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## G = C3×D10.D4order 480 = 25·3·5

### Direct product of C3 and D10.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×D10.D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — D5×C2×C6 — C3×C22⋊F5 — C3×D10.D4
 Lower central C5 — C10 — C2×C10 — C3×D10.D4
 Upper central C1 — C6 — C2×C6 — C2×C12

Generators and relations for C3×D10.D4
G = < a,b,c,d,e | a3=b10=c2=d4=1, e2=b-1c, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=ebe-1=b3, dcd-1=b7c, ece-1=b2c, ede-1=b4cd-1 >

Subgroups: 536 in 104 conjugacy classes, 32 normal (24 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, C12, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, C20, F5, D10, D10, C2×C10, C2×C12, C2×C12, C3×D4, C22×C6, C3×D5, C30, C30, C23⋊C4, D20, C2×C20, C2×F5, C22×D5, C3×C22⋊C4, C6×D4, C60, C3×F5, C6×D5, C6×D5, C2×C30, C22⋊F5, C2×D20, C3×C23⋊C4, C3×D20, C2×C60, C6×F5, D5×C2×C6, D10.D4, C3×C22⋊F5, C6×D20, C3×D10.D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, F5, C2×C12, C3×D4, C23⋊C4, C2×F5, C3×C22⋊C4, C3×F5, C22⋊F5, C3×C23⋊C4, C6×F5, D10.D4, C3×C22⋊F5, C3×D10.D4

Smallest permutation representation of C3×D10.D4
On 120 points
Generators in S120
(1 47 27)(2 48 28)(3 49 29)(4 50 30)(5 41 21)(6 42 22)(7 43 23)(8 44 24)(9 45 25)(10 46 26)(11 51 31)(12 52 32)(13 53 33)(14 54 34)(15 55 35)(16 56 36)(17 57 37)(18 58 38)(19 59 39)(20 60 40)(61 101 81)(62 102 82)(63 103 83)(64 104 84)(65 105 85)(66 106 86)(67 107 87)(68 108 88)(69 109 89)(70 110 90)(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 10)(2 9)(3 8)(4 7)(5 6)(11 12)(13 20)(14 19)(15 18)(16 17)(21 22)(23 30)(24 29)(25 28)(26 27)(31 32)(33 40)(34 39)(35 38)(36 37)(41 42)(43 50)(44 49)(45 48)(46 47)(51 52)(53 60)(54 59)(55 58)(56 57)(61 67)(62 66)(63 65)(68 70)(71 77)(72 76)(73 75)(78 80)(81 87)(82 86)(83 85)(88 90)(91 97)(92 96)(93 95)(98 100)(101 107)(102 106)(103 105)(108 110)(111 117)(112 116)(113 115)(118 120)
(1 62 12 72)(2 69 11 75)(3 66 20 78)(4 63 19 71)(5 70 18 74)(6 67 17 77)(7 64 16 80)(8 61 15 73)(9 68 14 76)(10 65 13 79)(21 90 38 94)(22 87 37 97)(23 84 36 100)(24 81 35 93)(25 88 34 96)(26 85 33 99)(27 82 32 92)(28 89 31 95)(29 86 40 98)(30 83 39 91)(41 110 58 114)(42 107 57 117)(43 104 56 120)(44 101 55 113)(45 108 54 116)(46 105 53 119)(47 102 52 112)(48 109 51 115)(49 106 60 118)(50 103 59 111)
(2 8 10 4)(3 5 9 7)(11 20 13 14)(12 17)(15 18 19 16)(21 25 23 29)(24 26 30 28)(31 40 33 34)(32 37)(35 38 39 36)(41 45 43 49)(44 46 50 48)(51 60 53 54)(52 57)(55 58 59 56)(61 75 68 74)(62 72 67 77)(63 79 66 80)(64 76 65 73)(69 71 70 78)(81 95 88 94)(82 92 87 97)(83 99 86 100)(84 96 85 93)(89 91 90 98)(101 115 108 114)(102 112 107 117)(103 119 106 120)(104 116 105 113)(109 111 110 118)

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

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

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

57 conjugacy classes

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

57 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 4 4 4 4 type + + + + + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 D4 C3×D4 F5 C23⋊C4 C2×F5 C3×F5 C22⋊F5 C3×C23⋊C4 C6×F5 D10.D4 C3×C22⋊F5 C3×D10.D4 kernel C3×D10.D4 C3×C22⋊F5 C6×D20 D10.D4 C2×C60 D5×C2×C6 C22⋊F5 C2×D20 C2×C20 C22×D5 C6×D5 D10 C2×C12 C15 C2×C6 C2×C4 C6 C5 C22 C3 C2 C1 # reps 1 2 1 2 2 2 4 2 4 4 2 4 1 1 1 2 2 2 2 4 4 8

Matrix representation of C3×D10.D4 in GL4(𝔽61) generated by

 13 0 0 0 0 13 0 0 0 0 13 0 0 0 0 13
,
 0 0 60 0 0 0 0 60 1 1 1 1 60 0 0 0
,
 0 0 1 0 0 1 0 0 1 0 0 0 60 60 60 60
,
 8 16 49 28 33 12 45 53 33 41 49 21 8 41 20 53
,
 7 0 14 14 14 14 0 7 47 54 47 0 54 7 7 54
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[0,0,1,60,0,0,1,0,60,0,1,0,0,60,1,0],[0,0,1,60,0,1,0,60,1,0,0,60,0,0,0,60],[8,33,33,8,16,12,41,41,49,45,49,20,28,53,21,53],[7,14,47,54,0,14,54,7,14,0,47,7,14,7,0,54] >;

C3×D10.D4 in GAP, Magma, Sage, TeX

C_3\times D_{10}.D_4
% in TeX

G:=Group("C3xD10.D4");
// GroupNames label

G:=SmallGroup(480,279);
// by ID

G=gap.SmallGroup(480,279);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,344,850,2524,9414,1595]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^10=c^2=d^4=1,e^2=b^-1*c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d^-1=e*b*e^-1=b^3,d*c*d^-1=b^7*c,e*c*e^-1=b^2*c,e*d*e^-1=b^4*c*d^-1>;
// generators/relations

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