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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C15×SD16
 Chief series C1 — C2 — C4 — C20 — C60 — Q8×C15 — C15×SD16
 Lower central C1 — C2 — C4 — C15×SD16
 Upper central C1 — C30 — C60 — C15×SD16

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

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

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

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

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

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

105 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 5A 5B 5C 5D 6A 6B 6C 6D 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 12A 12B 12C 12D 15A ··· 15H 20A 20B 20C 20D 20E 20F 20G 20H 24A 24B 24C 24D 30A ··· 30H 30I ··· 30P 40A ··· 40H 60A ··· 60H 60I ··· 60P 120A ··· 120P order 1 2 2 3 3 4 4 5 5 5 5 6 6 6 6 8 8 10 10 10 10 10 10 10 10 12 12 12 12 15 ··· 15 20 20 20 20 20 20 20 20 24 24 24 24 30 ··· 30 30 ··· 30 40 ··· 40 60 ··· 60 60 ··· 60 120 ··· 120 size 1 1 4 1 1 2 4 1 1 1 1 1 1 4 4 2 2 1 1 1 1 4 4 4 4 2 2 4 4 1 ··· 1 2 2 2 2 4 4 4 4 2 2 2 2 1 ··· 1 4 ··· 4 2 ··· 2 2 ··· 2 4 ··· 4 2 ··· 2

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 type + + + + + image C1 C2 C2 C2 C3 C5 C6 C6 C6 C10 C10 C10 C15 C30 C30 C30 D4 SD16 C3×D4 C5×D4 C3×SD16 C5×SD16 D4×C15 C15×SD16 kernel C15×SD16 C120 D4×C15 Q8×C15 C5×SD16 C3×SD16 C40 C5×D4 C5×Q8 C24 C3×D4 C3×Q8 SD16 C8 D4 Q8 C30 C15 C10 C6 C5 C3 C2 C1 # reps 1 1 1 1 2 4 2 2 2 4 4 4 8 8 8 8 1 2 2 4 4 8 8 16

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

 225 0 0 0 98 0 0 0 98
,
 240 0 0 0 0 203 0 19 203
,
 240 0 0 0 1 0 0 1 240
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

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