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G = C5×S3×SD16order 480 = 25·3·5

Direct product of C5, S3 and SD16

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5×S3×SD16, C4027D6, C12032C22, C60.220C23, C85(S3×C10), (S3×D4).C10, (S3×C8)⋊4C10, C245(C2×C10), (S3×Q8)⋊1C10, Q82(S3×C10), (C5×Q8)⋊17D6, C32(C10×SD16), (S3×C40)⋊13C2, C24⋊C25C10, D4.S33C10, (C5×D4).26D6, D4.2(S3×C10), D6.13(C5×D4), C6.30(D4×C10), C1521(C2×SD16), Q82S31C10, (C3×SD16)⋊3C10, Dic62(C2×C10), (S3×C10).49D4, D12.2(C2×C10), C10.184(S3×D4), C30.366(C2×D4), Dic3.4(C5×D4), (C15×SD16)⋊11C2, C12.4(C22×C10), (C5×Dic3).31D4, (Q8×C15)⋊16C22, (S3×C20).58C22, C20.193(C22×S3), (C5×Dic6)⋊17C22, (C5×D12).31C22, (D4×C15).31C22, (C5×S3×Q8)⋊8C2, C3⋊C86(C2×C10), C4.4(S3×C2×C10), (C5×S3×D4).2C2, C2.18(C5×S3×D4), (C5×C3⋊C8)⋊39C22, (C3×Q8)⋊1(C2×C10), (C5×C24⋊C2)⋊13C2, (C4×S3).9(C2×C10), (C5×D4.S3)⋊11C2, (C5×Q82S3)⋊9C2, (C3×D4).2(C2×C10), SmallGroup(480,792)

Series: Derived Chief Lower central Upper central

C1C12 — C5×S3×SD16
C1C3C6C12C60S3×C20C5×S3×D4 — C5×S3×SD16
C3C6C12 — C5×S3×SD16
C1C10C20C5×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 [×4], C3, C4, C4 [×3], C22 [×5], C5, S3 [×2], S3, C6, C6, C8, C8, C2×C4 [×2], D4, D4 [×2], Q8, Q8 [×2], C23, C10, C10 [×4], Dic3, Dic3, C12, C12, D6, D6 [×3], C2×C6, C15, C2×C8, SD16, SD16 [×3], C2×D4, C2×Q8, C20, C20 [×3], C2×C10 [×5], C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3 [×2], C5×S3, C30, C30, C2×SD16, C40, C40, C2×C20 [×2], C5×D4, C5×D4 [×2], C5×Q8, C5×Q8 [×2], 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 [×3], C2×C30, C2×C40, C5×SD16, C5×SD16 [×3], 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 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], SD16 [×2], C2×D4, C2×C10 [×7], C22×S3, C5×S3, C2×SD16, C5×D4 [×2], C22×C10, S3×D4, S3×C10 [×3], C5×SD16 [×2], 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 61 91 51 29)(2 62 92 52 30)(3 63 93 53 31)(4 64 94 54 32)(5 57 95 55 25)(6 58 96 56 26)(7 59 89 49 27)(8 60 90 50 28)(9 81 110 20 40)(10 82 111 21 33)(11 83 112 22 34)(12 84 105 23 35)(13 85 106 24 36)(14 86 107 17 37)(15 87 108 18 38)(16 88 109 19 39)(41 114 77 98 72)(42 115 78 99 65)(43 116 79 100 66)(44 117 80 101 67)(45 118 73 102 68)(46 119 74 103 69)(47 120 75 104 70)(48 113 76 97 71)
(1 24 103)(2 17 104)(3 18 97)(4 19 98)(5 20 99)(6 21 100)(7 22 101)(8 23 102)(9 42 95)(10 43 96)(11 44 89)(12 45 90)(13 46 91)(14 47 92)(15 48 93)(16 41 94)(25 110 78)(26 111 79)(27 112 80)(28 105 73)(29 106 74)(30 107 75)(31 108 76)(32 109 77)(33 66 58)(34 67 59)(35 68 60)(36 69 61)(37 70 62)(38 71 63)(39 72 64)(40 65 57)(49 83 117)(50 84 118)(51 85 119)(52 86 120)(53 87 113)(54 88 114)(55 81 115)(56 82 116)
(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(17 104)(18 97)(19 98)(20 99)(21 100)(22 101)(23 102)(24 103)(33 66)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 65)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)(81 115)(82 116)(83 117)(84 118)(85 119)(86 120)(87 113)(88 114)
(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 19)(18 22)(21 23)(26 28)(27 31)(30 32)(33 35)(34 38)(37 39)(41 47)(43 45)(44 48)(49 53)(50 56)(52 54)(58 60)(59 63)(62 64)(66 68)(67 71)(70 72)(73 79)(75 77)(76 80)(82 84)(83 87)(86 88)(89 93)(90 96)(92 94)(97 101)(98 104)(100 102)(105 111)(107 109)(108 112)(113 117)(114 120)(116 118)

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

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

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

105 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D5A5B5C5D6A6B8A8B8C8D10A10B10C10D10E···10L10M10N10O10P10Q10R10S10T12A12B15A15B15C15D20A20B20C20D20E20F20G20H20I20J20K20L20M20N20O20P24A24B30A30B30C30D30E30F30G30H40A···40H40I···40P60A60B60C60D60E60F60G60H120A···120H
order1222223444455556688881010101010···101010101010101010121215151515202020202020202020202020202020202424303030303030303040···4040···406060606060606060120···120
size1133412224612111128226611113···34444121212124822222222444466661212121244222288882···26···6444488884···4

105 irreducible representations

dim1111111111111111222222222222224444
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C5C10C10C10C10C10C10C10S3D4D4D6D6D6SD16C5×S3C5×D4C5×D4S3×C10S3×C10S3×C10C5×SD16S3×D4S3×SD16C5×S3×D4C5×S3×SD16
kernelC5×S3×SD16S3×C40C5×C24⋊C2C5×D4.S3C5×Q82S3C15×SD16C5×S3×D4C5×S3×Q8S3×SD16S3×C8C24⋊C2D4.S3Q82S3C3×SD16S3×D4S3×Q8C5×SD16C5×Dic3S3×C10C40C5×D4C5×Q8C5×S3SD16Dic3D6C8D4Q8S3C10C5C2C1
# reps11111111444444441111114444444161248

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

205000
020500
002050
000205
,
24024000
1000
0010
0001
,
1000
24024000
0010
0001
,
240000
024000
0022219
00222222
,
1000
0100
0001
0010
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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