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## G = C5×C8⋊S3order 240 = 24·3·5

### Direct product of C5 and C8⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×C8⋊S3
 Chief series C1 — C3 — C6 — C12 — C60 — S3×C20 — C5×C8⋊S3
 Lower central C3 — C6 — C5×C8⋊S3
 Upper central C1 — C20 — C40

Generators and relations for C5×C8⋊S3
G = < a,b,c,d | a5=b8=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Smallest permutation representation of C5×C8⋊S3
On 120 points
Generators in S120
(1 61 26 51 93)(2 62 27 52 94)(3 63 28 53 95)(4 64 29 54 96)(5 57 30 55 89)(6 58 31 56 90)(7 59 32 49 91)(8 60 25 50 92)(9 85 110 22 36)(10 86 111 23 37)(11 87 112 24 38)(12 88 105 17 39)(13 81 106 18 40)(14 82 107 19 33)(15 83 108 20 34)(16 84 109 21 35)(41 115 74 99 65)(42 116 75 100 66)(43 117 76 101 67)(44 118 77 102 68)(45 119 78 103 69)(46 120 79 104 70)(47 113 80 97 71)(48 114 73 98 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 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)
(1 103 18)(2 104 19)(3 97 20)(4 98 21)(5 99 22)(6 100 23)(7 101 24)(8 102 17)(9 30 41)(10 31 42)(11 32 43)(12 25 44)(13 26 45)(14 27 46)(15 28 47)(16 29 48)(33 62 70)(34 63 71)(35 64 72)(36 57 65)(37 58 66)(38 59 67)(39 60 68)(40 61 69)(49 117 87)(50 118 88)(51 119 81)(52 120 82)(53 113 83)(54 114 84)(55 115 85)(56 116 86)(73 109 96)(74 110 89)(75 111 90)(76 112 91)(77 105 92)(78 106 93)(79 107 94)(80 108 95)
(2 6)(4 8)(9 41)(10 46)(11 43)(12 48)(13 45)(14 42)(15 47)(16 44)(17 98)(18 103)(19 100)(20 97)(21 102)(22 99)(23 104)(24 101)(25 29)(27 31)(33 66)(34 71)(35 68)(36 65)(37 70)(38 67)(39 72)(40 69)(50 54)(52 56)(58 62)(60 64)(73 105)(74 110)(75 107)(76 112)(77 109)(78 106)(79 111)(80 108)(81 119)(82 116)(83 113)(84 118)(85 115)(86 120)(87 117)(88 114)(90 94)(92 96)

G:=sub<Sym(120)| (1,61,26,51,93)(2,62,27,52,94)(3,63,28,53,95)(4,64,29,54,96)(5,57,30,55,89)(6,58,31,56,90)(7,59,32,49,91)(8,60,25,50,92)(9,85,110,22,36)(10,86,111,23,37)(11,87,112,24,38)(12,88,105,17,39)(13,81,106,18,40)(14,82,107,19,33)(15,83,108,20,34)(16,84,109,21,35)(41,115,74,99,65)(42,116,75,100,66)(43,117,76,101,67)(44,118,77,102,68)(45,119,78,103,69)(46,120,79,104,70)(47,113,80,97,71)(48,114,73,98,72), (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)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120), (1,103,18)(2,104,19)(3,97,20)(4,98,21)(5,99,22)(6,100,23)(7,101,24)(8,102,17)(9,30,41)(10,31,42)(11,32,43)(12,25,44)(13,26,45)(14,27,46)(15,28,47)(16,29,48)(33,62,70)(34,63,71)(35,64,72)(36,57,65)(37,58,66)(38,59,67)(39,60,68)(40,61,69)(49,117,87)(50,118,88)(51,119,81)(52,120,82)(53,113,83)(54,114,84)(55,115,85)(56,116,86)(73,109,96)(74,110,89)(75,111,90)(76,112,91)(77,105,92)(78,106,93)(79,107,94)(80,108,95), (2,6)(4,8)(9,41)(10,46)(11,43)(12,48)(13,45)(14,42)(15,47)(16,44)(17,98)(18,103)(19,100)(20,97)(21,102)(22,99)(23,104)(24,101)(25,29)(27,31)(33,66)(34,71)(35,68)(36,65)(37,70)(38,67)(39,72)(40,69)(50,54)(52,56)(58,62)(60,64)(73,105)(74,110)(75,107)(76,112)(77,109)(78,106)(79,111)(80,108)(81,119)(82,116)(83,113)(84,118)(85,115)(86,120)(87,117)(88,114)(90,94)(92,96)>;

G:=Group( (1,61,26,51,93)(2,62,27,52,94)(3,63,28,53,95)(4,64,29,54,96)(5,57,30,55,89)(6,58,31,56,90)(7,59,32,49,91)(8,60,25,50,92)(9,85,110,22,36)(10,86,111,23,37)(11,87,112,24,38)(12,88,105,17,39)(13,81,106,18,40)(14,82,107,19,33)(15,83,108,20,34)(16,84,109,21,35)(41,115,74,99,65)(42,116,75,100,66)(43,117,76,101,67)(44,118,77,102,68)(45,119,78,103,69)(46,120,79,104,70)(47,113,80,97,71)(48,114,73,98,72), (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)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120), (1,103,18)(2,104,19)(3,97,20)(4,98,21)(5,99,22)(6,100,23)(7,101,24)(8,102,17)(9,30,41)(10,31,42)(11,32,43)(12,25,44)(13,26,45)(14,27,46)(15,28,47)(16,29,48)(33,62,70)(34,63,71)(35,64,72)(36,57,65)(37,58,66)(38,59,67)(39,60,68)(40,61,69)(49,117,87)(50,118,88)(51,119,81)(52,120,82)(53,113,83)(54,114,84)(55,115,85)(56,116,86)(73,109,96)(74,110,89)(75,111,90)(76,112,91)(77,105,92)(78,106,93)(79,107,94)(80,108,95), (2,6)(4,8)(9,41)(10,46)(11,43)(12,48)(13,45)(14,42)(15,47)(16,44)(17,98)(18,103)(19,100)(20,97)(21,102)(22,99)(23,104)(24,101)(25,29)(27,31)(33,66)(34,71)(35,68)(36,65)(37,70)(38,67)(39,72)(40,69)(50,54)(52,56)(58,62)(60,64)(73,105)(74,110)(75,107)(76,112)(77,109)(78,106)(79,111)(80,108)(81,119)(82,116)(83,113)(84,118)(85,115)(86,120)(87,117)(88,114)(90,94)(92,96) );

G=PermutationGroup([[(1,61,26,51,93),(2,62,27,52,94),(3,63,28,53,95),(4,64,29,54,96),(5,57,30,55,89),(6,58,31,56,90),(7,59,32,49,91),(8,60,25,50,92),(9,85,110,22,36),(10,86,111,23,37),(11,87,112,24,38),(12,88,105,17,39),(13,81,106,18,40),(14,82,107,19,33),(15,83,108,20,34),(16,84,109,21,35),(41,115,74,99,65),(42,116,75,100,66),(43,117,76,101,67),(44,118,77,102,68),(45,119,78,103,69),(46,120,79,104,70),(47,113,80,97,71),(48,114,73,98,72)], [(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),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120)], [(1,103,18),(2,104,19),(3,97,20),(4,98,21),(5,99,22),(6,100,23),(7,101,24),(8,102,17),(9,30,41),(10,31,42),(11,32,43),(12,25,44),(13,26,45),(14,27,46),(15,28,47),(16,29,48),(33,62,70),(34,63,71),(35,64,72),(36,57,65),(37,58,66),(38,59,67),(39,60,68),(40,61,69),(49,117,87),(50,118,88),(51,119,81),(52,120,82),(53,113,83),(54,114,84),(55,115,85),(56,116,86),(73,109,96),(74,110,89),(75,111,90),(76,112,91),(77,105,92),(78,106,93),(79,107,94),(80,108,95)], [(2,6),(4,8),(9,41),(10,46),(11,43),(12,48),(13,45),(14,42),(15,47),(16,44),(17,98),(18,103),(19,100),(20,97),(21,102),(22,99),(23,104),(24,101),(25,29),(27,31),(33,66),(34,71),(35,68),(36,65),(37,70),(38,67),(39,72),(40,69),(50,54),(52,56),(58,62),(60,64),(73,105),(74,110),(75,107),(76,112),(77,109),(78,106),(79,111),(80,108),(81,119),(82,116),(83,113),(84,118),(85,115),(86,120),(87,117),(88,114),(90,94),(92,96)]])

C5×C8⋊S3 is a maximal subgroup of
C40⋊D6  C401D6  D40⋊S3  Dic20⋊S3  C40.34D6  C40.35D6  C40.2D6  C5×S3×M4(2)

90 conjugacy classes

 class 1 2A 2B 3 4A 4B 4C 5A 5B 5C 5D 6 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 10G 10H 12A 12B 15A 15B 15C 15D 20A ··· 20H 20I 20J 20K 20L 24A 24B 24C 24D 30A 30B 30C 30D 40A ··· 40H 40I ··· 40P 60A ··· 60H 120A ··· 120P order 1 2 2 3 4 4 4 5 5 5 5 6 8 8 8 8 10 10 10 10 10 10 10 10 12 12 15 15 15 15 20 ··· 20 20 20 20 20 24 24 24 24 30 30 30 30 40 ··· 40 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 6 2 1 1 6 1 1 1 1 2 2 2 6 6 1 1 1 1 6 6 6 6 2 2 2 2 2 2 1 ··· 1 6 6 6 6 2 2 2 2 2 2 2 2 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C4 C5 C10 C10 C10 C20 C20 S3 D6 M4(2) C4×S3 C5×S3 C8⋊S3 S3×C10 C5×M4(2) S3×C20 C5×C8⋊S3 kernel C5×C8⋊S3 C5×C3⋊C8 C120 S3×C20 C5×Dic3 S3×C10 C8⋊S3 C3⋊C8 C24 C4×S3 Dic3 D6 C40 C20 C15 C10 C8 C5 C4 C3 C2 C1 # reps 1 1 1 1 2 2 4 4 4 4 8 8 1 1 2 2 4 4 4 8 8 16

Matrix representation of C5×C8⋊S3 in GL4(𝔽241) generated by

 91 0 0 0 0 91 0 0 0 0 98 0 0 0 0 98
,
 0 120 0 0 113 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 240 240
,
 1 0 0 0 0 240 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,98,0,0,0,0,98],[0,113,0,0,120,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,240,0,0,1,240],[1,0,0,0,0,240,0,0,0,0,0,1,0,0,1,0] >;

C5×C8⋊S3 in GAP, Magma, Sage, TeX

C_5\times C_8\rtimes S_3
% in TeX

G:=Group("C5xC8:S3");
// GroupNames label

G:=SmallGroup(240,50);
// by ID

G=gap.SmallGroup(240,50);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-3,505,127,69,5765]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations

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