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## G = C3×C15⋊Q8order 360 = 23·32·5

### Direct product of C3 and C15⋊Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C3×C15⋊Q8
 Chief series C1 — C5 — C15 — C30 — C3×C30 — C32×Dic5 — C3×C15⋊Q8
 Lower central C15 — C30 — C3×C15⋊Q8
 Upper central C1 — C6

Generators and relations for C3×C15⋊Q8
G = < a,b,c,d | a3=b15=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=b11, dbd-1=b4, dcd-1=c-1 >

Subgroups: 172 in 54 conjugacy classes, 28 normal (all characteristic)
C1, C2, C3, C3, C4, C5, C6, C6, Q8, C32, C10, Dic3, Dic3, C12, C15, C15, C3×C6, Dic5, Dic5, C20, Dic6, C3×Q8, C30, C30, C3×Dic3, C3×Dic3, C3×C12, Dic10, C3×C15, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, C3×Dic6, C3×C30, C15⋊Q8, C3×Dic10, C32×Dic5, Dic3×C15, C3×Dic15, C3×C15⋊Q8
Quotients: C1, C2, C3, C22, S3, C6, Q8, D5, D6, C2×C6, C3×S3, D10, Dic6, C3×Q8, C3×D5, S3×C6, Dic10, S3×D5, C6×D5, C3×Dic6, C15⋊Q8, C3×Dic10, C3×S3×D5, C3×C15⋊Q8

Smallest permutation representation of C3×C15⋊Q8
On 120 points
Generators in S120
(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)(61 71 66)(62 72 67)(63 73 68)(64 74 69)(65 75 70)(76 86 81)(77 87 82)(78 88 83)(79 89 84)(80 90 85)(91 96 101)(92 97 102)(93 98 103)(94 99 104)(95 100 105)(106 111 116)(107 112 117)(108 113 118)(109 114 119)(110 115 120)
(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 59 21 31)(2 55 22 42)(3 51 23 38)(4 47 24 34)(5 58 25 45)(6 54 26 41)(7 50 27 37)(8 46 28 33)(9 57 29 44)(10 53 30 40)(11 49 16 36)(12 60 17 32)(13 56 18 43)(14 52 19 39)(15 48 20 35)(61 93 78 119)(62 104 79 115)(63 100 80 111)(64 96 81 107)(65 92 82 118)(66 103 83 114)(67 99 84 110)(68 95 85 106)(69 91 86 117)(70 102 87 113)(71 98 88 109)(72 94 89 120)(73 105 90 116)(74 101 76 112)(75 97 77 108)
(1 83 21 66)(2 87 22 70)(3 76 23 74)(4 80 24 63)(5 84 25 67)(6 88 26 71)(7 77 27 75)(8 81 28 64)(9 85 29 68)(10 89 30 72)(11 78 16 61)(12 82 17 65)(13 86 18 69)(14 90 19 73)(15 79 20 62)(31 114 59 103)(32 118 60 92)(33 107 46 96)(34 111 47 100)(35 115 48 104)(36 119 49 93)(37 108 50 97)(38 112 51 101)(39 116 52 105)(40 120 53 94)(41 109 54 98)(42 113 55 102)(43 117 56 91)(44 106 57 95)(45 110 58 99)

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

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

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

63 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 4C 5A 5B 6A 6B 6C 6D 6E 10A 10B 12A 12B 12C ··· 12J 12K 12L 15A 15B 15C 15D 15E ··· 15J 20A 20B 20C 20D 30A 30B 30C 30D 30E ··· 30J 60A ··· 60H order 1 2 3 3 3 3 3 4 4 4 5 5 6 6 6 6 6 10 10 12 12 12 ··· 12 12 12 15 15 15 15 15 ··· 15 20 20 20 20 30 30 30 30 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 6 10 30 2 2 1 1 2 2 2 2 2 6 6 10 ··· 10 30 30 2 2 2 2 4 ··· 4 6 6 6 6 2 2 2 2 4 ··· 4 6 ··· 6

63 irreducible representations

 dim 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 C3 C6 C6 C6 S3 Q8 D5 D6 C3×S3 D10 Dic6 C3×Q8 C3×D5 S3×C6 Dic10 C6×D5 C3×Dic6 C3×Dic10 S3×D5 C15⋊Q8 C3×S3×D5 C3×C15⋊Q8 kernel C3×C15⋊Q8 C32×Dic5 Dic3×C15 C3×Dic15 C15⋊Q8 C5×Dic3 C3×Dic5 Dic15 C3×Dic5 C3×C15 C3×Dic3 C30 Dic5 C3×C6 C15 C15 Dic3 C10 C32 C6 C5 C3 C6 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 1 2 1 2 2 2 2 4 2 4 4 4 8 2 2 4 4

Matrix representation of C3×C15⋊Q8 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 47 0 0 0 0 47
,
 60 44 0 0 17 44 0 0 0 0 47 0 0 0 25 13
,
 1 0 0 0 0 1 0 0 0 0 34 5 0 0 37 27
,
 8 20 0 0 6 53 0 0 0 0 11 0 0 0 9 50
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,47,0,0,0,0,47],[60,17,0,0,44,44,0,0,0,0,47,25,0,0,0,13],[1,0,0,0,0,1,0,0,0,0,34,37,0,0,5,27],[8,6,0,0,20,53,0,0,0,0,11,9,0,0,0,50] >;

C3×C15⋊Q8 in GAP, Magma, Sage, TeX

C_3\times C_{15}\rtimes Q_8
% in TeX

G:=Group("C3xC15:Q8");
// GroupNames label

G:=SmallGroup(360,64);
// by ID

G=gap.SmallGroup(360,64);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,72,169,79,730,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^15=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^11,d*b*d^-1=b^4,d*c*d^-1=c^-1>;
// generators/relations

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