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G = C3×C15⋊Q8order 360 = 23·32·5

Direct product of C3 and C15⋊Q8

direct product, metabelian, supersoluble, monomial

Aliases: C3×C15⋊Q8, C154Dic6, C30.34D6, C324Dic10, Dic15.2C6, C15⋊(C3×Q8), (C3×C15)⋊3Q8, C6.7(C6×D5), C51(C3×Dic6), C10.7(S3×C6), C30.7(C2×C6), C6.34(S3×D5), Dic3.(C3×D5), C31(C3×Dic10), (C3×C6).19D10, (C3×C30).7C22, (C3×Dic3).2D5, (C3×Dic5).6S3, (C3×Dic5).1C6, (C5×Dic3).1C6, Dic5.1(C3×S3), (Dic3×C15).2C2, (C3×Dic15).2C2, (C32×Dic5).1C2, C2.7(C3×S3×D5), SmallGroup(360,64)

Series: Derived Chief Lower central Upper central

C1C30 — C3×C15⋊Q8
C1C5C15C30C3×C30C32×Dic5 — C3×C15⋊Q8
C15C30 — C3×C15⋊Q8
C1C6

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 3A3B3C3D3E4A4B4C5A5B6A6B6C6D6E10A10B12A12B12C···12J12K12L15A15B15C15D15E···15J20A20B20C20D30A30B30C30D30E···30J60A···60H
order123333344455666661010121212···1212121515151515···15202020203030303030···3060···60
size1111222610302211222226610···10303022224···4666622224···46···6

63 irreducible representations

dim11111111222222222222224444
type+++++-+++--+-
imageC1C2C2C2C3C6C6C6S3Q8D5D6C3×S3D10Dic6C3×Q8C3×D5S3×C6Dic10C6×D5C3×Dic6C3×Dic10S3×D5C15⋊Q8C3×S3×D5C3×C15⋊Q8
kernelC3×C15⋊Q8C32×Dic5Dic3×C15C3×Dic15C15⋊Q8C5×Dic3C3×Dic5Dic15C3×Dic5C3×C15C3×Dic3C30Dic5C3×C6C15C15Dic3C10C32C6C5C3C6C3C2C1
# reps11112222112122224244482244

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

1000
0100
00470
00047
,
604400
174400
00470
002513
,
1000
0100
00345
003727
,
82000
65300
00110
00950
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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