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## G = C5×S3×SD16order 480 = 25·3·5

### Direct product of C5, S3 and SD16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×S3×SD16
 Chief series C1 — C3 — C6 — C12 — C60 — S3×C20 — C5×S3×D4 — C5×S3×SD16
 Lower central C3 — C6 — C12 — C5×S3×SD16
 Upper central C1 — C10 — C20 — C5×SD16

Generators and relations for C5×S3×SD16
G = < a,b,c,d,e | a5=b3=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d3 >

Subgroups: 372 in 136 conjugacy classes, 58 normal (54 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, C23, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8, SD16, SD16, C2×D4, C2×Q8, C20, C20, C2×C10, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3, C5×S3, C30, C30, C2×SD16, C40, C40, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, S3×C8, C24⋊C2, D4.S3, Q82S3, C3×SD16, S3×D4, S3×Q8, C5×Dic3, C5×Dic3, C60, C60, S3×C10, S3×C10, C2×C30, C2×C40, C5×SD16, C5×SD16, D4×C10, Q8×C10, S3×SD16, C5×C3⋊C8, C120, C5×Dic6, C5×Dic6, S3×C20, S3×C20, C5×D12, C5×C3⋊D4, D4×C15, Q8×C15, S3×C2×C10, C10×SD16, S3×C40, C5×C24⋊C2, C5×D4.S3, C5×Q82S3, C15×SD16, C5×S3×D4, C5×S3×Q8, C5×S3×SD16
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, SD16, C2×D4, C2×C10, C22×S3, C5×S3, C2×SD16, C5×D4, C22×C10, S3×D4, S3×C10, C5×SD16, D4×C10, S3×SD16, S3×C2×C10, C10×SD16, C5×S3×D4, C5×S3×SD16

Smallest permutation representation of C5×S3×SD16
On 120 points
Generators in S120
(1 58 31 56 88)(2 59 32 49 81)(3 60 25 50 82)(4 61 26 51 83)(5 62 27 52 84)(6 63 28 53 85)(7 64 29 54 86)(8 57 30 55 87)(9 46 110 18 40)(10 47 111 19 33)(11 48 112 20 34)(12 41 105 21 35)(13 42 106 22 36)(14 43 107 23 37)(15 44 108 24 38)(16 45 109 17 39)(65 94 115 78 99)(66 95 116 79 100)(67 96 117 80 101)(68 89 118 73 102)(69 90 119 74 103)(70 91 120 75 104)(71 92 113 76 97)(72 93 114 77 98)
(1 22 103)(2 23 104)(3 24 97)(4 17 98)(5 18 99)(6 19 100)(7 20 101)(8 21 102)(9 94 27)(10 95 28)(11 96 29)(12 89 30)(13 90 31)(14 91 32)(15 92 25)(16 93 26)(33 66 63)(34 67 64)(35 68 57)(36 69 58)(37 70 59)(38 71 60)(39 72 61)(40 65 62)(41 118 55)(42 119 56)(43 120 49)(44 113 50)(45 114 51)(46 115 52)(47 116 53)(48 117 54)(73 87 105)(74 88 106)(75 81 107)(76 82 108)(77 83 109)(78 84 110)(79 85 111)(80 86 112)
(9 94)(10 95)(11 96)(12 89)(13 90)(14 91)(15 92)(16 93)(17 98)(18 99)(19 100)(20 101)(21 102)(22 103)(23 104)(24 97)(33 66)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 65)(41 118)(42 119)(43 120)(44 113)(45 114)(46 115)(47 116)(48 117)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)
(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)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(17 23)(19 21)(20 24)(25 29)(26 32)(28 30)(33 35)(34 38)(37 39)(41 47)(43 45)(44 48)(49 51)(50 54)(53 55)(57 63)(59 61)(60 64)(66 68)(67 71)(70 72)(73 79)(75 77)(76 80)(81 83)(82 86)(85 87)(89 95)(91 93)(92 96)(97 101)(98 104)(100 102)(105 111)(107 109)(108 112)(113 117)(114 120)(116 118)

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

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

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

105 conjugacy classes

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

105 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 2 2 2 2 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 D6 SD16 C5×S3 C5×D4 C5×D4 S3×C10 S3×C10 S3×C10 C5×SD16 S3×D4 S3×SD16 C5×S3×D4 C5×S3×SD16 kernel C5×S3×SD16 S3×C40 C5×C24⋊C2 C5×D4.S3 C5×Q8⋊2S3 C15×SD16 C5×S3×D4 C5×S3×Q8 S3×SD16 S3×C8 C24⋊C2 D4.S3 Q8⋊2S3 C3×SD16 S3×D4 S3×Q8 C5×SD16 C5×Dic3 S3×C10 C40 C5×D4 C5×Q8 C5×S3 SD16 Dic3 D6 C8 D4 Q8 S3 C10 C5 C2 C1 # reps 1 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 1 1 1 1 1 1 4 4 4 4 4 4 4 16 1 2 4 8

Matrix representation of C5×S3×SD16 in GL4(𝔽241) generated by

 205 0 0 0 0 205 0 0 0 0 205 0 0 0 0 205
,
 240 240 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 240 240 0 0 0 0 1 0 0 0 0 1
,
 240 0 0 0 0 240 0 0 0 0 222 19 0 0 222 222
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(241))| [205,0,0,0,0,205,0,0,0,0,205,0,0,0,0,205],[240,1,0,0,240,0,0,0,0,0,1,0,0,0,0,1],[1,240,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,0,240,0,0,0,0,222,222,0,0,19,222],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;

C5×S3×SD16 in GAP, Magma, Sage, TeX

C_5\times S_3\times {\rm SD}_{16}
% in TeX

G:=Group("C5xS3xSD16");
// GroupNames label

G:=SmallGroup(480,792);
// by ID

G=gap.SmallGroup(480,792);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,471,436,2111,1068,102,15686]);
// Polycyclic

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

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