direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4○D4×D15, D4⋊7D30, Q8⋊7D30, D60⋊27C22, C60.88C23, C30.64C24, D30.29C23, Dic30⋊25C22, Dic15.31C23, (C2×C4)⋊7D30, (C5×D4)⋊23D6, (C2×C20)⋊13D6, (C5×Q8)⋊23D6, (D4×D15)⋊13C2, (C3×D4)⋊23D10, (C2×C12)⋊13D10, (C2×C60)⋊9C22, (Q8×D15)⋊13C2, (C3×Q8)⋊20D10, D4⋊2D15⋊13C2, Q8⋊3D15⋊13C2, C6.64(C23×D5), (C4×D15)⋊19C22, (D4×C15)⋊25C22, C15⋊7D4⋊11C22, C10.64(S3×C23), (C2×C30).10C23, 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, C3⋊7(D5×C4○D4), C5⋊7(S3×C4○D4), (C2×C4×D15)⋊6C2, (C3×C4○D4)⋊3D5, (C5×C4○D4)⋊7S3, 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
Subgroups: 1796 in 328 conjugacy classes, 121 normal (26 characteristic)
C1, C2, C2 [×8], C3, C4, C4 [×3], C4 [×4], C22 [×3], C22 [×10], C5, S3 [×5], C6, C6 [×3], C2×C4 [×3], C2×C4 [×13], D4 [×3], D4 [×9], Q8, Q8 [×3], C23 [×3], D5 [×5], C10, C10 [×3], Dic3 [×4], C12, C12 [×3], D6 [×10], C2×C6 [×3], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4, C4○D4 [×7], Dic5 [×4], C20, C20 [×3], D10 [×10], C2×C10 [×3], Dic6 [×3], C4×S3 [×10], D12 [×3], C2×Dic3 [×3], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×3], D15 [×2], D15 [×3], C30, C30 [×3], C2×C4○D4, Dic10 [×3], C4×D5 [×10], D20 [×3], C2×Dic5 [×3], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C5×Q8, C22×D5 [×3], S3×C2×C4 [×3], C4○D12 [×3], S3×D4 [×3], D4⋊2S3 [×3], S3×Q8, Q8⋊3S3, C3×C4○D4, Dic15, Dic15 [×3], C60, C60 [×3], D30, D30 [×3], D30 [×6], C2×C30 [×3], C2×C4×D5 [×3], C4○D20 [×3], D4×D5 [×3], D4⋊2D5 [×3], Q8×D5, Q8⋊2D5, C5×C4○D4, S3×C4○D4, Dic30 [×3], C4×D15, C4×D15 [×9], D60 [×3], C2×Dic15 [×3], C15⋊7D4 [×6], C2×C60 [×3], D4×C15 [×3], Q8×C15, C22×D15 [×3], D5×C4○D4, C2×C4×D15 [×3], D60⋊11C2 [×3], D4×D15 [×3], D4⋊2D15 [×3], Q8×D15, Q8⋊3D15, C15×C4○D4, C4○D4×D15
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], D15, C2×C4○D4, C22×D5 [×7], S3×C23, D30 [×7], C23×D5, S3×C4○D4, C22×D15 [×7], D5×C4○D4, C23×D15, C4○D4×D15
Generators and relations
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 >
►On 120 points
(1 46 17 43)(2 47 18 44)(3 48 19 45)(4 49 20 31)(5 50 21 32)(6 51 22 33)(7 52 23 34)(8 53 24 35)(9 54 25 36)(10 55 26 37)(11 56 27 38)(12 57 28 39)(13 58 29 40)(14 59 30 41)(15 60 16 42)(61 106 87 98)(62 107 88 99)(63 108 89 100)(64 109 90 101)(65 110 76 102)(66 111 77 103)(67 112 78 104)(68 113 79 105)(69 114 80 91)(70 115 81 92)(71 116 82 93)(72 117 83 94)(73 118 84 95)(74 119 85 96)(75 120 86 97)
(1 62 17 88)(2 63 18 89)(3 64 19 90)(4 65 20 76)(5 66 21 77)(6 67 22 78)(7 68 23 79)(8 69 24 80)(9 70 25 81)(10 71 26 82)(11 72 27 83)(12 73 28 84)(13 74 29 85)(14 75 30 86)(15 61 16 87)(31 102 49 110)(32 103 50 111)(33 104 51 112)(34 105 52 113)(35 91 53 114)(36 92 54 115)(37 93 55 116)(38 94 56 117)(39 95 57 118)(40 96 58 119)(41 97 59 120)(42 98 60 106)(43 99 46 107)(44 100 47 108)(45 101 48 109)
(61 87)(62 88)(63 89)(64 90)(65 76)(66 77)(67 78)(68 79)(69 80)(70 81)(71 82)(72 83)(73 84)(74 85)(75 86)(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 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)(41 47)(42 46)(43 60)(44 59)(45 58)(61 88)(62 87)(63 86)(64 85)(65 84)(66 83)(67 82)(68 81)(69 80)(70 79)(71 78)(72 77)(73 76)(74 90)(75 89)(91 114)(92 113)(93 112)(94 111)(95 110)(96 109)(97 108)(98 107)(99 106)(100 120)(101 119)(102 118)(103 117)(104 116)(105 115)
G:=sub<Sym(120)| (1,46,17,43)(2,47,18,44)(3,48,19,45)(4,49,20,31)(5,50,21,32)(6,51,22,33)(7,52,23,34)(8,53,24,35)(9,54,25,36)(10,55,26,37)(11,56,27,38)(12,57,28,39)(13,58,29,40)(14,59,30,41)(15,60,16,42)(61,106,87,98)(62,107,88,99)(63,108,89,100)(64,109,90,101)(65,110,76,102)(66,111,77,103)(67,112,78,104)(68,113,79,105)(69,114,80,91)(70,115,81,92)(71,116,82,93)(72,117,83,94)(73,118,84,95)(74,119,85,96)(75,120,86,97), (1,62,17,88)(2,63,18,89)(3,64,19,90)(4,65,20,76)(5,66,21,77)(6,67,22,78)(7,68,23,79)(8,69,24,80)(9,70,25,81)(10,71,26,82)(11,72,27,83)(12,73,28,84)(13,74,29,85)(14,75,30,86)(15,61,16,87)(31,102,49,110)(32,103,50,111)(33,104,51,112)(34,105,52,113)(35,91,53,114)(36,92,54,115)(37,93,55,116)(38,94,56,117)(39,95,57,118)(40,96,58,119)(41,97,59,120)(42,98,60,106)(43,99,46,107)(44,100,47,108)(45,101,48,109), (61,87)(62,88)(63,89)(64,90)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(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)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48)(41,47)(42,46)(43,60)(44,59)(45,58)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,90)(75,89)(91,114)(92,113)(93,112)(94,111)(95,110)(96,109)(97,108)(98,107)(99,106)(100,120)(101,119)(102,118)(103,117)(104,116)(105,115)>;
G:=Group( (1,46,17,43)(2,47,18,44)(3,48,19,45)(4,49,20,31)(5,50,21,32)(6,51,22,33)(7,52,23,34)(8,53,24,35)(9,54,25,36)(10,55,26,37)(11,56,27,38)(12,57,28,39)(13,58,29,40)(14,59,30,41)(15,60,16,42)(61,106,87,98)(62,107,88,99)(63,108,89,100)(64,109,90,101)(65,110,76,102)(66,111,77,103)(67,112,78,104)(68,113,79,105)(69,114,80,91)(70,115,81,92)(71,116,82,93)(72,117,83,94)(73,118,84,95)(74,119,85,96)(75,120,86,97), (1,62,17,88)(2,63,18,89)(3,64,19,90)(4,65,20,76)(5,66,21,77)(6,67,22,78)(7,68,23,79)(8,69,24,80)(9,70,25,81)(10,71,26,82)(11,72,27,83)(12,73,28,84)(13,74,29,85)(14,75,30,86)(15,61,16,87)(31,102,49,110)(32,103,50,111)(33,104,51,112)(34,105,52,113)(35,91,53,114)(36,92,54,115)(37,93,55,116)(38,94,56,117)(39,95,57,118)(40,96,58,119)(41,97,59,120)(42,98,60,106)(43,99,46,107)(44,100,47,108)(45,101,48,109), (61,87)(62,88)(63,89)(64,90)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(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)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48)(41,47)(42,46)(43,60)(44,59)(45,58)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,90)(75,89)(91,114)(92,113)(93,112)(94,111)(95,110)(96,109)(97,108)(98,107)(99,106)(100,120)(101,119)(102,118)(103,117)(104,116)(105,115) );
G=PermutationGroup([(1,46,17,43),(2,47,18,44),(3,48,19,45),(4,49,20,31),(5,50,21,32),(6,51,22,33),(7,52,23,34),(8,53,24,35),(9,54,25,36),(10,55,26,37),(11,56,27,38),(12,57,28,39),(13,58,29,40),(14,59,30,41),(15,60,16,42),(61,106,87,98),(62,107,88,99),(63,108,89,100),(64,109,90,101),(65,110,76,102),(66,111,77,103),(67,112,78,104),(68,113,79,105),(69,114,80,91),(70,115,81,92),(71,116,82,93),(72,117,83,94),(73,118,84,95),(74,119,85,96),(75,120,86,97)], [(1,62,17,88),(2,63,18,89),(3,64,19,90),(4,65,20,76),(5,66,21,77),(6,67,22,78),(7,68,23,79),(8,69,24,80),(9,70,25,81),(10,71,26,82),(11,72,27,83),(12,73,28,84),(13,74,29,85),(14,75,30,86),(15,61,16,87),(31,102,49,110),(32,103,50,111),(33,104,51,112),(34,105,52,113),(35,91,53,114),(36,92,54,115),(37,93,55,116),(38,94,56,117),(39,95,57,118),(40,96,58,119),(41,97,59,120),(42,98,60,106),(43,99,46,107),(44,100,47,108),(45,101,48,109)], [(61,87),(62,88),(63,89),(64,90),(65,76),(66,77),(67,78),(68,79),(69,80),(70,81),(71,82),(72,83),(73,84),(74,85),(75,86),(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),(31,57),(32,56),(33,55),(34,54),(35,53),(36,52),(37,51),(38,50),(39,49),(40,48),(41,47),(42,46),(43,60),(44,59),(45,58),(61,88),(62,87),(63,86),(64,85),(65,84),(66,83),(67,82),(68,81),(69,80),(70,79),(71,78),(72,77),(73,76),(74,90),(75,89),(91,114),(92,113),(93,112),(94,111),(95,110),(96,109),(97,108),(98,107),(99,106),(100,120),(101,119),(102,118),(103,117),(104,116),(105,115)])
Matrix representation ►G ⊆ 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] >;
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 |
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