metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C30)⋊9D4, C15⋊7C22≀C2, (S3×C10)⋊17D4, (C2×C10)⋊11D12, C5⋊5(D6⋊D4), D6⋊8(C5⋊D4), (C2×Dic5)⋊4D6, (S3×C23)⋊2D5, D6⋊Dic5⋊36C2, C10.165(S3×D4), C10.70(C2×D12), C30.250(C2×D4), C3⋊1(C24⋊2D5), C23.D5⋊13S3, C23.34(S3×D5), C22⋊3(C5⋊D12), (C6×Dic5)⋊8C22, (C22×C6).46D10, (C2×C30).212C23, (C22×S3).79D10, (C22×C10).112D6, (C22×D15)⋊7C22, (C2×Dic15)⋊14C22, (C22×C30).74C22, (C2×C6)⋊2(C5⋊D4), (S3×C22×C10)⋊2C2, C2.45(S3×C5⋊D4), C6.24(C2×C5⋊D4), (C2×C5⋊D12)⋊15C2, (C2×C15⋊7D4)⋊20C2, C2.25(C2×C5⋊D12), C22.241(C2×S3×D5), (S3×C2×C10).96C22, (C3×C23.D5)⋊15C2, (C2×C6).224(C22×D5), (C2×C10).224(C22×S3), SmallGroup(480,646)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C10)⋊11D12
G = < a,b,c,d | a2=b10=c12=d2=1, ab=ba, cac-1=dad=ab5, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 1324 in 260 conjugacy classes, 60 normal (24 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, C24, Dic5, D10, C2×C10, C2×C10, C2×C10, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3, D15, C30, C30, C30, C22≀C2, C2×Dic5, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C22×C10, D6⋊C4, C3×C22⋊C4, C2×D12, C2×C3⋊D4, S3×C23, C3×Dic5, Dic15, S3×C10, S3×C10, D30, C2×C30, C2×C30, C2×C30, C23.D5, C23.D5, C2×C5⋊D4, C23×C10, D6⋊D4, C5⋊D12, C6×Dic5, C2×Dic15, C15⋊7D4, S3×C2×C10, S3×C2×C10, C22×D15, C22×C30, C24⋊2D5, D6⋊Dic5, C3×C23.D5, C2×C5⋊D12, C2×C15⋊7D4, S3×C22×C10, (C2×C10)⋊11D12
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, D12, C22×S3, C22≀C2, C5⋊D4, C22×D5, C2×D12, S3×D4, S3×D5, C2×C5⋊D4, D6⋊D4, C5⋊D12, C2×S3×D5, C24⋊2D5, C2×C5⋊D12, S3×C5⋊D4, (C2×C10)⋊11D12
►On 120 points
Generators in S120
(1 7)(2 59)(3 9)(4 49)(5 11)(6 51)(8 53)(10 55)(12 57)(13 19)(14 98)(15 21)(16 100)(17 23)(18 102)(20 104)(22 106)(24 108)(25 31)(26 91)(27 33)(28 93)(29 35)(30 95)(32 85)(34 87)(36 89)(37 75)(38 44)(39 77)(40 46)(41 79)(42 48)(43 81)(45 83)(47 73)(50 56)(52 58)(54 60)(61 67)(62 119)(63 69)(64 109)(65 71)(66 111)(68 113)(70 115)(72 117)(74 80)(76 82)(78 84)(86 92)(88 94)(90 96)(97 103)(99 105)(101 107)(110 116)(112 118)(114 120)
(1 44 97 71 96 52 76 19 110 25)(2 26 111 20 77 53 85 72 98 45)(3 46 99 61 86 54 78 21 112 27)(4 28 113 22 79 55 87 62 100 47)(5 48 101 63 88 56 80 23 114 29)(6 30 115 24 81 57 89 64 102 37)(7 38 103 65 90 58 82 13 116 31)(8 32 117 14 83 59 91 66 104 39)(9 40 105 67 92 60 84 15 118 33)(10 34 119 16 73 49 93 68 106 41)(11 42 107 69 94 50 74 17 120 35)(12 36 109 18 75 51 95 70 108 43)
(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 51)(2 50)(3 49)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 54)(11 53)(12 52)(13 102)(14 101)(15 100)(16 99)(17 98)(18 97)(19 108)(20 107)(21 106)(22 105)(23 104)(24 103)(25 95)(26 94)(27 93)(28 92)(29 91)(30 90)(31 89)(32 88)(33 87)(34 86)(35 85)(36 96)(37 82)(38 81)(39 80)(40 79)(41 78)(42 77)(43 76)(44 75)(45 74)(46 73)(47 84)(48 83)(61 119)(62 118)(63 117)(64 116)(65 115)(66 114)(67 113)(68 112)(69 111)(70 110)(71 109)(72 120)
G:=sub<Sym(120)| (1,7)(2,59)(3,9)(4,49)(5,11)(6,51)(8,53)(10,55)(12,57)(13,19)(14,98)(15,21)(16,100)(17,23)(18,102)(20,104)(22,106)(24,108)(25,31)(26,91)(27,33)(28,93)(29,35)(30,95)(32,85)(34,87)(36,89)(37,75)(38,44)(39,77)(40,46)(41,79)(42,48)(43,81)(45,83)(47,73)(50,56)(52,58)(54,60)(61,67)(62,119)(63,69)(64,109)(65,71)(66,111)(68,113)(70,115)(72,117)(74,80)(76,82)(78,84)(86,92)(88,94)(90,96)(97,103)(99,105)(101,107)(110,116)(112,118)(114,120), (1,44,97,71,96,52,76,19,110,25)(2,26,111,20,77,53,85,72,98,45)(3,46,99,61,86,54,78,21,112,27)(4,28,113,22,79,55,87,62,100,47)(5,48,101,63,88,56,80,23,114,29)(6,30,115,24,81,57,89,64,102,37)(7,38,103,65,90,58,82,13,116,31)(8,32,117,14,83,59,91,66,104,39)(9,40,105,67,92,60,84,15,118,33)(10,34,119,16,73,49,93,68,106,41)(11,42,107,69,94,50,74,17,120,35)(12,36,109,18,75,51,95,70,108,43), (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), (1,51)(2,50)(3,49)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,53)(12,52)(13,102)(14,101)(15,100)(16,99)(17,98)(18,97)(19,108)(20,107)(21,106)(22,105)(23,104)(24,103)(25,95)(26,94)(27,93)(28,92)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,96)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,84)(48,83)(61,119)(62,118)(63,117)(64,116)(65,115)(66,114)(67,113)(68,112)(69,111)(70,110)(71,109)(72,120)>;
G:=Group( (1,7)(2,59)(3,9)(4,49)(5,11)(6,51)(8,53)(10,55)(12,57)(13,19)(14,98)(15,21)(16,100)(17,23)(18,102)(20,104)(22,106)(24,108)(25,31)(26,91)(27,33)(28,93)(29,35)(30,95)(32,85)(34,87)(36,89)(37,75)(38,44)(39,77)(40,46)(41,79)(42,48)(43,81)(45,83)(47,73)(50,56)(52,58)(54,60)(61,67)(62,119)(63,69)(64,109)(65,71)(66,111)(68,113)(70,115)(72,117)(74,80)(76,82)(78,84)(86,92)(88,94)(90,96)(97,103)(99,105)(101,107)(110,116)(112,118)(114,120), (1,44,97,71,96,52,76,19,110,25)(2,26,111,20,77,53,85,72,98,45)(3,46,99,61,86,54,78,21,112,27)(4,28,113,22,79,55,87,62,100,47)(5,48,101,63,88,56,80,23,114,29)(6,30,115,24,81,57,89,64,102,37)(7,38,103,65,90,58,82,13,116,31)(8,32,117,14,83,59,91,66,104,39)(9,40,105,67,92,60,84,15,118,33)(10,34,119,16,73,49,93,68,106,41)(11,42,107,69,94,50,74,17,120,35)(12,36,109,18,75,51,95,70,108,43), (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), (1,51)(2,50)(3,49)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,53)(12,52)(13,102)(14,101)(15,100)(16,99)(17,98)(18,97)(19,108)(20,107)(21,106)(22,105)(23,104)(24,103)(25,95)(26,94)(27,93)(28,92)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,96)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,84)(48,83)(61,119)(62,118)(63,117)(64,116)(65,115)(66,114)(67,113)(68,112)(69,111)(70,110)(71,109)(72,120) );
G=PermutationGroup([[(1,7),(2,59),(3,9),(4,49),(5,11),(6,51),(8,53),(10,55),(12,57),(13,19),(14,98),(15,21),(16,100),(17,23),(18,102),(20,104),(22,106),(24,108),(25,31),(26,91),(27,33),(28,93),(29,35),(30,95),(32,85),(34,87),(36,89),(37,75),(38,44),(39,77),(40,46),(41,79),(42,48),(43,81),(45,83),(47,73),(50,56),(52,58),(54,60),(61,67),(62,119),(63,69),(64,109),(65,71),(66,111),(68,113),(70,115),(72,117),(74,80),(76,82),(78,84),(86,92),(88,94),(90,96),(97,103),(99,105),(101,107),(110,116),(112,118),(114,120)], [(1,44,97,71,96,52,76,19,110,25),(2,26,111,20,77,53,85,72,98,45),(3,46,99,61,86,54,78,21,112,27),(4,28,113,22,79,55,87,62,100,47),(5,48,101,63,88,56,80,23,114,29),(6,30,115,24,81,57,89,64,102,37),(7,38,103,65,90,58,82,13,116,31),(8,32,117,14,83,59,91,66,104,39),(9,40,105,67,92,60,84,15,118,33),(10,34,119,16,73,49,93,68,106,41),(11,42,107,69,94,50,74,17,120,35),(12,36,109,18,75,51,95,70,108,43)], [(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)], [(1,51),(2,50),(3,49),(4,60),(5,59),(6,58),(7,57),(8,56),(9,55),(10,54),(11,53),(12,52),(13,102),(14,101),(15,100),(16,99),(17,98),(18,97),(19,108),(20,107),(21,106),(22,105),(23,104),(24,103),(25,95),(26,94),(27,93),(28,92),(29,91),(30,90),(31,89),(32,88),(33,87),(34,86),(35,85),(36,96),(37,82),(38,81),(39,80),(40,79),(41,78),(42,77),(43,76),(44,75),(45,74),(46,73),(47,84),(48,83),(61,119),(62,118),(63,117),(64,116),(65,115),(66,114),(67,113),(68,112),(69,111),(70,110),(71,109),(72,120)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10N | 10O | ··· | 10AD | 12A | 12B | 12C | 12D | 15A | 15B | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 60 | 2 | 20 | 20 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | D12 | C5⋊D4 | C5⋊D4 | S3×D4 | S3×D5 | C5⋊D12 | C2×S3×D5 | S3×C5⋊D4 |
| kernel | (C2×C10)⋊11D12 | D6⋊Dic5 | C3×C23.D5 | C2×C5⋊D12 | C2×C15⋊7D4 | S3×C22×C10 | C23.D5 | S3×C10 | C2×C30 | S3×C23 | C2×Dic5 | C22×C10 | C22×S3 | C22×C6 | C2×C10 | D6 | C2×C6 | C10 | C23 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 16 | 8 | 2 | 2 | 4 | 2 | 8 |
Matrix representation of (C2×C10)⋊11D12 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 41 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 25 | 0 | 0 | 0 | 0 |
| 17 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 59 | 19 |
| 0 | 0 | 0 | 0 | 48 | 1 |
| 60 | 25 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 59 | 19 |
| 0 | 0 | 0 | 0 | 48 | 2 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,41,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,17,0,0,0,0,25,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,59,48,0,0,0,0,19,1],[60,0,0,0,0,0,25,1,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,0,0,59,48,0,0,0,0,19,2] >;
(C2×C10)⋊11D12 in GAP, Magma, Sage, TeX
(C_2\times C_{10})\rtimes_{11}D_{12} % in TeX
G:=Group("(C2xC10):11D12"); // GroupNames label
G:=SmallGroup(480,646);
// by ID
G=gap.SmallGroup(480,646);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,422,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^12=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a*b^5,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations