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## G = C5×D8⋊S3order 480 = 25·3·5

### Direct product of C5 and D8⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×D8⋊S3
 Chief series C1 — C3 — C6 — C12 — C60 — S3×C20 — C5×S3×D4 — C5×D8⋊S3
 Lower central C3 — C6 — C12 — C5×D8⋊S3
 Upper central C1 — C10 — C20 — C5×D8

Generators and relations for C5×D8⋊S3
G = < a,b,c,d,e | a5=b8=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b5, cd=dc, ce=ec, ede=d-1 >

Subgroups: 388 in 136 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, C23, C10, C10, Dic3, Dic3, C12, D6, D6, C2×C6, C15, M4(2), D8, D8, SD16, C2×D4, C4○D4, C20, C20, C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C22×S3, C5×S3, C30, C30, C8⋊C22, C40, C40, C2×C20, C5×D4, C5×D4, C5×Q8, C22×C10, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D42S3, C5×Dic3, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C5×M4(2), C5×D8, C5×D8, C5×SD16, D4×C10, C5×C4○D4, D8⋊S3, C5×C3⋊C8, C120, C5×Dic6, S3×C20, C5×D12, C10×Dic3, C5×C3⋊D4, D4×C15, S3×C2×C10, C5×C8⋊C22, C5×C8⋊S3, C5×C24⋊C2, C5×D4⋊S3, C5×D4.S3, C15×D8, C5×S3×D4, C5×D42S3, C5×D8⋊S3
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, C22×S3, C5×S3, C8⋊C22, C5×D4, C22×C10, S3×D4, S3×C10, D4×C10, D8⋊S3, S3×C2×C10, C5×C8⋊C22, C5×S3×D4, C5×D8⋊S3

Smallest permutation representation of C5×D8⋊S3
On 120 points
Generators in S120
(1 62 31 52 24)(2 63 32 53 17)(3 64 25 54 18)(4 57 26 55 19)(5 58 27 56 20)(6 59 28 49 21)(7 60 29 50 22)(8 61 30 51 23)(9 85 110 35 45)(10 86 111 36 46)(11 87 112 37 47)(12 88 105 38 48)(13 81 106 39 41)(14 82 107 40 42)(15 83 108 33 43)(16 84 109 34 44)(65 93 118 77 102)(66 94 119 78 103)(67 95 120 79 104)(68 96 113 80 97)(69 89 114 73 98)(70 90 115 74 99)(71 91 116 75 100)(72 92 117 76 101)
(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 3)(4 8)(5 7)(9 11)(12 16)(13 15)(18 24)(19 23)(20 22)(25 31)(26 30)(27 29)(33 39)(34 38)(35 37)(41 43)(44 48)(45 47)(50 56)(51 55)(52 54)(57 61)(58 60)(62 64)(65 69)(66 68)(70 72)(73 77)(74 76)(78 80)(81 83)(84 88)(85 87)(89 93)(90 92)(94 96)(97 103)(98 102)(99 101)(105 109)(106 108)(110 112)(113 119)(114 118)(115 117)
(1 99 35)(2 100 36)(3 101 37)(4 102 38)(5 103 39)(6 104 40)(7 97 33)(8 98 34)(9 31 90)(10 32 91)(11 25 92)(12 26 93)(13 27 94)(14 28 95)(15 29 96)(16 30 89)(17 75 111)(18 76 112)(19 77 105)(20 78 106)(21 79 107)(22 80 108)(23 73 109)(24 74 110)(41 58 66)(42 59 67)(43 60 68)(44 61 69)(45 62 70)(46 63 71)(47 64 72)(48 57 65)(49 120 82)(50 113 83)(51 114 84)(52 115 85)(53 116 86)(54 117 87)(55 118 88)(56 119 81)
(2 6)(4 8)(9 90)(10 95)(11 92)(12 89)(13 94)(14 91)(15 96)(16 93)(17 21)(19 23)(26 30)(28 32)(33 97)(34 102)(35 99)(36 104)(37 101)(38 98)(39 103)(40 100)(41 66)(42 71)(43 68)(44 65)(45 70)(46 67)(47 72)(48 69)(49 53)(51 55)(57 61)(59 63)(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)

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

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

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

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

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 C10 C10 S3 D4 D4 D6 D6 C5×S3 C5×D4 C5×D4 S3×C10 S3×C10 C8⋊C22 S3×D4 D8⋊S3 C5×C8⋊C22 C5×S3×D4 C5×D8⋊S3 kernel C5×D8⋊S3 C5×C8⋊S3 C5×C24⋊C2 C5×D4⋊S3 C5×D4.S3 C15×D8 C5×S3×D4 C5×D4⋊2S3 D8⋊S3 C8⋊S3 C24⋊C2 D4⋊S3 D4.S3 C3×D8 S3×D4 D4⋊2S3 C5×D8 C5×Dic3 S3×C10 C40 C5×D4 D8 Dic3 D6 C8 D4 C15 C10 C5 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 1 1 1 1 2 4 4 4 4 8 1 1 2 4 4 8

Matrix representation of C5×D8⋊S3 in GL6(𝔽241)

 205 0 0 0 0 0 0 205 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
,
 240 239 0 0 0 0 1 1 0 0 0 0 0 0 47 147 194 94 0 0 94 194 147 47 0 0 47 147 47 147 0 0 94 194 94 194
,
 240 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 1 0 0 0 0 240 0 0 0 0 0 0 0 240 1 0 0 0 0 240 0
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 1 240 0 0 0 0 0 240 0 0 0 0 0 0 1 240 0 0 0 0 0 240

G:=sub<GL(6,GF(241))| [205,0,0,0,0,0,0,205,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,1,0,0,0,0,239,1,0,0,0,0,0,0,47,94,47,94,0,0,147,194,147,194,0,0,194,147,47,94,0,0,94,47,147,194],[240,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,0,0,0,0,1,0,0,0,0,0,0,0,240,240,0,0,0,0,1,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,240,240] >;

C5×D8⋊S3 in GAP, Magma, Sage, TeX

C_5\times D_8\rtimes S_3
% in TeX

G:=Group("C5xD8:S3");
// GroupNames label

G:=SmallGroup(480,790);
// by ID

G=gap.SmallGroup(480,790);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1766,471,2111,1068,102,15686]);
// Polycyclic

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

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