direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4○D4×D15, D4⋊7D30, Q8⋊7D30, D60⋊27C22, C30.64C24, C60.88C23, D30.29C23, Dic30⋊25C22, Dic15.31C23, (C2×C4)⋊7D30, (C5×D4)⋊23D6, (C2×C20)⋊13D6, (C5×Q8)⋊23D6, (C3×D4)⋊23D10, (D4×D15)⋊13C2, (C2×C12)⋊13D10, (C2×C60)⋊9C22, (C3×Q8)⋊20D10, (Q8×D15)⋊13C2, Q8⋊3D15⋊13C2, D4⋊2D15⋊13C2, C6.64(C23×D5), (D4×C15)⋊25C22, (C4×D15)⋊19C22, C15⋊7D4⋊11C22, (C2×C30).10C23, C10.64(S3×C23), D60⋊11C2⋊17C2, (Q8×C15)⋊22C22, C4.43(C22×D15), C2.12(C23×D15), C20.138(C22×S3), C12.136(C22×D5), C22.2(C22×D15), (C2×Dic15)⋊27C22, (C22×D15).93C22, C5⋊7(S3×C4○D4), C3⋊7(D5×C4○D4), (C2×C4×D15)⋊6C2, (C5×C4○D4)⋊7S3, (C3×C4○D4)⋊3D5, C15⋊29(C2×C4○D4), (C15×C4○D4)⋊3C2, (C2×C6).17(C22×D5), (C2×C10).18(C22×S3), SmallGroup(480,1175)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4○D4×D15
G = < a,b,c,d,e | a4=c2=d15=e2=1, b2=a2, ab=ba, ac=ca, ad=da, ae=ea, cbc=a2b, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1796 in 328 conjugacy classes, 121 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, D15, D15, C30, C30, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, S3×C2×C4, C4○D12, S3×D4, D4⋊2S3, S3×Q8, Q8⋊3S3, C3×C4○D4, Dic15, Dic15, C60, C60, D30, D30, D30, C2×C30, C2×C4×D5, C4○D20, D4×D5, D4⋊2D5, Q8×D5, Q8⋊2D5, C5×C4○D4, S3×C4○D4, Dic30, C4×D15, C4×D15, D60, C2×Dic15, C15⋊7D4, C2×C60, D4×C15, Q8×C15, C22×D15, D5×C4○D4, C2×C4×D15, D60⋊11C2, D4×D15, D4⋊2D15, Q8×D15, Q8⋊3D15, C15×C4○D4, C4○D4×D15
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, C24, D10, C22×S3, D15, C2×C4○D4, C22×D5, S3×C23, D30, C23×D5, S3×C4○D4, C22×D15, D5×C4○D4, C23×D15, C4○D4×D15
►On 120 points
Generators in S120
(1 54 17 39)(2 55 18 40)(3 56 19 41)(4 57 20 42)(5 58 21 43)(6 59 22 44)(7 60 23 45)(8 46 24 31)(9 47 25 32)(10 48 26 33)(11 49 27 34)(12 50 28 35)(13 51 29 36)(14 52 30 37)(15 53 16 38)(61 118 77 95)(62 119 78 96)(63 120 79 97)(64 106 80 98)(65 107 81 99)(66 108 82 100)(67 109 83 101)(68 110 84 102)(69 111 85 103)(70 112 86 104)(71 113 87 105)(72 114 88 91)(73 115 89 92)(74 116 90 93)(75 117 76 94)
(1 61 17 77)(2 62 18 78)(3 63 19 79)(4 64 20 80)(5 65 21 81)(6 66 22 82)(7 67 23 83)(8 68 24 84)(9 69 25 85)(10 70 26 86)(11 71 27 87)(12 72 28 88)(13 73 29 89)(14 74 30 90)(15 75 16 76)(31 102 46 110)(32 103 47 111)(33 104 48 112)(34 105 49 113)(35 91 50 114)(36 92 51 115)(37 93 52 116)(38 94 53 117)(39 95 54 118)(40 96 55 119)(41 97 56 120)(42 98 57 106)(43 99 58 107)(44 100 59 108)(45 101 60 109)
(61 77)(62 78)(63 79)(64 80)(65 81)(66 82)(67 83)(68 84)(69 85)(70 86)(71 87)(72 88)(73 89)(74 90)(75 76)(91 114)(92 115)(93 116)(94 117)(95 118)(96 119)(97 120)(98 106)(99 107)(100 108)(101 109)(102 110)(103 111)(104 112)(105 113)
(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 16)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(15 17)(31 46)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(61 76)(62 90)(63 89)(64 88)(65 87)(66 86)(67 85)(68 84)(69 83)(70 82)(71 81)(72 80)(73 79)(74 78)(75 77)(91 106)(92 120)(93 119)(94 118)(95 117)(96 116)(97 115)(98 114)(99 113)(100 112)(101 111)(102 110)(103 109)(104 108)(105 107)
G:=sub<Sym(120)| (1,54,17,39)(2,55,18,40)(3,56,19,41)(4,57,20,42)(5,58,21,43)(6,59,22,44)(7,60,23,45)(8,46,24,31)(9,47,25,32)(10,48,26,33)(11,49,27,34)(12,50,28,35)(13,51,29,36)(14,52,30,37)(15,53,16,38)(61,118,77,95)(62,119,78,96)(63,120,79,97)(64,106,80,98)(65,107,81,99)(66,108,82,100)(67,109,83,101)(68,110,84,102)(69,111,85,103)(70,112,86,104)(71,113,87,105)(72,114,88,91)(73,115,89,92)(74,116,90,93)(75,117,76,94), (1,61,17,77)(2,62,18,78)(3,63,19,79)(4,64,20,80)(5,65,21,81)(6,66,22,82)(7,67,23,83)(8,68,24,84)(9,69,25,85)(10,70,26,86)(11,71,27,87)(12,72,28,88)(13,73,29,89)(14,74,30,90)(15,75,16,76)(31,102,46,110)(32,103,47,111)(33,104,48,112)(34,105,49,113)(35,91,50,114)(36,92,51,115)(37,93,52,116)(38,94,53,117)(39,95,54,118)(40,96,55,119)(41,97,56,120)(42,98,57,106)(43,99,58,107)(44,100,59,108)(45,101,60,109), (61,77)(62,78)(63,79)(64,80)(65,81)(66,82)(67,83)(68,84)(69,85)(70,86)(71,87)(72,88)(73,89)(74,90)(75,76)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(97,120)(98,106)(99,107)(100,108)(101,109)(102,110)(103,111)(104,112)(105,113), (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), (1,16)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(31,46)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(61,76)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)(91,106)(92,120)(93,119)(94,118)(95,117)(96,116)(97,115)(98,114)(99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)>;
G:=Group( (1,54,17,39)(2,55,18,40)(3,56,19,41)(4,57,20,42)(5,58,21,43)(6,59,22,44)(7,60,23,45)(8,46,24,31)(9,47,25,32)(10,48,26,33)(11,49,27,34)(12,50,28,35)(13,51,29,36)(14,52,30,37)(15,53,16,38)(61,118,77,95)(62,119,78,96)(63,120,79,97)(64,106,80,98)(65,107,81,99)(66,108,82,100)(67,109,83,101)(68,110,84,102)(69,111,85,103)(70,112,86,104)(71,113,87,105)(72,114,88,91)(73,115,89,92)(74,116,90,93)(75,117,76,94), (1,61,17,77)(2,62,18,78)(3,63,19,79)(4,64,20,80)(5,65,21,81)(6,66,22,82)(7,67,23,83)(8,68,24,84)(9,69,25,85)(10,70,26,86)(11,71,27,87)(12,72,28,88)(13,73,29,89)(14,74,30,90)(15,75,16,76)(31,102,46,110)(32,103,47,111)(33,104,48,112)(34,105,49,113)(35,91,50,114)(36,92,51,115)(37,93,52,116)(38,94,53,117)(39,95,54,118)(40,96,55,119)(41,97,56,120)(42,98,57,106)(43,99,58,107)(44,100,59,108)(45,101,60,109), (61,77)(62,78)(63,79)(64,80)(65,81)(66,82)(67,83)(68,84)(69,85)(70,86)(71,87)(72,88)(73,89)(74,90)(75,76)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(97,120)(98,106)(99,107)(100,108)(101,109)(102,110)(103,111)(104,112)(105,113), (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), (1,16)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(31,46)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(61,76)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)(91,106)(92,120)(93,119)(94,118)(95,117)(96,116)(97,115)(98,114)(99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107) );
G=PermutationGroup([[(1,54,17,39),(2,55,18,40),(3,56,19,41),(4,57,20,42),(5,58,21,43),(6,59,22,44),(7,60,23,45),(8,46,24,31),(9,47,25,32),(10,48,26,33),(11,49,27,34),(12,50,28,35),(13,51,29,36),(14,52,30,37),(15,53,16,38),(61,118,77,95),(62,119,78,96),(63,120,79,97),(64,106,80,98),(65,107,81,99),(66,108,82,100),(67,109,83,101),(68,110,84,102),(69,111,85,103),(70,112,86,104),(71,113,87,105),(72,114,88,91),(73,115,89,92),(74,116,90,93),(75,117,76,94)], [(1,61,17,77),(2,62,18,78),(3,63,19,79),(4,64,20,80),(5,65,21,81),(6,66,22,82),(7,67,23,83),(8,68,24,84),(9,69,25,85),(10,70,26,86),(11,71,27,87),(12,72,28,88),(13,73,29,89),(14,74,30,90),(15,75,16,76),(31,102,46,110),(32,103,47,111),(33,104,48,112),(34,105,49,113),(35,91,50,114),(36,92,51,115),(37,93,52,116),(38,94,53,117),(39,95,54,118),(40,96,55,119),(41,97,56,120),(42,98,57,106),(43,99,58,107),(44,100,59,108),(45,101,60,109)], [(61,77),(62,78),(63,79),(64,80),(65,81),(66,82),(67,83),(68,84),(69,85),(70,86),(71,87),(72,88),(73,89),(74,90),(75,76),(91,114),(92,115),(93,116),(94,117),(95,118),(96,119),(97,120),(98,106),(99,107),(100,108),(101,109),(102,110),(103,111),(104,112),(105,113)], [(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)], [(1,16),(2,30),(3,29),(4,28),(5,27),(6,26),(7,25),(8,24),(9,23),(10,22),(11,21),(12,20),(13,19),(14,18),(15,17),(31,46),(32,60),(33,59),(34,58),(35,57),(36,56),(37,55),(38,54),(39,53),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(61,76),(62,90),(63,89),(64,88),(65,87),(66,86),(67,85),(68,84),(69,83),(70,82),(71,81),(72,80),(73,79),(74,78),(75,77),(91,106),(92,120),(93,119),(94,118),(95,117),(96,116),(97,115),(98,114),(99,113),(100,112),(101,111),(102,110),(103,109),(104,108),(105,107)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 6D | 10A | 10B | 10C | ··· | 10H | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 30A | 30B | 30C | 30D | 30E | ··· | 30P | 60A | ··· | 60H | 60I | ··· | 60T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 2 | 2 | 15 | 15 | 30 | 30 | 30 | 2 | 1 | 1 | 2 | 2 | 2 | 15 | 15 | 30 | 30 | 30 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | D15 | D30 | D30 | D30 | S3×C4○D4 | D5×C4○D4 | C4○D4×D15 |
| kernel | C4○D4×D15 | C2×C4×D15 | D60⋊11C2 | D4×D15 | D4⋊2D15 | Q8×D15 | Q8⋊3D15 | C15×C4○D4 | C5×C4○D4 | C3×C4○D4 | C2×C20 | C5×D4 | C5×Q8 | D15 | C2×C12 | C3×D4 | C3×Q8 | C4○D4 | C2×C4 | D4 | Q8 | C5 | C3 | C1 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 1 | 4 | 6 | 6 | 2 | 4 | 12 | 12 | 4 | 2 | 4 | 8 |
Matrix representation of C4○D4×D15 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 0 | 0 | 50 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 9 |
| 0 | 0 | 0 | 0 | 54 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 7 | 60 |
| 60 | 43 | 0 | 0 | 0 | 0 |
| 18 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 20 | 0 | 0 |
| 0 | 0 | 9 | 59 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 43 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 20 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,50,0,0,0,0,0,0,50],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,54,0,0,0,0,9,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,7,0,0,0,0,0,60],[60,18,0,0,0,0,43,18,0,0,0,0,0,0,1,9,0,0,0,0,20,59,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,43,1,0,0,0,0,0,0,1,0,0,0,0,0,20,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;
C4○D4×D15 in GAP, Magma, Sage, TeX
C_4\circ D_4\times D_{15} % in TeX
G:=Group("C4oD4xD15"); // GroupNames label
G:=SmallGroup(480,1175);
// by ID
G=gap.SmallGroup(480,1175);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,346,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^2=d^15=e^2=1,b^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=a^2*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations