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## G = C2×S3×D17order 408 = 23·3·17

### Direct product of C2, S3 and D17

Aliases: C2×S3×D17, C51⋊C23, C61D34, C341D6, C102⋊C22, D51⋊C22, D1025C2, (S3×C34)⋊3C2, (C6×D17)⋊3C2, (S3×C17)⋊C22, C171(C22×S3), (C3×D17)⋊C22, C31(C22×D17), SmallGroup(408,41)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C51 — C2×S3×D17
 Chief series C1 — C17 — C51 — C3×D17 — S3×D17 — C2×S3×D17
 Lower central C51 — C2×S3×D17
 Upper central C1 — C2

Generators and relations for C2×S3×D17
G = < a,b,c,d,e | a2=b3=c2=d17=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 716 in 64 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2, C3, C22, S3, S3, C6, C6, C23, D6, D6, C2×C6, C17, C22×S3, D17, D17, C34, C34, C51, D34, D34, C2×C34, S3×C17, C3×D17, D51, C102, C22×D17, S3×D17, C6×D17, S3×C34, D102, C2×S3×D17
Quotients: C1, C2, C22, S3, C23, D6, C22×S3, D17, D34, C22×D17, S3×D17, C2×S3×D17

Smallest permutation representation of C2×S3×D17
On 102 points
Generators in S102
(1 66)(2 67)(3 68)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 84)(19 85)(20 69)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 100)(36 101)(37 102)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)(49 97)(50 98)(51 99)
(1 23 51)(2 24 35)(3 25 36)(4 26 37)(5 27 38)(6 28 39)(7 29 40)(8 30 41)(9 31 42)(10 32 43)(11 33 44)(12 34 45)(13 18 46)(14 19 47)(15 20 48)(16 21 49)(17 22 50)(52 75 102)(53 76 86)(54 77 87)(55 78 88)(56 79 89)(57 80 90)(58 81 91)(59 82 92)(60 83 93)(61 84 94)(62 85 95)(63 69 96)(64 70 97)(65 71 98)(66 72 99)(67 73 100)(68 74 101)
(1 66)(2 67)(3 68)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 94)(19 95)(20 96)(21 97)(22 98)(23 99)(24 100)(25 101)(26 102)(27 86)(28 87)(29 88)(30 89)(31 90)(32 91)(33 92)(34 93)(35 73)(36 74)(37 75)(38 76)(39 77)(40 78)(41 79)(42 80)(43 81)(44 82)(45 83)(46 84)(47 85)(48 69)(49 70)(50 71)(51 72)
(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 97 98 99 100 101 102)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(18 27)(19 26)(20 25)(21 24)(22 23)(28 34)(29 33)(30 32)(35 49)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(50 51)(52 62)(53 61)(54 60)(55 59)(56 58)(63 68)(64 67)(65 66)(69 74)(70 73)(71 72)(75 85)(76 84)(77 83)(78 82)(79 81)(86 94)(87 93)(88 92)(89 91)(95 102)(96 101)(97 100)(98 99)

G:=sub<Sym(102)| (1,66)(2,67)(3,68)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,84)(19,85)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,100)(36,101)(37,102)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(49,97)(50,98)(51,99), (1,23,51)(2,24,35)(3,25,36)(4,26,37)(5,27,38)(6,28,39)(7,29,40)(8,30,41)(9,31,42)(10,32,43)(11,33,44)(12,34,45)(13,18,46)(14,19,47)(15,20,48)(16,21,49)(17,22,50)(52,75,102)(53,76,86)(54,77,87)(55,78,88)(56,79,89)(57,80,90)(58,81,91)(59,82,92)(60,83,93)(61,84,94)(62,85,95)(63,69,96)(64,70,97)(65,71,98)(66,72,99)(67,73,100)(68,74,101), (1,66)(2,67)(3,68)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,94)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,86)(28,87)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)(43,81)(44,82)(45,83)(46,84)(47,85)(48,69)(49,70)(50,71)(51,72), (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)(86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,27)(19,26)(20,25)(21,24)(22,23)(28,34)(29,33)(30,32)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(50,51)(52,62)(53,61)(54,60)(55,59)(56,58)(63,68)(64,67)(65,66)(69,74)(70,73)(71,72)(75,85)(76,84)(77,83)(78,82)(79,81)(86,94)(87,93)(88,92)(89,91)(95,102)(96,101)(97,100)(98,99)>;

G:=Group( (1,66)(2,67)(3,68)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,84)(19,85)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,100)(36,101)(37,102)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(49,97)(50,98)(51,99), (1,23,51)(2,24,35)(3,25,36)(4,26,37)(5,27,38)(6,28,39)(7,29,40)(8,30,41)(9,31,42)(10,32,43)(11,33,44)(12,34,45)(13,18,46)(14,19,47)(15,20,48)(16,21,49)(17,22,50)(52,75,102)(53,76,86)(54,77,87)(55,78,88)(56,79,89)(57,80,90)(58,81,91)(59,82,92)(60,83,93)(61,84,94)(62,85,95)(63,69,96)(64,70,97)(65,71,98)(66,72,99)(67,73,100)(68,74,101), (1,66)(2,67)(3,68)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,94)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,86)(28,87)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)(43,81)(44,82)(45,83)(46,84)(47,85)(48,69)(49,70)(50,71)(51,72), (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)(86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,27)(19,26)(20,25)(21,24)(22,23)(28,34)(29,33)(30,32)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(50,51)(52,62)(53,61)(54,60)(55,59)(56,58)(63,68)(64,67)(65,66)(69,74)(70,73)(71,72)(75,85)(76,84)(77,83)(78,82)(79,81)(86,94)(87,93)(88,92)(89,91)(95,102)(96,101)(97,100)(98,99) );

G=PermutationGroup([[(1,66),(2,67),(3,68),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,84),(19,85),(20,69),(21,70),(22,71),(23,72),(24,73),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,80),(32,81),(33,82),(34,83),(35,100),(36,101),(37,102),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96),(49,97),(50,98),(51,99)], [(1,23,51),(2,24,35),(3,25,36),(4,26,37),(5,27,38),(6,28,39),(7,29,40),(8,30,41),(9,31,42),(10,32,43),(11,33,44),(12,34,45),(13,18,46),(14,19,47),(15,20,48),(16,21,49),(17,22,50),(52,75,102),(53,76,86),(54,77,87),(55,78,88),(56,79,89),(57,80,90),(58,81,91),(59,82,92),(60,83,93),(61,84,94),(62,85,95),(63,69,96),(64,70,97),(65,71,98),(66,72,99),(67,73,100),(68,74,101)], [(1,66),(2,67),(3,68),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,94),(19,95),(20,96),(21,97),(22,98),(23,99),(24,100),(25,101),(26,102),(27,86),(28,87),(29,88),(30,89),(31,90),(32,91),(33,92),(34,93),(35,73),(36,74),(37,75),(38,76),(39,77),(40,78),(41,79),(42,80),(43,81),(44,82),(45,83),(46,84),(47,85),(48,69),(49,70),(50,71),(51,72)], [(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),(86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(18,27),(19,26),(20,25),(21,24),(22,23),(28,34),(29,33),(30,32),(35,49),(36,48),(37,47),(38,46),(39,45),(40,44),(41,43),(50,51),(52,62),(53,61),(54,60),(55,59),(56,58),(63,68),(64,67),(65,66),(69,74),(70,73),(71,72),(75,85),(76,84),(77,83),(78,82),(79,81),(86,94),(87,93),(88,92),(89,91),(95,102),(96,101),(97,100),(98,99)]])

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 6A 6B 6C 17A ··· 17H 34A ··· 34H 34I ··· 34X 51A ··· 51H 102A ··· 102H order 1 2 2 2 2 2 2 2 3 6 6 6 17 ··· 17 34 ··· 34 34 ··· 34 51 ··· 51 102 ··· 102 size 1 1 3 3 17 17 51 51 2 2 34 34 2 ··· 2 2 ··· 2 6 ··· 6 4 ··· 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D6 D6 D17 D34 D34 S3×D17 C2×S3×D17 kernel C2×S3×D17 S3×D17 C6×D17 S3×C34 D102 D34 D17 C34 D6 S3 C6 C2 C1 # reps 1 4 1 1 1 1 2 1 8 16 8 8 8

Matrix representation of C2×S3×D17 in GL4(𝔽103) generated by

 102 0 0 0 0 102 0 0 0 0 102 0 0 0 0 102
,
 1 0 0 0 0 1 0 0 0 0 101 69 0 0 94 1
,
 102 0 0 0 0 102 0 0 0 0 102 0 0 0 94 1
,
 4 1 0 0 25 58 0 0 0 0 1 0 0 0 0 1
,
 33 5 0 0 9 70 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(103))| [102,0,0,0,0,102,0,0,0,0,102,0,0,0,0,102],[1,0,0,0,0,1,0,0,0,0,101,94,0,0,69,1],[102,0,0,0,0,102,0,0,0,0,102,94,0,0,0,1],[4,25,0,0,1,58,0,0,0,0,1,0,0,0,0,1],[33,9,0,0,5,70,0,0,0,0,1,0,0,0,0,1] >;

C2×S3×D17 in GAP, Magma, Sage, TeX

C_2\times S_3\times D_{17}
% in TeX

G:=Group("C2xS3xD17");
// GroupNames label

G:=SmallGroup(408,41);
// by ID

G=gap.SmallGroup(408,41);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-17,168,9604]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^17=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,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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