direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×M4(2)⋊S3, C60.224D4, C20.63D12, C12.46(C5×D4), C4.11(C5×D12), (C22×S3).C20, (C2×D12).6C10, (C2×C20).213D6, C15⋊9(C4.D4), (C5×M4(2))⋊7S3, M4(2)⋊3(C5×S3), C4.Dic3⋊2C10, C22.4(S3×C20), C10.55(D6⋊C4), (C10×D12).16C2, (C3×M4(2))⋊7C10, C20.114(C3⋊D4), C30.97(C22⋊C4), (C15×M4(2))⋊17C2, (C2×C60).342C22, (S3×C2×C10).7C4, C2.9(C5×D6⋊C4), C3⋊1(C5×C4.D4), (C2×C4).1(S3×C10), (C2×C6).2(C2×C20), C4.21(C5×C3⋊D4), C6.8(C5×C22⋊C4), (C2×C10).63(C4×S3), (C2×C12).12(C2×C10), (C2×C30).126(C2×C4), (C5×C4.Dic3)⋊14C2, SmallGroup(480,142)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×M4(2)⋊S3
G = < a,b,c,d,e | a5=b8=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b5, bd=db, ebe=bc, cd=dc, ce=ec, ede=d-1 >
Subgroups: 292 in 92 conjugacy classes, 38 normal (34 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C22, C22 [×4], C5, S3 [×2], C6, C6, C8 [×2], C2×C4, D4 [×2], C23 [×2], C10, C10 [×3], C12 [×2], D6 [×4], C2×C6, C15, M4(2), M4(2), C2×D4, C20 [×2], C2×C10, C2×C10 [×4], C3⋊C8, C24, D12 [×2], C2×C12, C22×S3 [×2], C5×S3 [×2], C30, C30, C4.D4, C40 [×2], C2×C20, C5×D4 [×2], C22×C10 [×2], C4.Dic3, C3×M4(2), C2×D12, C60 [×2], S3×C10 [×4], C2×C30, C5×M4(2), C5×M4(2), D4×C10, M4(2)⋊S3, C5×C3⋊C8, C120, C5×D12 [×2], C2×C60, S3×C2×C10 [×2], C5×C4.D4, C5×C4.Dic3, C15×M4(2), C10×D12, C5×M4(2)⋊S3
Quotients: C1, C2 [×3], C4 [×2], C22, C5, S3, C2×C4, D4 [×2], C10 [×3], D6, C22⋊C4, C20 [×2], C2×C10, C4×S3, D12, C3⋊D4, C5×S3, C4.D4, C2×C20, C5×D4 [×2], D6⋊C4, S3×C10, C5×C22⋊C4, M4(2)⋊S3, S3×C20, C5×D12, C5×C3⋊D4, C5×C4.D4, C5×D6⋊C4, C5×M4(2)⋊S3
►On 120 points
Generators in S120
(1 15 41 108 24)(2 16 42 109 17)(3 9 43 110 18)(4 10 44 111 19)(5 11 45 112 20)(6 12 46 105 21)(7 13 47 106 22)(8 14 48 107 23)(25 35 62 85 49)(26 36 63 86 50)(27 37 64 87 51)(28 38 57 88 52)(29 39 58 81 53)(30 40 59 82 54)(31 33 60 83 55)(32 34 61 84 56)(65 93 118 77 102)(66 94 119 78 103)(67 95 120 79 104)(68 96 113 80 97)(69 89 114 73 98)(70 90 115 74 99)(71 91 116 75 100)(72 92 117 76 101)
(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)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(42 46)(44 48)(49 53)(51 55)(58 62)(60 64)(65 69)(67 71)(73 77)(75 79)(81 85)(83 87)(89 93)(91 95)(98 102)(100 104)(105 109)(107 111)(114 118)(116 120)
(1 99 26)(2 100 27)(3 101 28)(4 102 29)(5 103 30)(6 104 31)(7 97 32)(8 98 25)(9 72 38)(10 65 39)(11 66 40)(12 67 33)(13 68 34)(14 69 35)(15 70 36)(16 71 37)(17 75 51)(18 76 52)(19 77 53)(20 78 54)(21 79 55)(22 80 56)(23 73 49)(24 74 50)(41 90 63)(42 91 64)(43 92 57)(44 93 58)(45 94 59)(46 95 60)(47 96 61)(48 89 62)(81 111 118)(82 112 119)(83 105 120)(84 106 113)(85 107 114)(86 108 115)(87 109 116)(88 110 117)
(2 6)(3 7)(9 13)(12 16)(17 21)(18 22)(25 98)(26 99)(27 104)(28 97)(29 102)(30 103)(31 100)(32 101)(33 71)(34 72)(35 69)(36 70)(37 67)(38 68)(39 65)(40 66)(42 46)(43 47)(49 73)(50 74)(51 79)(52 80)(53 77)(54 78)(55 75)(56 76)(57 96)(58 93)(59 94)(60 91)(61 92)(62 89)(63 90)(64 95)(81 118)(82 119)(83 116)(84 117)(85 114)(86 115)(87 120)(88 113)(105 109)(106 110)
G:=sub<Sym(120)| (1,15,41,108,24)(2,16,42,109,17)(3,9,43,110,18)(4,10,44,111,19)(5,11,45,112,20)(6,12,46,105,21)(7,13,47,106,22)(8,14,48,107,23)(25,35,62,85,49)(26,36,63,86,50)(27,37,64,87,51)(28,38,57,88,52)(29,39,58,81,53)(30,40,59,82,54)(31,33,60,83,55)(32,34,61,84,56)(65,93,118,77,102)(66,94,119,78,103)(67,95,120,79,104)(68,96,113,80,97)(69,89,114,73,98)(70,90,115,74,99)(71,91,116,75,100)(72,92,117,76,101), (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), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(42,46)(44,48)(49,53)(51,55)(58,62)(60,64)(65,69)(67,71)(73,77)(75,79)(81,85)(83,87)(89,93)(91,95)(98,102)(100,104)(105,109)(107,111)(114,118)(116,120), (1,99,26)(2,100,27)(3,101,28)(4,102,29)(5,103,30)(6,104,31)(7,97,32)(8,98,25)(9,72,38)(10,65,39)(11,66,40)(12,67,33)(13,68,34)(14,69,35)(15,70,36)(16,71,37)(17,75,51)(18,76,52)(19,77,53)(20,78,54)(21,79,55)(22,80,56)(23,73,49)(24,74,50)(41,90,63)(42,91,64)(43,92,57)(44,93,58)(45,94,59)(46,95,60)(47,96,61)(48,89,62)(81,111,118)(82,112,119)(83,105,120)(84,106,113)(85,107,114)(86,108,115)(87,109,116)(88,110,117), (2,6)(3,7)(9,13)(12,16)(17,21)(18,22)(25,98)(26,99)(27,104)(28,97)(29,102)(30,103)(31,100)(32,101)(33,71)(34,72)(35,69)(36,70)(37,67)(38,68)(39,65)(40,66)(42,46)(43,47)(49,73)(50,74)(51,79)(52,80)(53,77)(54,78)(55,75)(56,76)(57,96)(58,93)(59,94)(60,91)(61,92)(62,89)(63,90)(64,95)(81,118)(82,119)(83,116)(84,117)(85,114)(86,115)(87,120)(88,113)(105,109)(106,110)>;
G:=Group( (1,15,41,108,24)(2,16,42,109,17)(3,9,43,110,18)(4,10,44,111,19)(5,11,45,112,20)(6,12,46,105,21)(7,13,47,106,22)(8,14,48,107,23)(25,35,62,85,49)(26,36,63,86,50)(27,37,64,87,51)(28,38,57,88,52)(29,39,58,81,53)(30,40,59,82,54)(31,33,60,83,55)(32,34,61,84,56)(65,93,118,77,102)(66,94,119,78,103)(67,95,120,79,104)(68,96,113,80,97)(69,89,114,73,98)(70,90,115,74,99)(71,91,116,75,100)(72,92,117,76,101), (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), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(42,46)(44,48)(49,53)(51,55)(58,62)(60,64)(65,69)(67,71)(73,77)(75,79)(81,85)(83,87)(89,93)(91,95)(98,102)(100,104)(105,109)(107,111)(114,118)(116,120), (1,99,26)(2,100,27)(3,101,28)(4,102,29)(5,103,30)(6,104,31)(7,97,32)(8,98,25)(9,72,38)(10,65,39)(11,66,40)(12,67,33)(13,68,34)(14,69,35)(15,70,36)(16,71,37)(17,75,51)(18,76,52)(19,77,53)(20,78,54)(21,79,55)(22,80,56)(23,73,49)(24,74,50)(41,90,63)(42,91,64)(43,92,57)(44,93,58)(45,94,59)(46,95,60)(47,96,61)(48,89,62)(81,111,118)(82,112,119)(83,105,120)(84,106,113)(85,107,114)(86,108,115)(87,109,116)(88,110,117), (2,6)(3,7)(9,13)(12,16)(17,21)(18,22)(25,98)(26,99)(27,104)(28,97)(29,102)(30,103)(31,100)(32,101)(33,71)(34,72)(35,69)(36,70)(37,67)(38,68)(39,65)(40,66)(42,46)(43,47)(49,73)(50,74)(51,79)(52,80)(53,77)(54,78)(55,75)(56,76)(57,96)(58,93)(59,94)(60,91)(61,92)(62,89)(63,90)(64,95)(81,118)(82,119)(83,116)(84,117)(85,114)(86,115)(87,120)(88,113)(105,109)(106,110) );
G=PermutationGroup([(1,15,41,108,24),(2,16,42,109,17),(3,9,43,110,18),(4,10,44,111,19),(5,11,45,112,20),(6,12,46,105,21),(7,13,47,106,22),(8,14,48,107,23),(25,35,62,85,49),(26,36,63,86,50),(27,37,64,87,51),(28,38,57,88,52),(29,39,58,81,53),(30,40,59,82,54),(31,33,60,83,55),(32,34,61,84,56),(65,93,118,77,102),(66,94,119,78,103),(67,95,120,79,104),(68,96,113,80,97),(69,89,114,73,98),(70,90,115,74,99),(71,91,116,75,100),(72,92,117,76,101)], [(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)], [(2,6),(4,8),(10,14),(12,16),(17,21),(19,23),(25,29),(27,31),(33,37),(35,39),(42,46),(44,48),(49,53),(51,55),(58,62),(60,64),(65,69),(67,71),(73,77),(75,79),(81,85),(83,87),(89,93),(91,95),(98,102),(100,104),(105,109),(107,111),(114,118),(116,120)], [(1,99,26),(2,100,27),(3,101,28),(4,102,29),(5,103,30),(6,104,31),(7,97,32),(8,98,25),(9,72,38),(10,65,39),(11,66,40),(12,67,33),(13,68,34),(14,69,35),(15,70,36),(16,71,37),(17,75,51),(18,76,52),(19,77,53),(20,78,54),(21,79,55),(22,80,56),(23,73,49),(24,74,50),(41,90,63),(42,91,64),(43,92,57),(44,93,58),(45,94,59),(46,95,60),(47,96,61),(48,89,62),(81,111,118),(82,112,119),(83,105,120),(84,106,113),(85,107,114),(86,108,115),(87,109,116),(88,110,117)], [(2,6),(3,7),(9,13),(12,16),(17,21),(18,22),(25,98),(26,99),(27,104),(28,97),(29,102),(30,103),(31,100),(32,101),(33,71),(34,72),(35,69),(36,70),(37,67),(38,68),(39,65),(40,66),(42,46),(43,47),(49,73),(50,74),(51,79),(52,80),(53,77),(54,78),(55,75),(56,76),(57,96),(58,93),(59,94),(60,91),(61,92),(62,89),(63,90),(64,95),(81,118),(82,119),(83,116),(84,117),(85,114),(86,115),(87,120),(88,113),(105,109),(106,110)])
105 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10P | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | ··· | 40H | 40I | ··· | 40P | 60A | ··· | 60H | 60I | 60J | 60K | 60L | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 12 | 12 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | ··· | 12 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | S3 | D4 | D6 | D12 | C3⋊D4 | C4×S3 | C5×S3 | C5×D4 | S3×C10 | C5×D12 | C5×C3⋊D4 | S3×C20 | C4.D4 | M4(2)⋊S3 | C5×C4.D4 | C5×M4(2)⋊S3 |
| kernel | C5×M4(2)⋊S3 | C5×C4.Dic3 | C15×M4(2) | C10×D12 | S3×C2×C10 | M4(2)⋊S3 | C4.Dic3 | C3×M4(2) | C2×D12 | C22×S3 | C5×M4(2) | C60 | C2×C20 | C20 | C20 | C2×C10 | M4(2) | C12 | C2×C4 | C4 | C4 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 16 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 8 | 4 | 8 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C5×M4(2)⋊S3 ►in GL4(𝔽241) generated by
| 98 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 90 | 81 | 226 | 153 |
| 205 | 106 | 120 | 34 |
| 198 | 86 | 47 | 16 |
| 155 | 43 | 29 | 239 |
| 1 | 0 | 125 | 189 |
| 0 | 1 | 72 | 8 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 240 | 1 | 10 | 88 |
| 240 | 0 | 98 | 231 |
| 0 | 0 | 240 | 1 |
| 0 | 0 | 240 | 0 |
| 1 | 240 | 205 | 127 |
| 0 | 240 | 169 | 36 |
| 0 | 0 | 1 | 240 |
| 0 | 0 | 0 | 240 |
G:=sub<GL(4,GF(241))| [98,0,0,0,0,98,0,0,0,0,98,0,0,0,0,98],[90,205,198,155,81,106,86,43,226,120,47,29,153,34,16,239],[1,0,0,0,0,1,0,0,125,72,240,0,189,8,0,240],[240,240,0,0,1,0,0,0,10,98,240,240,88,231,1,0],[1,0,0,0,240,240,0,0,205,169,1,0,127,36,240,240] >;
C5×M4(2)⋊S3 in GAP, Magma, Sage, TeX
C_5\times M_{4(2})\rtimes S_3 % in TeX
G:=Group("C5xM4(2):S3"); // GroupNames label
G:=SmallGroup(480,142);
// by ID
G=gap.SmallGroup(480,142);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,589,148,2803,136,2111,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^8=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^5,b*d=d*b,e*b*e=b*c,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations