direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×D6⋊D4, D6⋊4(C5×D4), D6⋊C4⋊4C10, (C2×C30)⋊19D4, (C2×C20)⋊19D6, C6.5(D4×C10), C15⋊13C22≀C2, (S3×C10)⋊19D4, (C2×D12)⋊2C10, (C2×C10)⋊10D12, C2.7(C10×D12), C22⋊3(C5×D12), (C10×D12)⋊18C2, (S3×C23)⋊1C10, (C2×C60)⋊26C22, C10.76(C2×D12), C10.171(S3×D4), C30.292(C2×D4), C23.20(S3×C10), (C22×C10).90D6, (C2×C30).402C23, (C10×Dic3)⋊18C22, (C22×C30).117C22, C2.7(C5×S3×D4), (C2×C6)⋊1(C5×D4), (C2×C4)⋊1(S3×C10), C3⋊1(C5×C22≀C2), (C2×C12)⋊1(C2×C10), (C5×D6⋊C4)⋊20C2, (C2×C3⋊D4)⋊1C10, C22⋊C4⋊2(C5×S3), (S3×C22×C10)⋊7C2, (C10×C3⋊D4)⋊16C2, (C3×C22⋊C4)⋊3C10, (C5×C22⋊C4)⋊10S3, (S3×C2×C10)⋊13C22, C22.41(S3×C2×C10), (C15×C22⋊C4)⋊17C2, (C22×S3)⋊1(C2×C10), (C2×Dic3)⋊1(C2×C10), (C22×C6).12(C2×C10), (C2×C6).23(C22×C10), (C2×C10).336(C22×S3), SmallGroup(480,761)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D6⋊D4
G = < a,b,c,d,e | a5=b6=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b3c, ece=bc, ede=d-1 >
Subgroups: 756 in 260 conjugacy classes, 74 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C10, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22⋊C4, C2×D4, C24, C20, C2×C10, C2×C10, C2×C10, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×S3, C22×S3, C22×C6, C5×S3, C30, C30, C30, C22≀C2, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, D6⋊C4, C3×C22⋊C4, C2×D12, C2×C3⋊D4, S3×C23, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C2×C30, C2×C30, C5×C22⋊C4, C5×C22⋊C4, D4×C10, C23×C10, D6⋊D4, C5×D12, C10×Dic3, C5×C3⋊D4, C2×C60, S3×C2×C10, S3×C2×C10, S3×C2×C10, C22×C30, C5×C22≀C2, C5×D6⋊C4, C15×C22⋊C4, C10×D12, C10×C3⋊D4, S3×C22×C10, C5×D6⋊D4
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, D12, C22×S3, C5×S3, C22≀C2, C5×D4, C22×C10, C2×D12, S3×D4, S3×C10, D4×C10, D6⋊D4, C5×D12, S3×C2×C10, C5×C22≀C2, C10×D12, C5×S3×D4, C5×D6⋊D4
►On 120 points
Generators in S120
(1 61 49 37 25)(2 62 50 38 26)(3 63 51 39 27)(4 64 52 40 28)(5 65 53 41 29)(6 66 54 42 30)(7 111 99 87 75)(8 112 100 88 76)(9 113 101 89 77)(10 114 102 90 78)(11 109 97 85 73)(12 110 98 86 74)(13 56 44 32 20)(14 57 45 33 21)(15 58 46 34 22)(16 59 47 35 23)(17 60 48 36 24)(18 55 43 31 19)(67 115 103 91 79)(68 116 104 92 80)(69 117 105 93 81)(70 118 106 94 82)(71 119 107 95 83)(72 120 108 96 84)
(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 6)(2 5)(3 4)(7 11)(8 10)(13 18)(14 17)(15 16)(19 20)(21 24)(22 23)(25 30)(26 29)(27 28)(31 32)(33 36)(34 35)(37 42)(38 41)(39 40)(43 44)(45 48)(46 47)(49 54)(50 53)(51 52)(55 56)(57 60)(58 59)(61 66)(62 65)(63 64)(67 71)(68 70)(73 75)(76 78)(79 83)(80 82)(85 87)(88 90)(91 95)(92 94)(97 99)(100 102)(103 107)(104 106)(109 111)(112 114)(115 119)(116 118)
(1 71 16 73)(2 72 17 74)(3 67 18 75)(4 68 13 76)(5 69 14 77)(6 70 15 78)(7 63 115 55)(8 64 116 56)(9 65 117 57)(10 66 118 58)(11 61 119 59)(12 62 120 60)(19 87 27 79)(20 88 28 80)(21 89 29 81)(22 90 30 82)(23 85 25 83)(24 86 26 84)(31 99 39 91)(32 100 40 92)(33 101 41 93)(34 102 42 94)(35 97 37 95)(36 98 38 96)(43 111 51 103)(44 112 52 104)(45 113 53 105)(46 114 54 106)(47 109 49 107)(48 110 50 108)
(1 73)(2 78)(3 77)(4 76)(5 75)(6 74)(7 65)(8 64)(9 63)(10 62)(11 61)(12 66)(13 68)(14 67)(15 72)(16 71)(17 70)(18 69)(19 81)(20 80)(21 79)(22 84)(23 83)(24 82)(25 85)(26 90)(27 89)(28 88)(29 87)(30 86)(31 93)(32 92)(33 91)(34 96)(35 95)(36 94)(37 97)(38 102)(39 101)(40 100)(41 99)(42 98)(43 105)(44 104)(45 103)(46 108)(47 107)(48 106)(49 109)(50 114)(51 113)(52 112)(53 111)(54 110)(55 117)(56 116)(57 115)(58 120)(59 119)(60 118)
G:=sub<Sym(120)| (1,61,49,37,25)(2,62,50,38,26)(3,63,51,39,27)(4,64,52,40,28)(5,65,53,41,29)(6,66,54,42,30)(7,111,99,87,75)(8,112,100,88,76)(9,113,101,89,77)(10,114,102,90,78)(11,109,97,85,73)(12,110,98,86,74)(13,56,44,32,20)(14,57,45,33,21)(15,58,46,34,22)(16,59,47,35,23)(17,60,48,36,24)(18,55,43,31,19)(67,115,103,91,79)(68,116,104,92,80)(69,117,105,93,81)(70,118,106,94,82)(71,119,107,95,83)(72,120,108,96,84), (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), (1,6)(2,5)(3,4)(7,11)(8,10)(13,18)(14,17)(15,16)(19,20)(21,24)(22,23)(25,30)(26,29)(27,28)(31,32)(33,36)(34,35)(37,42)(38,41)(39,40)(43,44)(45,48)(46,47)(49,54)(50,53)(51,52)(55,56)(57,60)(58,59)(61,66)(62,65)(63,64)(67,71)(68,70)(73,75)(76,78)(79,83)(80,82)(85,87)(88,90)(91,95)(92,94)(97,99)(100,102)(103,107)(104,106)(109,111)(112,114)(115,119)(116,118), (1,71,16,73)(2,72,17,74)(3,67,18,75)(4,68,13,76)(5,69,14,77)(6,70,15,78)(7,63,115,55)(8,64,116,56)(9,65,117,57)(10,66,118,58)(11,61,119,59)(12,62,120,60)(19,87,27,79)(20,88,28,80)(21,89,29,81)(22,90,30,82)(23,85,25,83)(24,86,26,84)(31,99,39,91)(32,100,40,92)(33,101,41,93)(34,102,42,94)(35,97,37,95)(36,98,38,96)(43,111,51,103)(44,112,52,104)(45,113,53,105)(46,114,54,106)(47,109,49,107)(48,110,50,108), (1,73)(2,78)(3,77)(4,76)(5,75)(6,74)(7,65)(8,64)(9,63)(10,62)(11,61)(12,66)(13,68)(14,67)(15,72)(16,71)(17,70)(18,69)(19,81)(20,80)(21,79)(22,84)(23,83)(24,82)(25,85)(26,90)(27,89)(28,88)(29,87)(30,86)(31,93)(32,92)(33,91)(34,96)(35,95)(36,94)(37,97)(38,102)(39,101)(40,100)(41,99)(42,98)(43,105)(44,104)(45,103)(46,108)(47,107)(48,106)(49,109)(50,114)(51,113)(52,112)(53,111)(54,110)(55,117)(56,116)(57,115)(58,120)(59,119)(60,118)>;
G:=Group( (1,61,49,37,25)(2,62,50,38,26)(3,63,51,39,27)(4,64,52,40,28)(5,65,53,41,29)(6,66,54,42,30)(7,111,99,87,75)(8,112,100,88,76)(9,113,101,89,77)(10,114,102,90,78)(11,109,97,85,73)(12,110,98,86,74)(13,56,44,32,20)(14,57,45,33,21)(15,58,46,34,22)(16,59,47,35,23)(17,60,48,36,24)(18,55,43,31,19)(67,115,103,91,79)(68,116,104,92,80)(69,117,105,93,81)(70,118,106,94,82)(71,119,107,95,83)(72,120,108,96,84), (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), (1,6)(2,5)(3,4)(7,11)(8,10)(13,18)(14,17)(15,16)(19,20)(21,24)(22,23)(25,30)(26,29)(27,28)(31,32)(33,36)(34,35)(37,42)(38,41)(39,40)(43,44)(45,48)(46,47)(49,54)(50,53)(51,52)(55,56)(57,60)(58,59)(61,66)(62,65)(63,64)(67,71)(68,70)(73,75)(76,78)(79,83)(80,82)(85,87)(88,90)(91,95)(92,94)(97,99)(100,102)(103,107)(104,106)(109,111)(112,114)(115,119)(116,118), (1,71,16,73)(2,72,17,74)(3,67,18,75)(4,68,13,76)(5,69,14,77)(6,70,15,78)(7,63,115,55)(8,64,116,56)(9,65,117,57)(10,66,118,58)(11,61,119,59)(12,62,120,60)(19,87,27,79)(20,88,28,80)(21,89,29,81)(22,90,30,82)(23,85,25,83)(24,86,26,84)(31,99,39,91)(32,100,40,92)(33,101,41,93)(34,102,42,94)(35,97,37,95)(36,98,38,96)(43,111,51,103)(44,112,52,104)(45,113,53,105)(46,114,54,106)(47,109,49,107)(48,110,50,108), (1,73)(2,78)(3,77)(4,76)(5,75)(6,74)(7,65)(8,64)(9,63)(10,62)(11,61)(12,66)(13,68)(14,67)(15,72)(16,71)(17,70)(18,69)(19,81)(20,80)(21,79)(22,84)(23,83)(24,82)(25,85)(26,90)(27,89)(28,88)(29,87)(30,86)(31,93)(32,92)(33,91)(34,96)(35,95)(36,94)(37,97)(38,102)(39,101)(40,100)(41,99)(42,98)(43,105)(44,104)(45,103)(46,108)(47,107)(48,106)(49,109)(50,114)(51,113)(52,112)(53,111)(54,110)(55,117)(56,116)(57,115)(58,120)(59,119)(60,118) );
G=PermutationGroup([[(1,61,49,37,25),(2,62,50,38,26),(3,63,51,39,27),(4,64,52,40,28),(5,65,53,41,29),(6,66,54,42,30),(7,111,99,87,75),(8,112,100,88,76),(9,113,101,89,77),(10,114,102,90,78),(11,109,97,85,73),(12,110,98,86,74),(13,56,44,32,20),(14,57,45,33,21),(15,58,46,34,22),(16,59,47,35,23),(17,60,48,36,24),(18,55,43,31,19),(67,115,103,91,79),(68,116,104,92,80),(69,117,105,93,81),(70,118,106,94,82),(71,119,107,95,83),(72,120,108,96,84)], [(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)], [(1,6),(2,5),(3,4),(7,11),(8,10),(13,18),(14,17),(15,16),(19,20),(21,24),(22,23),(25,30),(26,29),(27,28),(31,32),(33,36),(34,35),(37,42),(38,41),(39,40),(43,44),(45,48),(46,47),(49,54),(50,53),(51,52),(55,56),(57,60),(58,59),(61,66),(62,65),(63,64),(67,71),(68,70),(73,75),(76,78),(79,83),(80,82),(85,87),(88,90),(91,95),(92,94),(97,99),(100,102),(103,107),(104,106),(109,111),(112,114),(115,119),(116,118)], [(1,71,16,73),(2,72,17,74),(3,67,18,75),(4,68,13,76),(5,69,14,77),(6,70,15,78),(7,63,115,55),(8,64,116,56),(9,65,117,57),(10,66,118,58),(11,61,119,59),(12,62,120,60),(19,87,27,79),(20,88,28,80),(21,89,29,81),(22,90,30,82),(23,85,25,83),(24,86,26,84),(31,99,39,91),(32,100,40,92),(33,101,41,93),(34,102,42,94),(35,97,37,95),(36,98,38,96),(43,111,51,103),(44,112,52,104),(45,113,53,105),(46,114,54,106),(47,109,49,107),(48,110,50,108)], [(1,73),(2,78),(3,77),(4,76),(5,75),(6,74),(7,65),(8,64),(9,63),(10,62),(11,61),(12,66),(13,68),(14,67),(15,72),(16,71),(17,70),(18,69),(19,81),(20,80),(21,79),(22,84),(23,83),(24,82),(25,85),(26,90),(27,89),(28,88),(29,87),(30,86),(31,93),(32,92),(33,91),(34,96),(35,95),(36,94),(37,97),(38,102),(39,101),(40,100),(41,99),(42,98),(43,105),(44,104),(45,103),(46,108),(47,107),(48,106),(49,109),(50,114),(51,113),(52,112),(53,111),(54,110),(55,117),(56,116),(57,115),(58,120),(59,119),(60,118)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AJ | 10AK | 10AL | 10AM | 10AN | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 30A | ··· | 30L | 30M | ··· | 30T | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 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 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 2 | 4 | 4 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
120 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 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | S3 | D4 | D4 | D6 | D6 | D12 | C5×S3 | C5×D4 | C5×D4 | S3×C10 | S3×C10 | C5×D12 | S3×D4 | C5×S3×D4 |
| kernel | C5×D6⋊D4 | C5×D6⋊C4 | C15×C22⋊C4 | C10×D12 | C10×C3⋊D4 | S3×C22×C10 | D6⋊D4 | D6⋊C4 | C3×C22⋊C4 | C2×D12 | C2×C3⋊D4 | S3×C23 | C5×C22⋊C4 | S3×C10 | C2×C30 | C2×C20 | C22×C10 | C2×C10 | C22⋊C4 | D6 | C2×C6 | C2×C4 | C23 | C22 | C10 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 8 | 4 | 8 | 4 | 4 | 1 | 4 | 2 | 2 | 1 | 4 | 4 | 16 | 8 | 8 | 4 | 16 | 2 | 8 |
Matrix representation of C5×D6⋊D4 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 58 | 0 |
| 0 | 0 | 0 | 58 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 60 |
| 0 | 0 | 1 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 60 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 23 | 15 |
| 0 | 0 | 46 | 38 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 46 | 38 |
| 0 | 0 | 23 | 15 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,58,0,0,0,0,58],[60,0,0,0,0,60,0,0,0,0,1,1,0,0,60,0],[60,0,0,0,0,1,0,0,0,0,1,1,0,0,0,60],[0,1,0,0,1,0,0,0,0,0,23,46,0,0,15,38],[0,1,0,0,1,0,0,0,0,0,46,23,0,0,38,15] >;
C5×D6⋊D4 in GAP, Magma, Sage, TeX
C_5\times D_6\rtimes D_4
% in TeX
G:=Group("C5xD6:D4"); // GroupNames label
G:=SmallGroup(480,761);
// by ID
G=gap.SmallGroup(480,761);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,926,891,226,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^6=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^3*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations