direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C4○D12, D12⋊5C10, C20.60D6, Dic6⋊5C10, C30.51C23, C60.68C22, (C2×C20)⋊7S3, (S3×C20)⋊9C2, (C4×S3)⋊4C10, (C2×C12)⋊4C10, (C2×C60)⋊11C2, C3⋊D4⋊3C10, (C5×D12)⋊11C2, C15⋊13(C4○D4), C4.16(S3×C10), D6.1(C2×C10), (C2×C10).20D6, C12.16(C2×C10), (C5×Dic6)⋊11C2, C22.2(S3×C10), C6.4(C22×C10), C10.41(C22×S3), (C2×C30).50C22, Dic3.2(C2×C10), (S3×C10).12C22, (C5×Dic3).14C22, C3⋊1(C5×C4○D4), (C2×C4)⋊3(C5×S3), C2.5(S3×C2×C10), (C5×C3⋊D4)⋊7C2, (C2×C6).11(C2×C10), SmallGroup(240,168)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4○D12
G = < a,b,c,d | a5=b4=d2=1, c6=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c5 >
Subgroups: 152 in 80 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C10, C10, Dic3, C12, D6, C2×C6, C15, C4○D4, C20, C20, C2×C10, C2×C10, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C5×S3, C30, C30, C2×C20, C2×C20, C5×D4, C5×Q8, C4○D12, C5×Dic3, C60, S3×C10, C2×C30, C5×C4○D4, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, C5×C4○D12
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, C4○D4, C2×C10, C22×S3, C5×S3, C22×C10, C4○D12, S3×C10, C5×C4○D4, S3×C2×C10, C5×C4○D12
►On 120 points
Generators in S120
(1 54 100 28 92)(2 55 101 29 93)(3 56 102 30 94)(4 57 103 31 95)(5 58 104 32 96)(6 59 105 33 85)(7 60 106 34 86)(8 49 107 35 87)(9 50 108 36 88)(10 51 97 25 89)(11 52 98 26 90)(12 53 99 27 91)(13 37 115 74 61)(14 38 116 75 62)(15 39 117 76 63)(16 40 118 77 64)(17 41 119 78 65)(18 42 120 79 66)(19 43 109 80 67)(20 44 110 81 68)(21 45 111 82 69)(22 46 112 83 70)(23 47 113 84 71)(24 48 114 73 72)
(1 114 7 120)(2 115 8 109)(3 116 9 110)(4 117 10 111)(5 118 11 112)(6 119 12 113)(13 35 19 29)(14 36 20 30)(15 25 21 31)(16 26 22 32)(17 27 23 33)(18 28 24 34)(37 87 43 93)(38 88 44 94)(39 89 45 95)(40 90 46 96)(41 91 47 85)(42 92 48 86)(49 80 55 74)(50 81 56 75)(51 82 57 76)(52 83 58 77)(53 84 59 78)(54 73 60 79)(61 107 67 101)(62 108 68 102)(63 97 69 103)(64 98 70 104)(65 99 71 105)(66 100 72 106)
(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 3)(4 12)(5 11)(6 10)(7 9)(14 24)(15 23)(16 22)(17 21)(18 20)(25 33)(26 32)(27 31)(28 30)(34 36)(38 48)(39 47)(40 46)(41 45)(42 44)(50 60)(51 59)(52 58)(53 57)(54 56)(62 72)(63 71)(64 70)(65 69)(66 68)(73 75)(76 84)(77 83)(78 82)(79 81)(85 89)(86 88)(90 96)(91 95)(92 94)(97 105)(98 104)(99 103)(100 102)(106 108)(110 120)(111 119)(112 118)(113 117)(114 116)
G:=sub<Sym(120)| (1,54,100,28,92)(2,55,101,29,93)(3,56,102,30,94)(4,57,103,31,95)(5,58,104,32,96)(6,59,105,33,85)(7,60,106,34,86)(8,49,107,35,87)(9,50,108,36,88)(10,51,97,25,89)(11,52,98,26,90)(12,53,99,27,91)(13,37,115,74,61)(14,38,116,75,62)(15,39,117,76,63)(16,40,118,77,64)(17,41,119,78,65)(18,42,120,79,66)(19,43,109,80,67)(20,44,110,81,68)(21,45,111,82,69)(22,46,112,83,70)(23,47,113,84,71)(24,48,114,73,72), (1,114,7,120)(2,115,8,109)(3,116,9,110)(4,117,10,111)(5,118,11,112)(6,119,12,113)(13,35,19,29)(14,36,20,30)(15,25,21,31)(16,26,22,32)(17,27,23,33)(18,28,24,34)(37,87,43,93)(38,88,44,94)(39,89,45,95)(40,90,46,96)(41,91,47,85)(42,92,48,86)(49,80,55,74)(50,81,56,75)(51,82,57,76)(52,83,58,77)(53,84,59,78)(54,73,60,79)(61,107,67,101)(62,108,68,102)(63,97,69,103)(64,98,70,104)(65,99,71,105)(66,100,72,106), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(14,24)(15,23)(16,22)(17,21)(18,20)(25,33)(26,32)(27,31)(28,30)(34,36)(38,48)(39,47)(40,46)(41,45)(42,44)(50,60)(51,59)(52,58)(53,57)(54,56)(62,72)(63,71)(64,70)(65,69)(66,68)(73,75)(76,84)(77,83)(78,82)(79,81)(85,89)(86,88)(90,96)(91,95)(92,94)(97,105)(98,104)(99,103)(100,102)(106,108)(110,120)(111,119)(112,118)(113,117)(114,116)>;
G:=Group( (1,54,100,28,92)(2,55,101,29,93)(3,56,102,30,94)(4,57,103,31,95)(5,58,104,32,96)(6,59,105,33,85)(7,60,106,34,86)(8,49,107,35,87)(9,50,108,36,88)(10,51,97,25,89)(11,52,98,26,90)(12,53,99,27,91)(13,37,115,74,61)(14,38,116,75,62)(15,39,117,76,63)(16,40,118,77,64)(17,41,119,78,65)(18,42,120,79,66)(19,43,109,80,67)(20,44,110,81,68)(21,45,111,82,69)(22,46,112,83,70)(23,47,113,84,71)(24,48,114,73,72), (1,114,7,120)(2,115,8,109)(3,116,9,110)(4,117,10,111)(5,118,11,112)(6,119,12,113)(13,35,19,29)(14,36,20,30)(15,25,21,31)(16,26,22,32)(17,27,23,33)(18,28,24,34)(37,87,43,93)(38,88,44,94)(39,89,45,95)(40,90,46,96)(41,91,47,85)(42,92,48,86)(49,80,55,74)(50,81,56,75)(51,82,57,76)(52,83,58,77)(53,84,59,78)(54,73,60,79)(61,107,67,101)(62,108,68,102)(63,97,69,103)(64,98,70,104)(65,99,71,105)(66,100,72,106), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(14,24)(15,23)(16,22)(17,21)(18,20)(25,33)(26,32)(27,31)(28,30)(34,36)(38,48)(39,47)(40,46)(41,45)(42,44)(50,60)(51,59)(52,58)(53,57)(54,56)(62,72)(63,71)(64,70)(65,69)(66,68)(73,75)(76,84)(77,83)(78,82)(79,81)(85,89)(86,88)(90,96)(91,95)(92,94)(97,105)(98,104)(99,103)(100,102)(106,108)(110,120)(111,119)(112,118)(113,117)(114,116) );
G=PermutationGroup([[(1,54,100,28,92),(2,55,101,29,93),(3,56,102,30,94),(4,57,103,31,95),(5,58,104,32,96),(6,59,105,33,85),(7,60,106,34,86),(8,49,107,35,87),(9,50,108,36,88),(10,51,97,25,89),(11,52,98,26,90),(12,53,99,27,91),(13,37,115,74,61),(14,38,116,75,62),(15,39,117,76,63),(16,40,118,77,64),(17,41,119,78,65),(18,42,120,79,66),(19,43,109,80,67),(20,44,110,81,68),(21,45,111,82,69),(22,46,112,83,70),(23,47,113,84,71),(24,48,114,73,72)], [(1,114,7,120),(2,115,8,109),(3,116,9,110),(4,117,10,111),(5,118,11,112),(6,119,12,113),(13,35,19,29),(14,36,20,30),(15,25,21,31),(16,26,22,32),(17,27,23,33),(18,28,24,34),(37,87,43,93),(38,88,44,94),(39,89,45,95),(40,90,46,96),(41,91,47,85),(42,92,48,86),(49,80,55,74),(50,81,56,75),(51,82,57,76),(52,83,58,77),(53,84,59,78),(54,73,60,79),(61,107,67,101),(62,108,68,102),(63,97,69,103),(64,98,70,104),(65,99,71,105),(66,100,72,106)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(14,24),(15,23),(16,22),(17,21),(18,20),(25,33),(26,32),(27,31),(28,30),(34,36),(38,48),(39,47),(40,46),(41,45),(42,44),(50,60),(51,59),(52,58),(53,57),(54,56),(62,72),(63,71),(64,70),(65,69),(66,68),(73,75),(76,84),(77,83),(78,82),(79,81),(85,89),(86,88),(90,96),(91,95),(92,94),(97,105),(98,104),(99,103),(100,102),(106,108),(110,120),(111,119),(112,118),(113,117),(114,116)]])
C5×C4○D12 is a maximal subgroup of
C60.98D4 C60.99D4 D12.2Dic5 D12.Dic5 D20.34D6 C60.36D4 C20.60D12 C60.38D4 D12.37D10 D12.33D10 D20.39D6 C30.C24 D20⋊24D6 D20⋊25D6 D20⋊29D6 C5×S3×C4○D4
C5×C4○D12 is a maximal quotient of
C20×Dic6 C20×D12 C20×C3⋊D4
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10P | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | ··· | 20T | 30A | ··· | 30L | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 6 | 6 | 2 | 1 | 1 | 2 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | S3 | D6 | D6 | C4○D4 | C5×S3 | C4○D12 | S3×C10 | S3×C10 | C5×C4○D4 | C5×C4○D12 |
| kernel | C5×C4○D12 | C5×Dic6 | S3×C20 | C5×D12 | C5×C3⋊D4 | C2×C60 | C4○D12 | Dic6 | C4×S3 | D12 | C3⋊D4 | C2×C12 | C2×C20 | C20 | C2×C10 | C15 | C2×C4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 8 | 4 | 1 | 2 | 1 | 2 | 4 | 4 | 8 | 4 | 8 | 16 |
Matrix representation of C5×C4○D12 ►in GL2(𝔽61) generated by
| 20 | 0 |
| 0 | 20 |
| 11 | 0 |
| 0 | 11 |
| 23 | 23 |
| 38 | 46 |
| 0 | 60 |
| 60 | 0 |
G:=sub<GL(2,GF(61))| [20,0,0,20],[11,0,0,11],[23,38,23,46],[0,60,60,0] >;
C5×C4○D12 in GAP, Magma, Sage, TeX
C_5\times C_4\circ D_{12} % in TeX
G:=Group("C5xC4oD12"); // GroupNames label
G:=SmallGroup(240,168);
// by ID
G=gap.SmallGroup(240,168);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-3,247,794,5765]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=d^2=1,c^6=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^5>;
// generators/relations