metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊5D15, C23.31D30, (C2×C30)⋊30D4, (C23×C6)⋊5D5, C15⋊17C22≀C2, (C23×C30)⋊3C2, (C23×C10)⋊9S3, C5⋊3(C24⋊4S3), C30.395(C2×D4), C3⋊3(C24⋊2D5), C22⋊4(C15⋊7D4), C30.38D4⋊13C2, (C2×C30).317C23, (C2×Dic15)⋊3C22, (C22×C6).123D10, (C22×C10).141D6, (C22×D15)⋊2C22, C22.66(C22×D15), (C22×C30).146C22, (C2×C15⋊7D4)⋊8C2, (C2×C6)⋊14(C5⋊D4), C6.122(C2×C5⋊D4), C2.26(C2×C15⋊7D4), (C2×C10)⋊18(C3⋊D4), C10.122(C2×C3⋊D4), (C2×C6).313(C22×D5), (C2×C10).312(C22×S3), SmallGroup(480,918)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊5D15
G = < a,b,c,d,e,f | a2=b2=c2=d2=e15=f2=1, ab=ba, ac=ca, faf=ad=da, ae=ea, fbf=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >
Subgroups: 1204 in 260 conjugacy classes, 71 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, C24, Dic5, D10, C2×C10, C2×C10, C2×C10, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C22×C6, D15, C30, C30, C22≀C2, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C22×C10, C6.D4, C2×C3⋊D4, C23×C6, Dic15, D30, C2×C30, C2×C30, C2×C30, C23.D5, C2×C5⋊D4, C23×C10, C24⋊4S3, C2×Dic15, C15⋊7D4, C22×D15, C22×C30, C22×C30, C24⋊2D5, C30.38D4, C2×C15⋊7D4, C23×C30, C24⋊5D15
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, D15, C22≀C2, C5⋊D4, C22×D5, C2×C3⋊D4, D30, C2×C5⋊D4, C24⋊4S3, C15⋊7D4, C22×D15, C24⋊2D5, C2×C15⋊7D4, C24⋊5D15
►On 120 points
Generators in S120
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 16)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)(41 60)(42 46)(43 47)(44 48)(45 49)
(61 94)(62 95)(63 96)(64 97)(65 98)(66 99)(67 100)(68 101)(69 102)(70 103)(71 104)(72 105)(73 91)(74 92)(75 93)(76 109)(77 110)(78 111)(79 112)(80 113)(81 114)(82 115)(83 116)(84 117)(85 118)(86 119)(87 120)(88 106)(89 107)(90 108)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 31)(13 32)(14 33)(15 34)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 46)(25 47)(26 48)(27 49)(28 50)(29 51)(30 52)(61 94)(62 95)(63 96)(64 97)(65 98)(66 99)(67 100)(68 101)(69 102)(70 103)(71 104)(72 105)(73 91)(74 92)(75 93)(76 109)(77 110)(78 111)(79 112)(80 113)(81 114)(82 115)(83 116)(84 117)(85 118)(86 119)(87 120)(88 106)(89 107)(90 108)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 16)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)(41 60)(42 46)(43 47)(44 48)(45 49)(61 81)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 76)(72 77)(73 78)(74 79)(75 80)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(101 106)(102 107)(103 108)(104 109)(105 110)
(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 71)(2 70)(3 69)(4 68)(5 67)(6 66)(7 65)(8 64)(9 63)(10 62)(11 61)(12 75)(13 74)(14 73)(15 72)(16 77)(17 76)(18 90)(19 89)(20 88)(21 87)(22 86)(23 85)(24 84)(25 83)(26 82)(27 81)(28 80)(29 79)(30 78)(31 93)(32 92)(33 91)(34 105)(35 104)(36 103)(37 102)(38 101)(39 100)(40 99)(41 98)(42 97)(43 96)(44 95)(45 94)(46 117)(47 116)(48 115)(49 114)(50 113)(51 112)(52 111)(53 110)(54 109)(55 108)(56 107)(57 106)(58 120)(59 119)(60 118)
G:=sub<Sym(120)| (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,16)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(41,60)(42,46)(43,47)(44,48)(45,49), (61,94)(62,95)(63,96)(64,97)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(73,91)(74,92)(75,93)(76,109)(77,110)(78,111)(79,112)(80,113)(81,114)(82,115)(83,116)(84,117)(85,118)(86,119)(87,120)(88,106)(89,107)(90,108), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,31)(13,32)(14,33)(15,34)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(61,94)(62,95)(63,96)(64,97)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(73,91)(74,92)(75,93)(76,109)(77,110)(78,111)(79,112)(80,113)(81,114)(82,115)(83,116)(84,117)(85,118)(86,119)(87,120)(88,106)(89,107)(90,108), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,16)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(41,60)(42,46)(43,47)(44,48)(45,49)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,76)(72,77)(73,78)(74,79)(75,80)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(101,106)(102,107)(103,108)(104,109)(105,110), (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), (1,71)(2,70)(3,69)(4,68)(5,67)(6,66)(7,65)(8,64)(9,63)(10,62)(11,61)(12,75)(13,74)(14,73)(15,72)(16,77)(17,76)(18,90)(19,89)(20,88)(21,87)(22,86)(23,85)(24,84)(25,83)(26,82)(27,81)(28,80)(29,79)(30,78)(31,93)(32,92)(33,91)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,94)(46,117)(47,116)(48,115)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,108)(56,107)(57,106)(58,120)(59,119)(60,118)>;
G:=Group( (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,16)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(41,60)(42,46)(43,47)(44,48)(45,49), (61,94)(62,95)(63,96)(64,97)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(73,91)(74,92)(75,93)(76,109)(77,110)(78,111)(79,112)(80,113)(81,114)(82,115)(83,116)(84,117)(85,118)(86,119)(87,120)(88,106)(89,107)(90,108), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,31)(13,32)(14,33)(15,34)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(61,94)(62,95)(63,96)(64,97)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(73,91)(74,92)(75,93)(76,109)(77,110)(78,111)(79,112)(80,113)(81,114)(82,115)(83,116)(84,117)(85,118)(86,119)(87,120)(88,106)(89,107)(90,108), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,16)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(41,60)(42,46)(43,47)(44,48)(45,49)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,76)(72,77)(73,78)(74,79)(75,80)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(101,106)(102,107)(103,108)(104,109)(105,110), (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), (1,71)(2,70)(3,69)(4,68)(5,67)(6,66)(7,65)(8,64)(9,63)(10,62)(11,61)(12,75)(13,74)(14,73)(15,72)(16,77)(17,76)(18,90)(19,89)(20,88)(21,87)(22,86)(23,85)(24,84)(25,83)(26,82)(27,81)(28,80)(29,79)(30,78)(31,93)(32,92)(33,91)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,94)(46,117)(47,116)(48,115)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,108)(56,107)(57,106)(58,120)(59,119)(60,118) );
G=PermutationGroup([[(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,16),(31,50),(32,51),(33,52),(34,53),(35,54),(36,55),(37,56),(38,57),(39,58),(40,59),(41,60),(42,46),(43,47),(44,48),(45,49)], [(61,94),(62,95),(63,96),(64,97),(65,98),(66,99),(67,100),(68,101),(69,102),(70,103),(71,104),(72,105),(73,91),(74,92),(75,93),(76,109),(77,110),(78,111),(79,112),(80,113),(81,114),(82,115),(83,116),(84,117),(85,118),(86,119),(87,120),(88,106),(89,107),(90,108)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,41),(8,42),(9,43),(10,44),(11,45),(12,31),(13,32),(14,33),(15,34),(16,53),(17,54),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,46),(25,47),(26,48),(27,49),(28,50),(29,51),(30,52),(61,94),(62,95),(63,96),(64,97),(65,98),(66,99),(67,100),(68,101),(69,102),(70,103),(71,104),(72,105),(73,91),(74,92),(75,93),(76,109),(77,110),(78,111),(79,112),(80,113),(81,114),(82,115),(83,116),(84,117),(85,118),(86,119),(87,120),(88,106),(89,107),(90,108)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,16),(31,50),(32,51),(33,52),(34,53),(35,54),(36,55),(37,56),(38,57),(39,58),(40,59),(41,60),(42,46),(43,47),(44,48),(45,49),(61,81),(62,82),(63,83),(64,84),(65,85),(66,86),(67,87),(68,88),(69,89),(70,90),(71,76),(72,77),(73,78),(74,79),(75,80),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116),(97,117),(98,118),(99,119),(100,120),(101,106),(102,107),(103,108),(104,109),(105,110)], [(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)], [(1,71),(2,70),(3,69),(4,68),(5,67),(6,66),(7,65),(8,64),(9,63),(10,62),(11,61),(12,75),(13,74),(14,73),(15,72),(16,77),(17,76),(18,90),(19,89),(20,88),(21,87),(22,86),(23,85),(24,84),(25,83),(26,82),(27,81),(28,80),(29,79),(30,78),(31,93),(32,92),(33,91),(34,105),(35,104),(36,103),(37,102),(38,101),(39,100),(40,99),(41,98),(42,97),(43,96),(44,95),(45,94),(46,117),(47,116),(48,115),(49,114),(50,113),(51,112),(52,111),(53,110),(54,109),(55,108),(56,107),(57,106),(58,120),(59,119),(60,118)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 3 | 4A | 4B | 4C | 5A | 5B | 6A | ··· | 6O | 10A | ··· | 10AD | 15A | 15B | 15C | 15D | 30A | ··· | 30BH |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 60 | 2 | 60 | 60 | 60 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D10 | C3⋊D4 | D15 | C5⋊D4 | D30 | C15⋊7D4 |
| kernel | C24⋊5D15 | C30.38D4 | C2×C15⋊7D4 | C23×C30 | C23×C10 | C2×C30 | C23×C6 | C22×C10 | C22×C6 | C2×C10 | C24 | C2×C6 | C23 | C22 |
| # reps | 1 | 3 | 3 | 1 | 1 | 6 | 2 | 3 | 6 | 12 | 4 | 24 | 12 | 48 |
Matrix representation of C24⋊5D15 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 22 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 |
| 0 | 0 | 20 | 0 |
| 0 | 0 | 0 | 58 |
| 0 | 25 | 0 | 0 |
| 22 | 0 | 0 | 0 |
| 0 | 0 | 0 | 58 |
| 0 | 0 | 20 | 0 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[22,0,0,0,0,25,0,0,0,0,20,0,0,0,0,58],[0,22,0,0,25,0,0,0,0,0,0,20,0,0,58,0] >;
C24⋊5D15 in GAP, Magma, Sage, TeX
C_2^4\rtimes_5D_{15} % in TeX
G:=Group("C2^4:5D15"); // GroupNames label
G:=SmallGroup(480,918);
// by ID
G=gap.SmallGroup(480,918);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,254,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^15=f^2=1,a*b=b*a,a*c=c*a,f*a*f=a*d=d*a,a*e=e*a,f*b*f=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations