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## G = C15⋊SD16order 240 = 24·3·5

### 4th semidirect product of C15 and SD16 acting via SD16/C4=C22

Aliases: C6.8D20, C30.7D4, C154SD16, D60.3C2, C12.6D10, C20.24D6, Dic101S3, C60.10C22, C3⋊C83D5, C4.3(S3×D5), C32(C40⋊C2), C51(Q82S3), (C3×Dic10)⋊1C2, C10.3(C3⋊D4), C2.6(C3⋊D20), (C5×C3⋊C8)⋊3C2, SmallGroup(240,19)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C15⋊SD16
 Chief series C1 — C5 — C15 — C30 — C60 — C3×Dic10 — C15⋊SD16
 Lower central C15 — C30 — C60 — C15⋊SD16
 Upper central C1 — C2 — C4

Generators and relations for C15⋊SD16
G = < a,b,c | a15=b8=c2=1, bab-1=a11, cac=a-1, 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 87 31 114 24 66 54 100)(2 83 32 110 25 62 55 96)(3 79 33 106 26 73 56 92)(4 90 34 117 27 69 57 103)(5 86 35 113 28 65 58 99)(6 82 36 109 29 61 59 95)(7 78 37 120 30 72 60 91)(8 89 38 116 16 68 46 102)(9 85 39 112 17 64 47 98)(10 81 40 108 18 75 48 94)(11 77 41 119 19 71 49 105)(12 88 42 115 20 67 50 101)(13 84 43 111 21 63 51 97)(14 80 44 107 22 74 52 93)(15 76 45 118 23 70 53 104)
(1 31)(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 32)(16 47)(17 46)(18 60)(19 59)(20 58)(21 57)(22 56)(23 55)(24 54)(25 53)(26 52)(27 51)(28 50)(29 49)(30 48)(61 77)(62 76)(63 90)(64 89)(65 88)(66 87)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(73 80)(74 79)(75 78)(91 94)(92 93)(95 105)(96 104)(97 103)(98 102)(99 101)(106 107)(108 120)(109 119)(110 118)(111 117)(112 116)(113 115)

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

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

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

C15⋊SD16 is a maximal subgroup of
S3×C40⋊C2  C401D6  Dic20⋊S3  D1205C2  D2019D6  D20.31D6  C12.D20  Dic103D6  Dic10⋊D6  C60.16C23  D125D10  D5×Q82S3  C60.C23  C60.39C23  D12.D10
C15⋊SD16 is a maximal quotient of
D6012C4  C6.Dic20  C60.Q8

36 conjugacy classes

 class 1 2A 2B 3 4A 4B 5A 5B 6 8A 8B 10A 10B 12A 12B 12C 15A 15B 20A 20B 20C 20D 30A 30B 40A ··· 40H 60A 60B 60C 60D order 1 2 2 3 4 4 5 5 6 8 8 10 10 12 12 12 15 15 20 20 20 20 30 30 40 ··· 40 60 60 60 60 size 1 1 60 2 2 20 2 2 2 6 6 2 2 4 20 20 4 4 2 2 2 2 4 4 6 ··· 6 4 4 4 4

36 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 D4 D5 D6 SD16 D10 C3⋊D4 D20 C40⋊C2 Q8⋊2S3 S3×D5 C3⋊D20 C15⋊SD16 kernel C15⋊SD16 C5×C3⋊C8 C3×Dic10 D60 Dic10 C30 C3⋊C8 C20 C15 C12 C10 C6 C3 C5 C4 C2 C1 # reps 1 1 1 1 1 1 2 1 2 2 2 4 8 1 2 2 4

Matrix representation of C15⋊SD16 in GL4(𝔽241) generated by

 240 189 0 0 52 52 0 0 0 0 1 5 0 0 144 239
,
 166 72 0 0 169 37 0 0 0 0 210 218 0 0 199 31
,
 41 85 0 0 122 200 0 0 0 0 240 0 0 0 97 1
G:=sub<GL(4,GF(241))| [240,52,0,0,189,52,0,0,0,0,1,144,0,0,5,239],[166,169,0,0,72,37,0,0,0,0,210,199,0,0,218,31],[41,122,0,0,85,200,0,0,0,0,240,97,0,0,0,1] >;

C15⋊SD16 in GAP, Magma, Sage, TeX

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

G:=Group("C15:SD16");
// GroupNames label

G:=SmallGroup(240,19);
// by ID

G=gap.SmallGroup(240,19);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,73,31,218,50,490,6917]);
// Polycyclic

G:=Group<a,b,c|a^15=b^8=c^2=1,b*a*b^-1=a^11,c*a*c=a^-1,c*b*c=b^3>;
// generators/relations

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