direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C5⋊D12, C30⋊3D4, D6⋊6D10, C10⋊2D12, Dic5⋊4D6, D30⋊9C22, C30.23C23, C15⋊6(C2×D4), C5⋊3(C2×D12), C6⋊1(C5⋊D4), (C2×Dic5)⋊4S3, (C6×Dic5)⋊6C2, (C2×C10).18D6, (C2×C6).18D10, (C22×S3)⋊2D5, (S3×C10)⋊6C22, (C22×D15)⋊5C2, C22.16(S3×D5), C6.23(C22×D5), C10.23(C22×S3), (C2×C30).17C22, (C3×Dic5)⋊7C22, C3⋊1(C2×C5⋊D4), (S3×C2×C10)⋊2C2, C2.23(C2×S3×D5), SmallGroup(240,147)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C5⋊D12
G = < a,b,c,d | a2=b5=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 528 in 108 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, D5, C10, C10, C10, C12, D6, D6, C2×C6, C15, C2×D4, Dic5, D10, C2×C10, C2×C10, D12, C2×C12, C22×S3, C22×S3, C5×S3, D15, C30, C30, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C2×D12, C3×Dic5, S3×C10, S3×C10, D30, D30, C2×C30, C2×C5⋊D4, C5⋊D12, C6×Dic5, S3×C2×C10, C22×D15, C2×C5⋊D12
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, D12, C22×S3, C5⋊D4, C22×D5, C2×D12, S3×D5, C2×C5⋊D4, C5⋊D12, C2×S3×D5, C2×C5⋊D12
►On 120 points
Generators in S120
(1 20)(2 21)(3 22)(4 23)(5 24)(6 13)(7 14)(8 15)(9 16)(10 17)(11 18)(12 19)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 37)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 119)(62 120)(63 109)(64 110)(65 111)(66 112)(67 113)(68 114)(69 115)(70 116)(71 117)(72 118)(73 106)(74 107)(75 108)(76 97)(77 98)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)
(1 25 114 54 76)(2 77 55 115 26)(3 27 116 56 78)(4 79 57 117 28)(5 29 118 58 80)(6 81 59 119 30)(7 31 120 60 82)(8 83 49 109 32)(9 33 110 50 84)(10 73 51 111 34)(11 35 112 52 74)(12 75 53 113 36)(13 102 95 61 43)(14 44 62 96 103)(15 104 85 63 45)(16 46 64 86 105)(17 106 87 65 47)(18 48 66 88 107)(19 108 89 67 37)(20 38 68 90 97)(21 98 91 69 39)(22 40 70 92 99)(23 100 93 71 41)(24 42 72 94 101)
(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 9)(2 8)(3 7)(4 6)(10 12)(13 23)(14 22)(15 21)(16 20)(17 19)(25 84)(26 83)(27 82)(28 81)(29 80)(30 79)(31 78)(32 77)(33 76)(34 75)(35 74)(36 73)(37 106)(38 105)(39 104)(40 103)(41 102)(42 101)(43 100)(44 99)(45 98)(46 97)(47 108)(48 107)(49 115)(50 114)(51 113)(52 112)(53 111)(54 110)(55 109)(56 120)(57 119)(58 118)(59 117)(60 116)(61 93)(62 92)(63 91)(64 90)(65 89)(66 88)(67 87)(68 86)(69 85)(70 96)(71 95)(72 94)
G:=sub<Sym(120)| (1,20)(2,21)(3,22)(4,23)(5,24)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,37)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,119)(62,120)(63,109)(64,110)(65,111)(66,112)(67,113)(68,114)(69,115)(70,116)(71,117)(72,118)(73,106)(74,107)(75,108)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (1,25,114,54,76)(2,77,55,115,26)(3,27,116,56,78)(4,79,57,117,28)(5,29,118,58,80)(6,81,59,119,30)(7,31,120,60,82)(8,83,49,109,32)(9,33,110,50,84)(10,73,51,111,34)(11,35,112,52,74)(12,75,53,113,36)(13,102,95,61,43)(14,44,62,96,103)(15,104,85,63,45)(16,46,64,86,105)(17,106,87,65,47)(18,48,66,88,107)(19,108,89,67,37)(20,38,68,90,97)(21,98,91,69,39)(22,40,70,92,99)(23,100,93,71,41)(24,42,72,94,101), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,23)(14,22)(15,21)(16,20)(17,19)(25,84)(26,83)(27,82)(28,81)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,108)(48,107)(49,115)(50,114)(51,113)(52,112)(53,111)(54,110)(55,109)(56,120)(57,119)(58,118)(59,117)(60,116)(61,93)(62,92)(63,91)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,96)(71,95)(72,94)>;
G:=Group( (1,20)(2,21)(3,22)(4,23)(5,24)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,37)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,119)(62,120)(63,109)(64,110)(65,111)(66,112)(67,113)(68,114)(69,115)(70,116)(71,117)(72,118)(73,106)(74,107)(75,108)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (1,25,114,54,76)(2,77,55,115,26)(3,27,116,56,78)(4,79,57,117,28)(5,29,118,58,80)(6,81,59,119,30)(7,31,120,60,82)(8,83,49,109,32)(9,33,110,50,84)(10,73,51,111,34)(11,35,112,52,74)(12,75,53,113,36)(13,102,95,61,43)(14,44,62,96,103)(15,104,85,63,45)(16,46,64,86,105)(17,106,87,65,47)(18,48,66,88,107)(19,108,89,67,37)(20,38,68,90,97)(21,98,91,69,39)(22,40,70,92,99)(23,100,93,71,41)(24,42,72,94,101), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,23)(14,22)(15,21)(16,20)(17,19)(25,84)(26,83)(27,82)(28,81)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,108)(48,107)(49,115)(50,114)(51,113)(52,112)(53,111)(54,110)(55,109)(56,120)(57,119)(58,118)(59,117)(60,116)(61,93)(62,92)(63,91)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,96)(71,95)(72,94) );
G=PermutationGroup([[(1,20),(2,21),(3,22),(4,23),(5,24),(6,13),(7,14),(8,15),(9,16),(10,17),(11,18),(12,19),(25,38),(26,39),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,37),(49,85),(50,86),(51,87),(52,88),(53,89),(54,90),(55,91),(56,92),(57,93),(58,94),(59,95),(60,96),(61,119),(62,120),(63,109),(64,110),(65,111),(66,112),(67,113),(68,114),(69,115),(70,116),(71,117),(72,118),(73,106),(74,107),(75,108),(76,97),(77,98),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105)], [(1,25,114,54,76),(2,77,55,115,26),(3,27,116,56,78),(4,79,57,117,28),(5,29,118,58,80),(6,81,59,119,30),(7,31,120,60,82),(8,83,49,109,32),(9,33,110,50,84),(10,73,51,111,34),(11,35,112,52,74),(12,75,53,113,36),(13,102,95,61,43),(14,44,62,96,103),(15,104,85,63,45),(16,46,64,86,105),(17,106,87,65,47),(18,48,66,88,107),(19,108,89,67,37),(20,38,68,90,97),(21,98,91,69,39),(22,40,70,92,99),(23,100,93,71,41),(24,42,72,94,101)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,12),(13,23),(14,22),(15,21),(16,20),(17,19),(25,84),(26,83),(27,82),(28,81),(29,80),(30,79),(31,78),(32,77),(33,76),(34,75),(35,74),(36,73),(37,106),(38,105),(39,104),(40,103),(41,102),(42,101),(43,100),(44,99),(45,98),(46,97),(47,108),(48,107),(49,115),(50,114),(51,113),(52,112),(53,111),(54,110),(55,109),(56,120),(57,119),(58,118),(59,117),(60,116),(61,93),(62,92),(63,91),(64,90),(65,89),(66,88),(67,87),(68,86),(69,85),(70,96),(71,95),(72,94)]])
C2×C5⋊D12 is a maximal subgroup of
Dic5.D12 (C2×C20).D6 Dic5.8D12 D6⋊Dic5⋊C2 D30.35D4 Dic5⋊4D12 Dic15⋊14D4 Dic5⋊D12 D30⋊D4 D6.D20 D30.7D4 C15⋊22(C4×D4) D10⋊D12 C20⋊D12 D6⋊D20 C60⋊6D4 D30⋊12D4 C20⋊2D12 D30⋊4D4 D30⋊5D4 D30⋊7D4 Dic15⋊4D4 (C2×C10)⋊4D12 Dic15⋊5D4 (C2×C10)⋊11D12 D30⋊19D4 C2×D5×D12 D12⋊14D10 C2×S3×C5⋊D4
C2×C5⋊D12 is a maximal quotient of
C20.60D12 D60⋊36C22 C60.38D4 C20.D12 D12.33D10 C60.45D4 C60.69D4 C60.70D4 Dic5⋊Dic6 D30⋊9Q8 C20⋊D12 C60⋊6D4 C20⋊2D12 C10.(C2×D12) (C2×C10).D12 (C2×C10)⋊4D12 (C2×C10)⋊11D12 D30⋊19D4
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 15A | 15B | 30A | ··· | 30F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 30 | 30 | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | D10 | D10 | D12 | C5⋊D4 | S3×D5 | C5⋊D12 | C2×S3×D5 |
| kernel | C2×C5⋊D12 | C5⋊D12 | C6×Dic5 | S3×C2×C10 | C22×D15 | C2×Dic5 | C30 | C22×S3 | Dic5 | C2×C10 | D6 | C2×C6 | C10 | C6 | C22 | C2 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 8 | 2 | 4 | 2 |
Matrix representation of C2×C5⋊D12 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 43 | 60 |
| 0 | 0 | 1 | 0 |
| 0 | 60 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 31 | 17 |
| 0 | 0 | 8 | 30 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 43 | 60 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,43,1,0,0,60,0],[0,1,0,0,60,1,0,0,0,0,31,8,0,0,17,30],[0,1,0,0,1,0,0,0,0,0,1,43,0,0,0,60] >;
C2×C5⋊D12 in GAP, Magma, Sage, TeX
C_2\times C_5\rtimes D_{12} % in TeX
G:=Group("C2xC5:D12"); // GroupNames label
G:=SmallGroup(240,147);
// by ID
G=gap.SmallGroup(240,147);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,121,490,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^5=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations