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## G = S3×C5⋊C8order 240 = 24·3·5

### Direct product of S3 and C5⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — S3×C5⋊C8
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C3×C5⋊C8 — S3×C5⋊C8
 Lower central C15 — S3×C5⋊C8
 Upper central C1 — C2

Generators and relations for S3×C5⋊C8
G = < a,b,c,d | a3=b2=c5=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Character table of S3×C5⋊C8

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5 6 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 12A 12B 15 24A 24B 24C 24D 30 size 1 1 3 3 2 5 5 15 15 4 2 5 5 5 5 15 15 15 15 4 12 12 10 10 8 10 10 10 10 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ3 1 1 -1 -1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ5 1 1 1 1 1 -1 -1 -1 -1 1 1 i i -i -i -i i -i i 1 1 1 -1 -1 1 i -i -i i 1 linear of order 4 ρ6 1 1 1 1 1 -1 -1 -1 -1 1 1 -i -i i i i -i i -i 1 1 1 -1 -1 1 -i i i -i 1 linear of order 4 ρ7 1 1 -1 -1 1 -1 -1 1 1 1 1 i i -i -i i -i i -i 1 -1 -1 -1 -1 1 i -i -i i 1 linear of order 4 ρ8 1 1 -1 -1 1 -1 -1 1 1 1 1 -i -i i i -i i -i i 1 -1 -1 -1 -1 1 -i i i -i 1 linear of order 4 ρ9 1 -1 1 -1 1 i -i -i i 1 -1 ζ85 ζ8 ζ87 ζ83 ζ83 ζ8 ζ87 ζ85 -1 1 -1 -i i 1 ζ8 ζ87 ζ83 ζ85 -1 linear of order 8 ρ10 1 -1 -1 1 1 -i i -i i 1 -1 ζ83 ζ87 ζ8 ζ85 ζ8 ζ83 ζ85 ζ87 -1 -1 1 i -i 1 ζ87 ζ8 ζ85 ζ83 -1 linear of order 8 ρ11 1 -1 1 -1 1 -i i i -i 1 -1 ζ87 ζ83 ζ85 ζ8 ζ8 ζ83 ζ85 ζ87 -1 1 -1 i -i 1 ζ83 ζ85 ζ8 ζ87 -1 linear of order 8 ρ12 1 -1 1 -1 1 i -i -i i 1 -1 ζ8 ζ85 ζ83 ζ87 ζ87 ζ85 ζ83 ζ8 -1 1 -1 -i i 1 ζ85 ζ83 ζ87 ζ8 -1 linear of order 8 ρ13 1 -1 -1 1 1 i -i i -i 1 -1 ζ85 ζ8 ζ87 ζ83 ζ87 ζ85 ζ83 ζ8 -1 -1 1 -i i 1 ζ8 ζ87 ζ83 ζ85 -1 linear of order 8 ρ14 1 -1 -1 1 1 -i i -i i 1 -1 ζ87 ζ83 ζ85 ζ8 ζ85 ζ87 ζ8 ζ83 -1 -1 1 i -i 1 ζ83 ζ85 ζ8 ζ87 -1 linear of order 8 ρ15 1 -1 -1 1 1 i -i i -i 1 -1 ζ8 ζ85 ζ83 ζ87 ζ83 ζ8 ζ87 ζ85 -1 -1 1 -i i 1 ζ85 ζ83 ζ87 ζ8 -1 linear of order 8 ρ16 1 -1 1 -1 1 -i i i -i 1 -1 ζ83 ζ87 ζ8 ζ85 ζ85 ζ87 ζ8 ζ83 -1 1 -1 i -i 1 ζ87 ζ8 ζ85 ζ83 -1 linear of order 8 ρ17 2 2 0 0 -1 2 2 0 0 2 -1 2 2 2 2 0 0 0 0 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ18 2 2 0 0 -1 2 2 0 0 2 -1 -2 -2 -2 -2 0 0 0 0 2 0 0 -1 -1 -1 1 1 1 1 -1 orthogonal lifted from D6 ρ19 2 2 0 0 -1 -2 -2 0 0 2 -1 2i 2i -2i -2i 0 0 0 0 2 0 0 1 1 -1 -i i i -i -1 complex lifted from C4×S3 ρ20 2 2 0 0 -1 -2 -2 0 0 2 -1 -2i -2i 2i 2i 0 0 0 0 2 0 0 1 1 -1 i -i -i i -1 complex lifted from C4×S3 ρ21 2 -2 0 0 -1 2i -2i 0 0 2 1 2ζ85 2ζ8 2ζ87 2ζ83 0 0 0 0 -2 0 0 i -i -1 ζ85 ζ83 ζ87 ζ8 1 complex lifted from S3×C8 ρ22 2 -2 0 0 -1 2i -2i 0 0 2 1 2ζ8 2ζ85 2ζ83 2ζ87 0 0 0 0 -2 0 0 i -i -1 ζ8 ζ87 ζ83 ζ85 1 complex lifted from S3×C8 ρ23 2 -2 0 0 -1 -2i 2i 0 0 2 1 2ζ83 2ζ87 2ζ8 2ζ85 0 0 0 0 -2 0 0 -i i -1 ζ83 ζ85 ζ8 ζ87 1 complex lifted from S3×C8 ρ24 2 -2 0 0 -1 -2i 2i 0 0 2 1 2ζ87 2ζ83 2ζ85 2ζ8 0 0 0 0 -2 0 0 -i i -1 ζ87 ζ8 ζ85 ζ83 1 complex lifted from S3×C8 ρ25 4 4 4 4 4 0 0 0 0 -1 4 0 0 0 0 0 0 0 0 -1 -1 -1 0 0 -1 0 0 0 0 -1 orthogonal lifted from F5 ρ26 4 4 -4 -4 4 0 0 0 0 -1 4 0 0 0 0 0 0 0 0 -1 1 1 0 0 -1 0 0 0 0 -1 orthogonal lifted from C2×F5 ρ27 4 -4 4 -4 4 0 0 0 0 -1 -4 0 0 0 0 0 0 0 0 1 -1 1 0 0 -1 0 0 0 0 1 symplectic lifted from C5⋊C8, Schur index 2 ρ28 4 -4 -4 4 4 0 0 0 0 -1 -4 0 0 0 0 0 0 0 0 1 1 -1 0 0 -1 0 0 0 0 1 symplectic lifted from C5⋊C8, Schur index 2 ρ29 8 8 0 0 -4 0 0 0 0 -2 -4 0 0 0 0 0 0 0 0 -2 0 0 0 0 1 0 0 0 0 1 orthogonal lifted from S3×F5 ρ30 8 -8 0 0 -4 0 0 0 0 -2 4 0 0 0 0 0 0 0 0 2 0 0 0 0 1 0 0 0 0 -1 symplectic faithful, Schur index 2

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

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

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

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

S3×C5⋊C8 is a maximal subgroup of   D12.2F5  D12.F5  C5⋊C8.D6  D15⋊C8⋊C2
S3×C5⋊C8 is a maximal quotient of   C15⋊M5(2)  Dic5.22D12  C30.4M4(2)

Matrix representation of S3×C5⋊C8 in GL6(𝔽241)

 0 240 0 0 0 0 1 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 240 0 0 1 0 0 240 0 0 0 1 0 240 0 0 0 0 1 240
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 64 196 3 222 0 0 67 177 0 45 0 0 64 0 196 48 0 0 19 3 177 45

G:=sub<GL(6,GF(241))| [0,1,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,240,240,240,240],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,64,67,64,19,0,0,196,177,0,3,0,0,3,0,196,177,0,0,222,45,48,45] >;

S3×C5⋊C8 in GAP, Magma, Sage, TeX

S_3\times C_5\rtimes C_8
% in TeX

G:=Group("S3xC5:C8");
// GroupNames label

G:=SmallGroup(240,98);
// by ID

G=gap.SmallGroup(240,98);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,50,490,3461,1745]);
// Polycyclic

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

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