82,47)(4,83,48)(5,84,33)(6,85,34)(7,86,35)(8,87,36)(9,88,37)(10,89,38)(11,90,39)(12,91,40)(13,92,41)(14,93,42)(15,94,43)(16,95,44)(17,61,66)(18,62,67)(19,63,68)(20,64,69)(21,49,70)(22,50,71)(23,51,72)(24,52,73)(25,53,74)(26,54,75)(27,55,76)(28,56,77)(29,57,78)(30,58,79)(31,59,80)(32,60,65), (1,15,13,11,9,7,5,3)(2,16,14,12,10,8,6,4)(17,31,29,27,25,23,21,19)(18,32,30,28,26,24,22,20)(33,47,45,43,41,39,37,35)(34,48,46,44,42,40,38,36)(49,63,61,59,57,55,53,51)(50,64,62,60,58,56,54,52)(65,79,77,75,73,71,69,67)(66,80,78,76,74,72,70,68)(81,95,93,91,89,87,85,83)(82,96,94,92,90,88,86,84), (1,63,13,51,9,55,5,59)(2,50,6,62,10,58,14,54)(3,61,15,49,11,53,7,57)(4,64,8,60,12,56,16,52)(17,43,21,39,25,35,29,47)(18,38,30,42,26,46,22,34)(19,41,23,37,27,33,31,45)(20,36,32,40,28,44,24,48)(65,91,77,95,73,83,69,87)(66,94,70,90,74,86,78,82)(67,89,79,93,75,81,71,85)(68,92,72,88,76,84,80,96), (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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)>;

G:=Group( (1,96,45)(2,81,46)(3,82,47)(4,83,48)(5,84,33)(6,85,34)(7,86,35)(8,87,36)(9,88,37)(10,89,38)(11,90,39)(12,91,40)(13,92,41)(14,93,42)(15,94,43)(16,95,44)(17,61,66)(18,62,67)(19,63,68)(20,64,69)(21,49,70)(22,50,71)(23,51,72)(24,52,73)(25,53,74)(26,54,75)(27,55,76)(28,56,77)(29,57,78)(30,58,79)(31,59,80)(32,60,65), (1,15,13,11,9,7,5,3)(2,16,14,12,10,8,6,4)(17,31,29,27,25,23,21,19)(18,32,30,28,26,24,22,20)(33,47,45,43,41,39,37,35)(34,48,46,44,42,40,38,36)(49,63,61,59,57,55,53,51)(50,64,62,60,58,56,54,52)(65,79,77,75,73,71,69,67)(66,80,78,76,74,72,70,68)(81,95,93,91,89,87,85,83)(82,96,94,92,90,88,86,84), (1,63,13,51,9,55,5,59)(2,50,6,62,10,58,14,54)(3,61,15,49,11,53,7,57)(4,64,8,60,12,56,16,52)(17,43,21,39,25,35,29,47)(18,38,30,42,26,46,22,34)(19,41,23,37,27,33,31,45)(20,36,32,40,28,44,24,48)(65,91,77,95,73,83,69,87)(66,94,70,90,74,86,78,82)(67,89,79,93,75,81,71,85)(68,92,72,88,76,84,80,96), (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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96) );

G=PermutationGroup([[(1,96,45),(2,81,46),(3,82,47),(4,83,48),(5,84,33),(6,85,34),(7,86,35),(8,87,36),(9,88,37),(10,89,38),(11,90,39),(12,91,40),(13,92,41),(14,93,42),(15,94,43),(16,95,44),(17,61,66),(18,62,67),(19,63,68),(20,64,69),(21,49,70),(22,50,71),(23,51,72),(24,52,73),(25,53,74),(26,54,75),(27,55,76),(28,56,77),(29,57,78),(30,58,79),(31,59,80),(32,60,65)], [(1,15,13,11,9,7,5,3),(2,16,14,12,10,8,6,4),(17,31,29,27,25,23,21,19),(18,32,30,28,26,24,22,20),(33,47,45,43,41,39,37,35),(34,48,46,44,42,40,38,36),(49,63,61,59,57,55,53,51),(50,64,62,60,58,56,54,52),(65,79,77,75,73,71,69,67),(66,80,78,76,74,72,70,68),(81,95,93,91,89,87,85,83),(82,96,94,92,90,88,86,84)], [(1,63,13,51,9,55,5,59),(2,50,6,62,10,58,14,54),(3,61,15,49,11,53,7,57),(4,64,8,60,12,56,16,52),(17,43,21,39,25,35,29,47),(18,38,30,42,26,46,22,34),(19,41,23,37,27,33,31,45),(20,36,32,40,28,44,24,48),(65,91,77,95,73,83,69,87),(66,94,70,90,74,86,78,82),(67,89,79,93,75,81,71,85),(68,92,72,88,76,84,80,96)], [(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),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)]])

48 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 8A 8B 8C 8D 8E 12A 12B 12C 12D 12E 12F 12G 12H 16A 16B 16C 16D 24A 24B 24C 24D 24E 24F 24G 24H 24I 24J 48A ··· 48H order 1 2 2 3 3 4 4 4 4 6 6 6 6 8 8 8 8 8 12 12 12 12 12 12 12 12 16 16 16 16 24 24 24 24 24 24 24 24 24 24 48 ··· 48 size 1 1 2 1 1 2 2 8 8 1 1 2 2 2 2 4 8 8 2 2 2 2 8 8 8 8 4 4 4 4 2 2 2 2 4 4 8 8 8 8 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D4 D4 D8 SD16 C3×D4 C3×D4 C3×D8 C3×SD16 C8.17D4 C3×C8.17D4 kernel C3×C8.17D4 C3×C8.C4 C3×M5(2) C6×Q16 C8.17D4 C3×Q16 C8.C4 M5(2) C2×Q16 Q16 C24 C2×C12 C12 C2×C6 C8 C2×C4 C4 C22 C3 C1 # reps 1 1 1 1 2 4 2 2 2 8 1 1 2 2 2 2 4 4 2 4

Matrix representation of C3×C8.17D4 in GL4(𝔽7) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 5 0 2 6 6 5 5 6 6 1 1 2 2 2 6 2
,
 1 1 5 0 4 5 3 3 6 6 3 2 6 3 0 5
,
 4 5 1 6 4 1 4 3 2 3 0 6 5 0 5 2
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[5,6,6,2,0,5,1,2,2,5,1,6,6,6,2,2],[1,4,6,6,1,5,6,3,5,3,3,0,0,3,2,5],[4,4,2,5,5,1,3,0,1,4,0,5,6,3,6,2] >;

C3×C8.17D4 in GAP, Magma, Sage, TeX

C_3\times C_8._{17}D_4
% in TeX

G:=Group("C3xC8.17D4");
// GroupNames label

G:=SmallGroup(192,168);
// by ID

G=gap.SmallGroup(192,168);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,680,1683,2194,136,2111,172,6053,3036,124]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=1,c^4=b^4,d^2=c*b*c^-1=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations

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