direct product, non-abelian, soluble
Aliases: C5×Q8.A4, 2+ (1+4)⋊1C15, (C5×Q8).A4, Q8.(C5×A4), C4○D4.C30, C4.A4⋊3C10, C20.8(C2×A4), C4.2(C10×A4), Q8.2(C2×C30), C10.17(C22×A4), (C5×2+ (1+4))⋊1C3, SL2(𝔽3)⋊4(C2×C10), (C5×SL2(𝔽3))⋊12C22, C2.6(A4×C2×C10), (C5×C4.A4)⋊8C2, (C5×C4○D4).3C6, (C5×Q8).12(C2×C6), SmallGroup(480,1131)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — C5×Q8.A4 |
Subgroups: 278 in 92 conjugacy classes, 32 normal (14 characteristic)
C1, C2, C2 [×3], C3, C4 [×3], C4, C22 [×5], C5, C6, C2×C4 [×3], D4 [×6], Q8 [×2], C23 [×2], C10, C10 [×3], C12 [×3], C15, C2×D4 [×3], C4○D4 [×3], C4○D4, C20 [×3], C20, C2×C10 [×5], SL2(𝔽3), C3×Q8, C30, 2+ (1+4), C2×C20 [×3], C5×D4 [×6], C5×Q8 [×2], C22×C10 [×2], C4.A4 [×3], C60 [×3], D4×C10 [×3], C5×C4○D4 [×3], C5×C4○D4, Q8.A4, C5×SL2(𝔽3), Q8×C15, C5×2+ (1+4), C5×C4.A4 [×3], C5×Q8.A4
Quotients:
C1, C2 [×3], C3, C22, C5, C6 [×3], C10 [×3], A4, C2×C6, C15, C2×C10, C2×A4 [×3], C30 [×3], C22×A4, C5×A4, C2×C30, Q8.A4, C10×A4 [×3], A4×C2×C10, C5×Q8.A4
Generators and relations
G = < a,b,c,d,e,f | a5=b4=f3=1, c2=d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc-1=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=b2d, fdf-1=b2de, fef-1=d >
►On 120 points
Generators in S120
(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 54 12 29)(2 55 13 30)(3 51 14 26)(4 52 15 27)(5 53 11 28)(6 19 104 100)(7 20 105 96)(8 16 101 97)(9 17 102 98)(10 18 103 99)(21 115 119 34)(22 111 120 35)(23 112 116 31)(24 113 117 32)(25 114 118 33)(36 46 70 41)(37 47 66 42)(38 48 67 43)(39 49 68 44)(40 50 69 45)(56 81 94 65)(57 82 95 61)(58 83 91 62)(59 84 92 63)(60 85 93 64)(71 80 90 109)(72 76 86 110)(73 77 87 106)(74 78 88 107)(75 79 89 108)
(1 37 12 66)(2 38 13 67)(3 39 14 68)(4 40 15 69)(5 36 11 70)(6 116 104 23)(7 117 105 24)(8 118 101 25)(9 119 102 21)(10 120 103 22)(16 114 97 33)(17 115 98 34)(18 111 99 35)(19 112 100 31)(20 113 96 32)(26 49 51 44)(27 50 52 45)(28 46 53 41)(29 47 54 42)(30 48 55 43)(56 87 94 73)(57 88 95 74)(58 89 91 75)(59 90 92 71)(60 86 93 72)(61 107 82 78)(62 108 83 79)(63 109 84 80)(64 110 85 76)(65 106 81 77)
(1 47 12 42)(2 48 13 43)(3 49 14 44)(4 50 15 45)(5 46 11 41)(6 116 104 23)(7 117 105 24)(8 118 101 25)(9 119 102 21)(10 120 103 22)(16 33 97 114)(17 34 98 115)(18 35 99 111)(19 31 100 112)(20 32 96 113)(26 39 51 68)(27 40 52 69)(28 36 53 70)(29 37 54 66)(30 38 55 67)(56 81 94 65)(57 82 95 61)(58 83 91 62)(59 84 92 63)(60 85 93 64)(71 109 90 80)(72 110 86 76)(73 106 87 77)(74 107 88 78)(75 108 89 79)
(1 37 12 66)(2 38 13 67)(3 39 14 68)(4 40 15 69)(5 36 11 70)(6 19 104 100)(7 20 105 96)(8 16 101 97)(9 17 102 98)(10 18 103 99)(21 34 119 115)(22 35 120 111)(23 31 116 112)(24 32 117 113)(25 33 118 114)(26 44 51 49)(27 45 52 50)(28 41 53 46)(29 42 54 47)(30 43 55 48)(56 77 94 106)(57 78 95 107)(58 79 91 108)(59 80 92 109)(60 76 93 110)(61 74 82 88)(62 75 83 89)(63 71 84 90)(64 72 85 86)(65 73 81 87)
(1 81 7)(2 82 8)(3 83 9)(4 84 10)(5 85 6)(11 64 104)(12 65 105)(13 61 101)(14 62 102)(15 63 103)(16 55 95)(17 51 91)(18 52 92)(19 53 93)(20 54 94)(21 68 108)(22 69 109)(23 70 110)(24 66 106)(25 67 107)(26 58 98)(27 59 99)(28 60 100)(29 56 96)(30 57 97)(31 46 86)(32 47 87)(33 48 88)(34 49 89)(35 50 90)(36 76 116)(37 77 117)(38 78 118)(39 79 119)(40 80 120)(41 72 112)(42 73 113)(43 74 114)(44 75 115)(45 71 111)
G:=sub<Sym(120)| (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), (1,54,12,29)(2,55,13,30)(3,51,14,26)(4,52,15,27)(5,53,11,28)(6,19,104,100)(7,20,105,96)(8,16,101,97)(9,17,102,98)(10,18,103,99)(21,115,119,34)(22,111,120,35)(23,112,116,31)(24,113,117,32)(25,114,118,33)(36,46,70,41)(37,47,66,42)(38,48,67,43)(39,49,68,44)(40,50,69,45)(56,81,94,65)(57,82,95,61)(58,83,91,62)(59,84,92,63)(60,85,93,64)(71,80,90,109)(72,76,86,110)(73,77,87,106)(74,78,88,107)(75,79,89,108), (1,37,12,66)(2,38,13,67)(3,39,14,68)(4,40,15,69)(5,36,11,70)(6,116,104,23)(7,117,105,24)(8,118,101,25)(9,119,102,21)(10,120,103,22)(16,114,97,33)(17,115,98,34)(18,111,99,35)(19,112,100,31)(20,113,96,32)(26,49,51,44)(27,50,52,45)(28,46,53,41)(29,47,54,42)(30,48,55,43)(56,87,94,73)(57,88,95,74)(58,89,91,75)(59,90,92,71)(60,86,93,72)(61,107,82,78)(62,108,83,79)(63,109,84,80)(64,110,85,76)(65,106,81,77), (1,47,12,42)(2,48,13,43)(3,49,14,44)(4,50,15,45)(5,46,11,41)(6,116,104,23)(7,117,105,24)(8,118,101,25)(9,119,102,21)(10,120,103,22)(16,33,97,114)(17,34,98,115)(18,35,99,111)(19,31,100,112)(20,32,96,113)(26,39,51,68)(27,40,52,69)(28,36,53,70)(29,37,54,66)(30,38,55,67)(56,81,94,65)(57,82,95,61)(58,83,91,62)(59,84,92,63)(60,85,93,64)(71,109,90,80)(72,110,86,76)(73,106,87,77)(74,107,88,78)(75,108,89,79), (1,37,12,66)(2,38,13,67)(3,39,14,68)(4,40,15,69)(5,36,11,70)(6,19,104,100)(7,20,105,96)(8,16,101,97)(9,17,102,98)(10,18,103,99)(21,34,119,115)(22,35,120,111)(23,31,116,112)(24,32,117,113)(25,33,118,114)(26,44,51,49)(27,45,52,50)(28,41,53,46)(29,42,54,47)(30,43,55,48)(56,77,94,106)(57,78,95,107)(58,79,91,108)(59,80,92,109)(60,76,93,110)(61,74,82,88)(62,75,83,89)(63,71,84,90)(64,72,85,86)(65,73,81,87), (1,81,7)(2,82,8)(3,83,9)(4,84,10)(5,85,6)(11,64,104)(12,65,105)(13,61,101)(14,62,102)(15,63,103)(16,55,95)(17,51,91)(18,52,92)(19,53,93)(20,54,94)(21,68,108)(22,69,109)(23,70,110)(24,66,106)(25,67,107)(26,58,98)(27,59,99)(28,60,100)(29,56,96)(30,57,97)(31,46,86)(32,47,87)(33,48,88)(34,49,89)(35,50,90)(36,76,116)(37,77,117)(38,78,118)(39,79,119)(40,80,120)(41,72,112)(42,73,113)(43,74,114)(44,75,115)(45,71,111)>;
G:=Group( (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), (1,54,12,29)(2,55,13,30)(3,51,14,26)(4,52,15,27)(5,53,11,28)(6,19,104,100)(7,20,105,96)(8,16,101,97)(9,17,102,98)(10,18,103,99)(21,115,119,34)(22,111,120,35)(23,112,116,31)(24,113,117,32)(25,114,118,33)(36,46,70,41)(37,47,66,42)(38,48,67,43)(39,49,68,44)(40,50,69,45)(56,81,94,65)(57,82,95,61)(58,83,91,62)(59,84,92,63)(60,85,93,64)(71,80,90,109)(72,76,86,110)(73,77,87,106)(74,78,88,107)(75,79,89,108), (1,37,12,66)(2,38,13,67)(3,39,14,68)(4,40,15,69)(5,36,11,70)(6,116,104,23)(7,117,105,24)(8,118,101,25)(9,119,102,21)(10,120,103,22)(16,114,97,33)(17,115,98,34)(18,111,99,35)(19,112,100,31)(20,113,96,32)(26,49,51,44)(27,50,52,45)(28,46,53,41)(29,47,54,42)(30,48,55,43)(56,87,94,73)(57,88,95,74)(58,89,91,75)(59,90,92,71)(60,86,93,72)(61,107,82,78)(62,108,83,79)(63,109,84,80)(64,110,85,76)(65,106,81,77), (1,47,12,42)(2,48,13,43)(3,49,14,44)(4,50,15,45)(5,46,11,41)(6,116,104,23)(7,117,105,24)(8,118,101,25)(9,119,102,21)(10,120,103,22)(16,33,97,114)(17,34,98,115)(18,35,99,111)(19,31,100,112)(20,32,96,113)(26,39,51,68)(27,40,52,69)(28,36,53,70)(29,37,54,66)(30,38,55,67)(56,81,94,65)(57,82,95,61)(58,83,91,62)(59,84,92,63)(60,85,93,64)(71,109,90,80)(72,110,86,76)(73,106,87,77)(74,107,88,78)(75,108,89,79), (1,37,12,66)(2,38,13,67)(3,39,14,68)(4,40,15,69)(5,36,11,70)(6,19,104,100)(7,20,105,96)(8,16,101,97)(9,17,102,98)(10,18,103,99)(21,34,119,115)(22,35,120,111)(23,31,116,112)(24,32,117,113)(25,33,118,114)(26,44,51,49)(27,45,52,50)(28,41,53,46)(29,42,54,47)(30,43,55,48)(56,77,94,106)(57,78,95,107)(58,79,91,108)(59,80,92,109)(60,76,93,110)(61,74,82,88)(62,75,83,89)(63,71,84,90)(64,72,85,86)(65,73,81,87), (1,81,7)(2,82,8)(3,83,9)(4,84,10)(5,85,6)(11,64,104)(12,65,105)(13,61,101)(14,62,102)(15,63,103)(16,55,95)(17,51,91)(18,52,92)(19,53,93)(20,54,94)(21,68,108)(22,69,109)(23,70,110)(24,66,106)(25,67,107)(26,58,98)(27,59,99)(28,60,100)(29,56,96)(30,57,97)(31,46,86)(32,47,87)(33,48,88)(34,49,89)(35,50,90)(36,76,116)(37,77,117)(38,78,118)(39,79,119)(40,80,120)(41,72,112)(42,73,113)(43,74,114)(44,75,115)(45,71,111) );
G=PermutationGroup([(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)], [(1,54,12,29),(2,55,13,30),(3,51,14,26),(4,52,15,27),(5,53,11,28),(6,19,104,100),(7,20,105,96),(8,16,101,97),(9,17,102,98),(10,18,103,99),(21,115,119,34),(22,111,120,35),(23,112,116,31),(24,113,117,32),(25,114,118,33),(36,46,70,41),(37,47,66,42),(38,48,67,43),(39,49,68,44),(40,50,69,45),(56,81,94,65),(57,82,95,61),(58,83,91,62),(59,84,92,63),(60,85,93,64),(71,80,90,109),(72,76,86,110),(73,77,87,106),(74,78,88,107),(75,79,89,108)], [(1,37,12,66),(2,38,13,67),(3,39,14,68),(4,40,15,69),(5,36,11,70),(6,116,104,23),(7,117,105,24),(8,118,101,25),(9,119,102,21),(10,120,103,22),(16,114,97,33),(17,115,98,34),(18,111,99,35),(19,112,100,31),(20,113,96,32),(26,49,51,44),(27,50,52,45),(28,46,53,41),(29,47,54,42),(30,48,55,43),(56,87,94,73),(57,88,95,74),(58,89,91,75),(59,90,92,71),(60,86,93,72),(61,107,82,78),(62,108,83,79),(63,109,84,80),(64,110,85,76),(65,106,81,77)], [(1,47,12,42),(2,48,13,43),(3,49,14,44),(4,50,15,45),(5,46,11,41),(6,116,104,23),(7,117,105,24),(8,118,101,25),(9,119,102,21),(10,120,103,22),(16,33,97,114),(17,34,98,115),(18,35,99,111),(19,31,100,112),(20,32,96,113),(26,39,51,68),(27,40,52,69),(28,36,53,70),(29,37,54,66),(30,38,55,67),(56,81,94,65),(57,82,95,61),(58,83,91,62),(59,84,92,63),(60,85,93,64),(71,109,90,80),(72,110,86,76),(73,106,87,77),(74,107,88,78),(75,108,89,79)], [(1,37,12,66),(2,38,13,67),(3,39,14,68),(4,40,15,69),(5,36,11,70),(6,19,104,100),(7,20,105,96),(8,16,101,97),(9,17,102,98),(10,18,103,99),(21,34,119,115),(22,35,120,111),(23,31,116,112),(24,32,117,113),(25,33,118,114),(26,44,51,49),(27,45,52,50),(28,41,53,46),(29,42,54,47),(30,43,55,48),(56,77,94,106),(57,78,95,107),(58,79,91,108),(59,80,92,109),(60,76,93,110),(61,74,82,88),(62,75,83,89),(63,71,84,90),(64,72,85,86),(65,73,81,87)], [(1,81,7),(2,82,8),(3,83,9),(4,84,10),(5,85,6),(11,64,104),(12,65,105),(13,61,101),(14,62,102),(15,63,103),(16,55,95),(17,51,91),(18,52,92),(19,53,93),(20,54,94),(21,68,108),(22,69,109),(23,70,110),(24,66,106),(25,67,107),(26,58,98),(27,59,99),(28,60,100),(29,56,96),(30,57,97),(31,46,86),(32,47,87),(33,48,88),(34,49,89),(35,50,90),(36,76,116),(37,77,117),(38,78,118),(39,79,119),(40,80,120),(41,72,112),(42,73,113),(43,74,114),(44,75,115),(45,71,111)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 20 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 |
| 0 | 0 | 20 | 0 |
| 0 | 0 | 0 | 20 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 0 |
| 0 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 1 | 0 |
| 30 | 30 | 30 | 30 |
| 31 | 30 | 31 | 30 |
| 31 | 30 | 30 | 31 |
| 31 | 31 | 30 | 30 |
G:=sub<GL(4,GF(61))| [20,0,0,0,0,20,0,0,0,0,20,0,0,0,0,20],[0,0,60,0,0,0,0,1,1,0,0,0,0,60,0,0],[0,60,0,0,1,0,0,0,0,0,0,60,0,0,1,0],[0,0,0,60,0,0,1,0,0,60,0,0,1,0,0,0],[0,60,0,0,1,0,0,0,0,0,0,1,0,0,60,0],[30,31,31,31,30,30,30,31,30,31,30,30,30,30,31,30] >;
95 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 10A | 10B | 10C | 10D | 10E | ··· | 10P | 12A | ··· | 12F | 15A | ··· | 15H | 20A | ··· | 20L | 20M | 20N | 20O | 20P | 30A | ··· | 30H | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 6 | 6 | 6 | 4 | 4 | 2 | 2 | 2 | 6 | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 6 | ··· | 6 | 8 | ··· | 8 | 4 | ··· | 4 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 4 | ··· | 4 | 8 | ··· | 8 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | 4 | 4 |
| type | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 | A4 | C2×A4 | C5×A4 | C10×A4 | Q8.A4 | Q8.A4 | C5×Q8.A4 |
| kernel | C5×Q8.A4 | C5×C4.A4 | C5×2+ (1+4) | Q8.A4 | C5×C4○D4 | C4.A4 | 2+ (1+4) | C4○D4 | C5×Q8 | C20 | Q8 | C4 | C5 | C5 | C1 |
| # reps | 1 | 3 | 2 | 4 | 6 | 12 | 8 | 24 | 1 | 3 | 4 | 12 | 1 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_5\times Q_8.A_4
% in TeX
G:=Group("C5xQ8.A4"); // GroupNames label
G:=SmallGroup(480,1131);
// by ID
G=gap.SmallGroup(480,1131);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,1680,3389,1688,1068,172,1909,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^4=f^3=1,c^2=d^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,e*d*e^-1=b^2*d,f*d*f^-1=b^2*d*e,f*e*f^-1=d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀