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## G = C3×D10.3Q8order 480 = 25·3·5

### Direct product of C3 and D10.3Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×D10.3Q8
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — D5×C2×C6 — C2×C6×F5 — C3×D10.3Q8
 Lower central C5 — C10 — C3×D10.3Q8
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C3×D10.3Q8
G = < a,b,c,d,e | a3=b10=c2=d4=1, e2=b4cd2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b3, cd=dc, ece-1=b2c, ede-1=b5d-1 >

Subgroups: 536 in 152 conjugacy classes, 64 normal (36 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, D5, C10, C12, C2×C6, C2×C6, C15, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2×C12, C2×C12, C22×C6, C3×D5, C30, C2.C42, C4×D5, C2×Dic5, C2×C20, C2×F5, C2×F5, C22×D5, C22×C12, C3×Dic5, C60, C3×F5, C6×D5, C6×D5, C2×C30, C2×C4×D5, C22×F5, C3×C2.C42, D5×C12, C6×Dic5, C2×C60, C6×F5, C6×F5, D5×C2×C6, D10.3Q8, D5×C2×C12, C2×C6×F5, C3×D10.3Q8
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C12, C2×C6, C42, C22⋊C4, C4⋊C4, F5, C2×C12, C3×D4, C3×Q8, C2.C42, C2×F5, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C3×F5, C4×F5, C4⋊F5, C22⋊F5, C3×C2.C42, C6×F5, D10.3Q8, C12×F5, C3×C4⋊F5, C3×C22⋊F5, C3×D10.3Q8

Smallest permutation representation of C3×D10.3Q8
On 120 points
Generators in S120
(1 70 50)(2 61 41)(3 62 42)(4 63 43)(5 64 44)(6 65 45)(7 66 46)(8 67 47)(9 68 48)(10 69 49)(11 105 85)(12 106 86)(13 107 87)(14 108 88)(15 109 89)(16 110 90)(17 101 81)(18 102 82)(19 103 83)(20 104 84)(21 57 37)(22 58 38)(23 59 39)(24 60 40)(25 51 31)(26 52 32)(27 53 33)(28 54 34)(29 55 35)(30 56 36)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)
(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 25)(2 24)(3 23)(4 22)(5 21)(6 30)(7 29)(8 28)(9 27)(10 26)(11 116)(12 115)(13 114)(14 113)(15 112)(16 111)(17 120)(18 119)(19 118)(20 117)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(71 90)(72 89)(73 88)(74 87)(75 86)(76 85)(77 84)(78 83)(79 82)(80 81)(91 110)(92 109)(93 108)(94 107)(95 106)(96 105)(97 104)(98 103)(99 102)(100 101)
(1 77 21 90)(2 78 22 81)(3 79 23 82)(4 80 24 83)(5 71 25 84)(6 72 26 85)(7 73 27 86)(8 74 28 87)(9 75 29 88)(10 76 30 89)(11 65 112 52)(12 66 113 53)(13 67 114 54)(14 68 115 55)(15 69 116 56)(16 70 117 57)(17 61 118 58)(18 62 119 59)(19 63 120 60)(20 64 111 51)(31 104 44 91)(32 105 45 92)(33 106 46 93)(34 107 47 94)(35 108 48 95)(36 109 49 96)(37 110 50 97)(38 101 41 98)(39 102 42 99)(40 103 43 100)
(2 8 10 4)(3 5 9 7)(11 117)(12 114 20 120)(13 111 19 113)(14 118 18 116)(15 115 17 119)(16 112)(22 28 30 24)(23 25 29 27)(31 35 33 39)(34 36 40 38)(41 47 49 43)(42 44 48 46)(51 55 53 59)(54 56 60 58)(61 67 69 63)(62 64 68 66)(71 83 73 87)(72 90)(74 84 80 86)(75 81 79 89)(76 88 78 82)(77 85)(91 103 93 107)(92 110)(94 104 100 106)(95 101 99 109)(96 108 98 102)(97 105)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C ··· 4L 5 6A ··· 6F 6G ··· 6N 10A 10B 10C 12A 12B 12C 12D 12E ··· 12X 15A 15B 20A 20B 20C 20D 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 2 2 3 3 4 4 4 ··· 4 5 6 ··· 6 6 ··· 6 10 10 10 12 12 12 12 12 ··· 12 15 15 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 1 1 5 5 5 5 1 1 2 2 10 ··· 10 4 1 ··· 1 5 ··· 5 4 4 4 2 2 2 2 10 ··· 10 4 4 4 4 4 4 4 ··· 4 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 4 4 4 4 4 4 type + + + + - + + + image C1 C2 C2 C3 C4 C4 C4 C6 C6 C12 C12 C12 D4 Q8 C3×D4 C3×Q8 F5 C2×F5 C3×F5 C4×F5 C4⋊F5 C22⋊F5 C6×F5 C12×F5 C3×C4⋊F5 C3×C22⋊F5 kernel C3×D10.3Q8 D5×C2×C12 C2×C6×F5 D10.3Q8 C6×Dic5 C2×C60 C6×F5 C2×C4×D5 C22×F5 C2×Dic5 C2×C20 C2×F5 C6×D5 C6×D5 D10 D10 C2×C12 C2×C6 C2×C4 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 2 2 2 2 8 2 4 4 4 16 3 1 6 2 1 1 2 2 2 2 2 4 4 4

Matrix representation of C3×D10.3Q8 in GL6(𝔽61)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 60 0 0 0 0 1 0 60 0 0 0 1 0 0 0 0 60 1 0 0
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 1 60 0 0 0 1 0 60 0 0 1 0 0 60 0 0 0 0 0 60
,
 42 5 0 0 0 0 13 19 0 0 0 0 0 0 54 14 0 47 0 0 0 7 14 47 0 0 47 14 7 0 0 0 47 0 14 54
,
 1 0 0 0 0 0 32 60 0 0 0 0 0 0 1 0 60 0 0 0 0 0 60 1 0 0 0 1 60 0 0 0 0 0 60 0

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,0,0,1,1,1,1,0,0,60,0,0,0,0,0,0,60,0,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,60,60,60,60],[42,13,0,0,0,0,5,19,0,0,0,0,0,0,54,0,47,47,0,0,14,7,14,0,0,0,0,14,7,14,0,0,47,47,0,54],[1,32,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,60,60,60,60,0,0,0,1,0,0] >;

C3×D10.3Q8 in GAP, Magma, Sage, TeX

C_3\times D_{10}._3Q_8
% in TeX

G:=Group("C3xD10.3Q8");
// GroupNames label

G:=SmallGroup(480,286);
// by ID

G=gap.SmallGroup(480,286);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,701,176,9414,1595]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^10=c^2=d^4=1,e^2=b^4*c*d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=b^3,c*d=d*c,e*c*e^-1=b^2*c,e*d*e^-1=b^5*d^-1>;
// generators/relations

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