direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D10⋊C4, D10⋊1C12, C30.27D4, C6.17D20, (C2×C20)⋊1C6, (C2×C60)⋊1C2, (C6×D5)⋊3C4, (C2×C12)⋊1D5, C2.5(D5×C12), C2.2(C3×D20), C6.19(C4×D5), C10.6(C3×D4), C15⋊8(C22⋊C4), C30.44(C2×C4), (C6×Dic5)⋊7C2, (C2×Dic5)⋊1C6, (C2×C6).34D10, C22.6(C6×D5), C10.11(C2×C12), C6.22(C5⋊D4), (C22×D5).2C6, (C2×C30).35C22, (C2×C4)⋊1(C3×D5), C5⋊2(C3×C22⋊C4), (D5×C2×C6).3C2, C2.2(C3×C5⋊D4), (C2×C10).6(C2×C6), SmallGroup(240,43)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D10⋊C4
G = < a,b,c,d | a3=b10=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b5c >
Subgroups: 212 in 68 conjugacy classes, 34 normal (30 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, D10, D10, C2×C10, C2×C12, C2×C12, C22×C6, C3×D5, C30, C2×Dic5, C2×C20, C22×D5, C3×C22⋊C4, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, D10⋊C4, C6×Dic5, C2×C60, D5×C2×C6, C3×D10⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, D5, C12, C2×C6, C22⋊C4, D10, C2×C12, C3×D4, C3×D5, C4×D5, D20, C5⋊D4, C3×C22⋊C4, C6×D5, D10⋊C4, D5×C12, C3×D20, C3×C5⋊D4, C3×D10⋊C4
►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 77 17 67)(2 78 18 68)(3 79 19 69)(4 80 20 70)(5 71 11 61)(6 72 12 62)(7 73 13 63)(8 74 14 64)(9 75 15 65)(10 76 16 66)(21 91 31 81)(22 92 32 82)(23 93 33 83)(24 94 34 84)(25 95 35 85)(26 96 36 86)(27 97 37 87)(28 98 38 88)(29 99 39 89)(30 100 40 90)(41 111 51 101)(42 112 52 102)(43 113 53 103)(44 114 54 104)(45 115 55 105)(46 116 56 106)(47 117 57 107)(48 118 58 108)(49 119 59 109)(50 120 60 110)
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), (1,77,17,67)(2,78,18,68)(3,79,19,69)(4,80,20,70)(5,71,11,61)(6,72,12,62)(7,73,13,63)(8,74,14,64)(9,75,15,65)(10,76,16,66)(21,91,31,81)(22,92,32,82)(23,93,33,83)(24,94,34,84)(25,95,35,85)(26,96,36,86)(27,97,37,87)(28,98,38,88)(29,99,39,89)(30,100,40,90)(41,111,51,101)(42,112,52,102)(43,113,53,103)(44,114,54,104)(45,115,55,105)(46,116,56,106)(47,117,57,107)(48,118,58,108)(49,119,59,109)(50,120,60,110)>;
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), (1,77,17,67)(2,78,18,68)(3,79,19,69)(4,80,20,70)(5,71,11,61)(6,72,12,62)(7,73,13,63)(8,74,14,64)(9,75,15,65)(10,76,16,66)(21,91,31,81)(22,92,32,82)(23,93,33,83)(24,94,34,84)(25,95,35,85)(26,96,36,86)(27,97,37,87)(28,98,38,88)(29,99,39,89)(30,100,40,90)(41,111,51,101)(42,112,52,102)(43,113,53,103)(44,114,54,104)(45,115,55,105)(46,116,56,106)(47,117,57,107)(48,118,58,108)(49,119,59,109)(50,120,60,110) );
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)], [(1,77,17,67),(2,78,18,68),(3,79,19,69),(4,80,20,70),(5,71,11,61),(6,72,12,62),(7,73,13,63),(8,74,14,64),(9,75,15,65),(10,76,16,66),(21,91,31,81),(22,92,32,82),(23,93,33,83),(24,94,34,84),(25,95,35,85),(26,96,36,86),(27,97,37,87),(28,98,38,88),(29,99,39,89),(30,100,40,90),(41,111,51,101),(42,112,52,102),(43,113,53,103),(44,114,54,104),(45,115,55,105),(46,116,56,106),(47,117,57,107),(48,118,58,108),(49,119,59,109),(50,120,60,110)]])
C3×D10⋊C4 is a maximal subgroup of
Dic3.D20 D30.34D4 D30.D4 (C2×C12).D10 (C2×C60).C22 (C4×Dic3)⋊D5 (C4×Dic15)⋊C2 (D5×Dic3)⋊C4 D10.19(C4×S3) Dic3⋊4D20 Dic15⋊13D4 D10.16D12 D10.17D12 D10⋊1Dic6 D10⋊2Dic6 Dic15.D4 D10⋊4Dic6 C15⋊17(C4×D4) Dic15⋊9D4 D10⋊C4⋊S3 Dic15⋊2D4 D6⋊D20 (C2×Dic6)⋊D5 D6.9D20 D30⋊2D4 D30⋊12D4 Dic15.10D4 Dic15.31D4 D30.27D4 D30⋊4D4 D30⋊5D4 C12×D20 C3×D5×C22⋊C4 C12×C5⋊D4
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 1 | 1 | 2 | 2 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D4 | D5 | D10 | C3×D4 | C3×D5 | C4×D5 | D20 | C5⋊D4 | C6×D5 | D5×C12 | C3×D20 | C3×C5⋊D4 |
| kernel | C3×D10⋊C4 | C6×Dic5 | C2×C60 | D5×C2×C6 | D10⋊C4 | C6×D5 | C2×Dic5 | C2×C20 | C22×D5 | D10 | C30 | C2×C12 | C2×C6 | C10 | C2×C4 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
Matrix representation of C3×D10⋊C4 ►in GL3(𝔽61) generated by
| 1 | 0 | 0 |
| 0 | 47 | 0 |
| 0 | 0 | 47 |
| 1 | 0 | 0 |
| 0 | 18 | 44 |
| 0 | 18 | 0 |
| 60 | 0 | 0 |
| 0 | 60 | 1 |
| 0 | 0 | 1 |
| 11 | 0 | 0 |
| 0 | 31 | 16 |
| 0 | 1 | 30 |
G:=sub<GL(3,GF(61))| [1,0,0,0,47,0,0,0,47],[1,0,0,0,18,18,0,44,0],[60,0,0,0,60,0,0,1,1],[11,0,0,0,31,1,0,16,30] >;
C3×D10⋊C4 in GAP, Magma, Sage, TeX
C_3\times D_{10}\rtimes C_4 % in TeX
G:=Group("C3xD10:C4"); // GroupNames label
G:=SmallGroup(240,43);
// by ID
G=gap.SmallGroup(240,43);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-5,313,79,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^10=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^5*c>;
// generators/relations