direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×Q8⋊3Dic3, C60.241D4, C15⋊23C4≀C2, (C3×D4)⋊2C20, (C3×Q8)⋊2C20, (D4×C15)⋊14C4, C12.9(C2×C20), (Q8×C15)⋊14C4, Q8⋊3(C5×Dic3), (C5×D4)⋊8Dic3, D4⋊2(C5×Dic3), C12.56(C5×D4), (C2×C30).90D4, C60.179(C2×C4), (C5×Q8)⋊11Dic3, (C4×Dic3)⋊2C10, (C2×C20).352D6, C4.Dic3⋊4C10, C4.3(C10×Dic3), (Dic3×C20)⋊14C2, C20.53(C2×Dic3), C20.124(C3⋊D4), (C2×C60).349C22, C30.122(C22⋊C4), C10.38(C6.D4), C3⋊3(C5×C4≀C2), (C2×C6).3(C5×D4), C4○D4.3(C5×S3), (C5×C4○D4).8S3, C4.31(C5×C3⋊D4), (C2×C4).40(S3×C10), (C3×C4○D4).1C10, (C15×C4○D4).7C2, C6.18(C5×C22⋊C4), C22.3(C5×C3⋊D4), (C2×C12).19(C2×C10), (C5×C4.Dic3)⋊16C2, C2.8(C5×C6.D4), (C2×C10).39(C3⋊D4), SmallGroup(480,156)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Q8⋊3Dic3
G = < a,b,c,d,e | a5=b4=d6=1, c2=b2, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b2c, ece-1=b-1c, ede-1=d-1 >
Subgroups: 180 in 88 conjugacy classes, 42 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C10, C10, Dic3, C12, C12, C2×C6, C2×C6, C15, C42, M4(2), C4○D4, C20, C20, C2×C10, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C30, C30, C4≀C2, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C4.Dic3, C4×Dic3, C3×C4○D4, C5×Dic3, C60, C60, C2×C30, C2×C30, C4×C20, C5×M4(2), C5×C4○D4, Q8⋊3Dic3, C5×C3⋊C8, C10×Dic3, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, C5×C4≀C2, C5×C4.Dic3, Dic3×C20, C15×C4○D4, C5×Q8⋊3Dic3
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, C10, Dic3, D6, C22⋊C4, C20, C2×C10, C2×Dic3, C3⋊D4, C5×S3, C4≀C2, C2×C20, C5×D4, C6.D4, C5×Dic3, S3×C10, C5×C22⋊C4, Q8⋊3Dic3, C10×Dic3, C5×C3⋊D4, C5×C4≀C2, C5×C6.D4, C5×Q8⋊3Dic3
►On 120 points
Generators in S120
(1 49 37 25 13)(2 50 38 26 14)(3 51 39 27 15)(4 52 40 28 16)(5 53 41 29 17)(6 54 42 30 18)(7 55 43 31 19)(8 56 44 32 20)(9 57 45 33 21)(10 58 46 34 22)(11 59 47 35 23)(12 60 48 36 24)(61 109 97 85 73)(62 110 98 86 74)(63 111 99 87 75)(64 112 100 88 76)(65 113 101 89 77)(66 114 102 90 78)(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 10 4 7)(2 11 5 8)(3 12 6 9)(13 22 16 19)(14 23 17 20)(15 24 18 21)(25 34 28 31)(26 35 29 32)(27 36 30 33)(37 46 40 43)(38 47 41 44)(39 48 42 45)(49 58 52 55)(50 59 53 56)(51 60 54 57)(61 67 64 70)(62 68 65 71)(63 69 66 72)(73 79 76 82)(74 80 77 83)(75 81 78 84)(85 91 88 94)(86 92 89 95)(87 93 90 96)(97 103 100 106)(98 104 101 107)(99 105 102 108)(109 115 112 118)(110 116 113 119)(111 117 114 120)
(1 67 4 70)(2 71 5 68)(3 69 6 72)(7 64 10 61)(8 62 11 65)(9 66 12 63)(13 79 16 82)(14 83 17 80)(15 81 18 84)(19 76 22 73)(20 74 23 77)(21 78 24 75)(25 91 28 94)(26 95 29 92)(27 93 30 96)(31 88 34 85)(32 86 35 89)(33 90 36 87)(37 103 40 106)(38 107 41 104)(39 105 42 108)(43 100 46 97)(44 98 47 101)(45 102 48 99)(49 115 52 118)(50 119 53 116)(51 117 54 120)(55 112 58 109)(56 110 59 113)(57 114 60 111)
(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 4)(2 6)(3 5)(7 10)(8 12)(9 11)(13 16)(14 18)(15 17)(19 22)(20 24)(21 23)(25 28)(26 30)(27 29)(31 34)(32 36)(33 35)(37 40)(38 42)(39 41)(43 46)(44 48)(45 47)(49 52)(50 54)(51 53)(55 58)(56 60)(57 59)(61 67 64 70)(62 72 65 69)(63 71 66 68)(73 79 76 82)(74 84 77 81)(75 83 78 80)(85 91 88 94)(86 96 89 93)(87 95 90 92)(97 103 100 106)(98 108 101 105)(99 107 102 104)(109 115 112 118)(110 120 113 117)(111 119 114 116)
G:=sub<Sym(120)| (1,49,37,25,13)(2,50,38,26,14)(3,51,39,27,15)(4,52,40,28,16)(5,53,41,29,17)(6,54,42,30,18)(7,55,43,31,19)(8,56,44,32,20)(9,57,45,33,21)(10,58,46,34,22)(11,59,47,35,23)(12,60,48,36,24)(61,109,97,85,73)(62,110,98,86,74)(63,111,99,87,75)(64,112,100,88,76)(65,113,101,89,77)(66,114,102,90,78)(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), (1,10,4,7)(2,11,5,8)(3,12,6,9)(13,22,16,19)(14,23,17,20)(15,24,18,21)(25,34,28,31)(26,35,29,32)(27,36,30,33)(37,46,40,43)(38,47,41,44)(39,48,42,45)(49,58,52,55)(50,59,53,56)(51,60,54,57)(61,67,64,70)(62,68,65,71)(63,69,66,72)(73,79,76,82)(74,80,77,83)(75,81,78,84)(85,91,88,94)(86,92,89,95)(87,93,90,96)(97,103,100,106)(98,104,101,107)(99,105,102,108)(109,115,112,118)(110,116,113,119)(111,117,114,120), (1,67,4,70)(2,71,5,68)(3,69,6,72)(7,64,10,61)(8,62,11,65)(9,66,12,63)(13,79,16,82)(14,83,17,80)(15,81,18,84)(19,76,22,73)(20,74,23,77)(21,78,24,75)(25,91,28,94)(26,95,29,92)(27,93,30,96)(31,88,34,85)(32,86,35,89)(33,90,36,87)(37,103,40,106)(38,107,41,104)(39,105,42,108)(43,100,46,97)(44,98,47,101)(45,102,48,99)(49,115,52,118)(50,119,53,116)(51,117,54,120)(55,112,58,109)(56,110,59,113)(57,114,60,111), (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), (1,4)(2,6)(3,5)(7,10)(8,12)(9,11)(13,16)(14,18)(15,17)(19,22)(20,24)(21,23)(25,28)(26,30)(27,29)(31,34)(32,36)(33,35)(37,40)(38,42)(39,41)(43,46)(44,48)(45,47)(49,52)(50,54)(51,53)(55,58)(56,60)(57,59)(61,67,64,70)(62,72,65,69)(63,71,66,68)(73,79,76,82)(74,84,77,81)(75,83,78,80)(85,91,88,94)(86,96,89,93)(87,95,90,92)(97,103,100,106)(98,108,101,105)(99,107,102,104)(109,115,112,118)(110,120,113,117)(111,119,114,116)>;
G:=Group( (1,49,37,25,13)(2,50,38,26,14)(3,51,39,27,15)(4,52,40,28,16)(5,53,41,29,17)(6,54,42,30,18)(7,55,43,31,19)(8,56,44,32,20)(9,57,45,33,21)(10,58,46,34,22)(11,59,47,35,23)(12,60,48,36,24)(61,109,97,85,73)(62,110,98,86,74)(63,111,99,87,75)(64,112,100,88,76)(65,113,101,89,77)(66,114,102,90,78)(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), (1,10,4,7)(2,11,5,8)(3,12,6,9)(13,22,16,19)(14,23,17,20)(15,24,18,21)(25,34,28,31)(26,35,29,32)(27,36,30,33)(37,46,40,43)(38,47,41,44)(39,48,42,45)(49,58,52,55)(50,59,53,56)(51,60,54,57)(61,67,64,70)(62,68,65,71)(63,69,66,72)(73,79,76,82)(74,80,77,83)(75,81,78,84)(85,91,88,94)(86,92,89,95)(87,93,90,96)(97,103,100,106)(98,104,101,107)(99,105,102,108)(109,115,112,118)(110,116,113,119)(111,117,114,120), (1,67,4,70)(2,71,5,68)(3,69,6,72)(7,64,10,61)(8,62,11,65)(9,66,12,63)(13,79,16,82)(14,83,17,80)(15,81,18,84)(19,76,22,73)(20,74,23,77)(21,78,24,75)(25,91,28,94)(26,95,29,92)(27,93,30,96)(31,88,34,85)(32,86,35,89)(33,90,36,87)(37,103,40,106)(38,107,41,104)(39,105,42,108)(43,100,46,97)(44,98,47,101)(45,102,48,99)(49,115,52,118)(50,119,53,116)(51,117,54,120)(55,112,58,109)(56,110,59,113)(57,114,60,111), (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), (1,4)(2,6)(3,5)(7,10)(8,12)(9,11)(13,16)(14,18)(15,17)(19,22)(20,24)(21,23)(25,28)(26,30)(27,29)(31,34)(32,36)(33,35)(37,40)(38,42)(39,41)(43,46)(44,48)(45,47)(49,52)(50,54)(51,53)(55,58)(56,60)(57,59)(61,67,64,70)(62,72,65,69)(63,71,66,68)(73,79,76,82)(74,84,77,81)(75,83,78,80)(85,91,88,94)(86,96,89,93)(87,95,90,92)(97,103,100,106)(98,108,101,105)(99,107,102,104)(109,115,112,118)(110,120,113,117)(111,119,114,116) );
G=PermutationGroup([[(1,49,37,25,13),(2,50,38,26,14),(3,51,39,27,15),(4,52,40,28,16),(5,53,41,29,17),(6,54,42,30,18),(7,55,43,31,19),(8,56,44,32,20),(9,57,45,33,21),(10,58,46,34,22),(11,59,47,35,23),(12,60,48,36,24),(61,109,97,85,73),(62,110,98,86,74),(63,111,99,87,75),(64,112,100,88,76),(65,113,101,89,77),(66,114,102,90,78),(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)], [(1,10,4,7),(2,11,5,8),(3,12,6,9),(13,22,16,19),(14,23,17,20),(15,24,18,21),(25,34,28,31),(26,35,29,32),(27,36,30,33),(37,46,40,43),(38,47,41,44),(39,48,42,45),(49,58,52,55),(50,59,53,56),(51,60,54,57),(61,67,64,70),(62,68,65,71),(63,69,66,72),(73,79,76,82),(74,80,77,83),(75,81,78,84),(85,91,88,94),(86,92,89,95),(87,93,90,96),(97,103,100,106),(98,104,101,107),(99,105,102,108),(109,115,112,118),(110,116,113,119),(111,117,114,120)], [(1,67,4,70),(2,71,5,68),(3,69,6,72),(7,64,10,61),(8,62,11,65),(9,66,12,63),(13,79,16,82),(14,83,17,80),(15,81,18,84),(19,76,22,73),(20,74,23,77),(21,78,24,75),(25,91,28,94),(26,95,29,92),(27,93,30,96),(31,88,34,85),(32,86,35,89),(33,90,36,87),(37,103,40,106),(38,107,41,104),(39,105,42,108),(43,100,46,97),(44,98,47,101),(45,102,48,99),(49,115,52,118),(50,119,53,116),(51,117,54,120),(55,112,58,109),(56,110,59,113),(57,114,60,111)], [(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)], [(1,4),(2,6),(3,5),(7,10),(8,12),(9,11),(13,16),(14,18),(15,17),(19,22),(20,24),(21,23),(25,28),(26,30),(27,29),(31,34),(32,36),(33,35),(37,40),(38,42),(39,41),(43,46),(44,48),(45,47),(49,52),(50,54),(51,53),(55,58),(56,60),(57,59),(61,67,64,70),(62,72,65,69),(63,71,66,68),(73,79,76,82),(74,84,77,81),(75,83,78,80),(85,91,88,94),(86,96,89,93),(87,95,90,92),(97,103,100,106),(98,108,101,105),(99,107,102,104),(109,115,112,118),(110,120,113,117),(111,119,114,116)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | 20N | 20O | 20P | 20Q | ··· | 20AF | 30A | 30B | 30C | 30D | 30E | ··· | 30P | 40A | ··· | 40H | 60A | ··· | 60H | 60I | ··· | 60T |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 4 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | ··· | 2 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C10 | C20 | C20 | S3 | D4 | D4 | D6 | Dic3 | Dic3 | C3⋊D4 | C3⋊D4 | C5×S3 | C4≀C2 | C5×D4 | C5×D4 | S3×C10 | C5×Dic3 | C5×Dic3 | C5×C3⋊D4 | C5×C3⋊D4 | C5×C4≀C2 | Q8⋊3Dic3 | C5×Q8⋊3Dic3 |
| kernel | C5×Q8⋊3Dic3 | C5×C4.Dic3 | Dic3×C20 | C15×C4○D4 | D4×C15 | Q8×C15 | Q8⋊3Dic3 | C4.Dic3 | C4×Dic3 | C3×C4○D4 | C3×D4 | C3×Q8 | C5×C4○D4 | C60 | C2×C30 | C2×C20 | C5×D4 | C5×Q8 | C20 | C2×C10 | C4○D4 | C15 | C12 | C2×C6 | C2×C4 | D4 | Q8 | C4 | C22 | C3 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 16 | 2 | 8 |
Matrix representation of C5×Q8⋊3Dic3 ►in GL4(𝔽241) generated by
| 91 | 0 | 0 | 0 |
| 0 | 91 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 43 | 177 |
| 70 | 140 | 0 | 0 |
| 101 | 171 | 0 | 0 |
| 0 | 0 | 185 | 49 |
| 0 | 0 | 236 | 56 |
| 240 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 140 | 240 |
| 1 | 0 | 0 | 0 |
| 240 | 240 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 72 | 177 |
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,0,240,0,0,0,0,64,43,0,0,0,177],[70,101,0,0,140,171,0,0,0,0,185,236,0,0,49,56],[240,1,0,0,240,0,0,0,0,0,1,140,0,0,0,240],[1,240,0,0,0,240,0,0,0,0,240,72,0,0,0,177] >;
C5×Q8⋊3Dic3 in GAP, Magma, Sage, TeX
C_5\times Q_8\rtimes_3{\rm Dic}_3 % in TeX
G:=Group("C5xQ8:3Dic3"); // GroupNames label
G:=SmallGroup(480,156);
// by ID
G=gap.SmallGroup(480,156);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,589,136,4204,2111,102,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=d^6=1,c^2=b^2,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^-1>;
// generators/relations