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G = C15×SD16order 240 = 24·3·5

Direct product of C15 and SD16

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: C15×SD16, C82C30, C406C6, D4.C30, C246C10, Q82C30, C12014C2, C30.55D4, C60.78C22, (C5×Q8)⋊6C6, C4.2(C2×C30), (C3×Q8)⋊4C10, (C5×D4).2C6, C2.4(D4×C15), C6.15(C5×D4), C20.18(C2×C6), (Q8×C15)⋊10C2, (D4×C15).4C2, (C3×D4).2C10, C10.15(C3×D4), C12.18(C2×C10), SmallGroup(240,87)

Series: Derived Chief Lower central Upper central

C1C4 — C15×SD16
C1C2C4C20C60Q8×C15 — C15×SD16
C1C2C4 — C15×SD16
C1C30C60 — C15×SD16

Generators and relations for C15×SD16
 G = < a,b,c | a15=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

4C2
2C4
2C22
4C6
4C10
2C2×C6
2C12
2C2×C10
2C20
4C30
2C2×C30
2C60

Smallest permutation representation of C15×SD16
On 120 points
Generators in S120
(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)
(1 120 27 57 39 83 70 101)(2 106 28 58 40 84 71 102)(3 107 29 59 41 85 72 103)(4 108 30 60 42 86 73 104)(5 109 16 46 43 87 74 105)(6 110 17 47 44 88 75 91)(7 111 18 48 45 89 61 92)(8 112 19 49 31 90 62 93)(9 113 20 50 32 76 63 94)(10 114 21 51 33 77 64 95)(11 115 22 52 34 78 65 96)(12 116 23 53 35 79 66 97)(13 117 24 54 36 80 67 98)(14 118 25 55 37 81 68 99)(15 119 26 56 38 82 69 100)
(16 74)(17 75)(18 61)(19 62)(20 63)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 71)(29 72)(30 73)(46 109)(47 110)(48 111)(49 112)(50 113)(51 114)(52 115)(53 116)(54 117)(55 118)(56 119)(57 120)(58 106)(59 107)(60 108)(76 94)(77 95)(78 96)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 103)(86 104)(87 105)(88 91)(89 92)(90 93)

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

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

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

C15×SD16 is a maximal subgroup of   Q83D30  SD16⋊D15  D4.5D30

105 conjugacy classes

class 1 2A2B3A3B4A4B5A5B5C5D6A6B6C6D8A8B10A10B10C10D10E10F10G10H12A12B12C12D15A···15H20A20B20C20D20E20F20G20H24A24B24C24D30A···30H30I···30P40A···40H60A···60H60I···60P120A···120P
order1223344555566668810101010101010101212121215···1520202020202020202424242430···3030···3040···4060···6060···60120···120
size114112411111144221111444422441···12222444422221···14···42···22···24···42···2

105 irreducible representations

dim111111111111111122222222
type+++++
imageC1C2C2C2C3C5C6C6C6C10C10C10C15C30C30C30D4SD16C3×D4C5×D4C3×SD16C5×SD16D4×C15C15×SD16
kernelC15×SD16C120D4×C15Q8×C15C5×SD16C3×SD16C40C5×D4C5×Q8C24C3×D4C3×Q8SD16C8D4Q8C30C15C10C6C5C3C2C1
# reps1111242224448888122448816

Matrix representation of C15×SD16 in GL3(𝔽241) generated by

22500
0980
0098
,
24000
00203
019203
,
24000
010
01240
G:=sub<GL(3,GF(241))| [225,0,0,0,98,0,0,0,98],[240,0,0,0,0,19,0,203,203],[240,0,0,0,1,1,0,0,240] >;

C15×SD16 in GAP, Magma, Sage, TeX

C_{15}\times {\rm SD}_{16}
% in TeX

G:=Group("C15xSD16");
// GroupNames label

G:=SmallGroup(240,87);
// by ID

G=gap.SmallGroup(240,87);
# by ID

G:=PCGroup([6,-2,-2,-3,-5,-2,-2,720,745,5404,2710,88]);
// Polycyclic

G:=Group<a,b,c|a^15=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations

Export

Subgroup lattice of C15×SD16 in TeX

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