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 |
Generators and relations for C5×Q8.A4
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 >
Subgroups: 278 in 92 conjugacy classes, 32 normal (14 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C2×C4, D4, Q8, C23, C10, C10, C12, C15, C2×D4, C4○D4, C4○D4, C20, C20, C2×C10, SL2(𝔽3), C3×Q8, C30, 2+ 1+4, C2×C20, C5×D4, C5×Q8, C22×C10, C4.A4, C60, D4×C10, C5×C4○D4, C5×C4○D4, Q8.A4, C5×SL2(𝔽3), Q8×C15, C5×2+ 1+4, C5×C4.A4, C5×Q8.A4
Quotients: C1, C2, C3, C22, C5, C6, C10, A4, C2×C6, C15, C2×C10, C2×A4, C30, C22×A4, C5×A4, C2×C30, Q8.A4, C10×A4, A4×C2×C10, C5×Q8.A4
►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 42)(2 55 13 43)(3 51 14 44)(4 52 15 45)(5 53 11 41)(6 22 96 103)(7 23 97 104)(8 24 98 105)(9 25 99 101)(10 21 100 102)(16 113 117 32)(17 114 118 33)(18 115 119 34)(19 111 120 35)(20 112 116 31)(26 40 50 69)(27 36 46 70)(28 37 47 66)(29 38 48 67)(30 39 49 68)(56 63 85 93)(57 64 81 94)(58 65 82 95)(59 61 83 91)(60 62 84 92)(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 96 20)(7 117 97 16)(8 118 98 17)(9 119 99 18)(10 120 100 19)(21 111 102 35)(22 112 103 31)(23 113 104 32)(24 114 105 33)(25 115 101 34)(26 45 50 52)(27 41 46 53)(28 42 47 54)(29 43 48 55)(30 44 49 51)(56 110 85 76)(57 106 81 77)(58 107 82 78)(59 108 83 79)(60 109 84 80)(61 89 91 75)(62 90 92 71)(63 86 93 72)(64 87 94 73)(65 88 95 74)
(1 47 12 28)(2 48 13 29)(3 49 14 30)(4 50 15 26)(5 46 11 27)(6 116 96 20)(7 117 97 16)(8 118 98 17)(9 119 99 18)(10 120 100 19)(21 35 102 111)(22 31 103 112)(23 32 104 113)(24 33 105 114)(25 34 101 115)(36 53 70 41)(37 54 66 42)(38 55 67 43)(39 51 68 44)(40 52 69 45)(56 63 85 93)(57 64 81 94)(58 65 82 95)(59 61 83 91)(60 62 84 92)(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 22 96 103)(7 23 97 104)(8 24 98 105)(9 25 99 101)(10 21 100 102)(16 32 117 113)(17 33 118 114)(18 34 119 115)(19 35 120 111)(20 31 116 112)(26 52 50 45)(27 53 46 41)(28 54 47 42)(29 55 48 43)(30 51 49 44)(56 72 85 86)(57 73 81 87)(58 74 82 88)(59 75 83 89)(60 71 84 90)(61 79 91 108)(62 80 92 109)(63 76 93 110)(64 77 94 106)(65 78 95 107)
(1 81 7)(2 82 8)(3 83 9)(4 84 10)(5 85 6)(11 56 96)(12 57 97)(13 58 98)(14 59 99)(15 60 100)(16 66 106)(17 67 107)(18 68 108)(19 69 109)(20 70 110)(21 52 92)(22 53 93)(23 54 94)(24 55 95)(25 51 91)(26 71 111)(27 72 112)(28 73 113)(29 74 114)(30 75 115)(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 63 103)(42 64 104)(43 65 105)(44 61 101)(45 62 102)
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), (1,54,12,42)(2,55,13,43)(3,51,14,44)(4,52,15,45)(5,53,11,41)(6,22,96,103)(7,23,97,104)(8,24,98,105)(9,25,99,101)(10,21,100,102)(16,113,117,32)(17,114,118,33)(18,115,119,34)(19,111,120,35)(20,112,116,31)(26,40,50,69)(27,36,46,70)(28,37,47,66)(29,38,48,67)(30,39,49,68)(56,63,85,93)(57,64,81,94)(58,65,82,95)(59,61,83,91)(60,62,84,92)(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)(6,116,96,20)(7,117,97,16)(8,118,98,17)(9,119,99,18)(10,120,100,19)(21,111,102,35)(22,112,103,31)(23,113,104,32)(24,114,105,33)(25,115,101,34)(26,45,50,52)(27,41,46,53)(28,42,47,54)(29,43,48,55)(30,44,49,51)(56,110,85,76)(57,106,81,77)(58,107,82,78)(59,108,83,79)(60,109,84,80)(61,89,91,75)(62,90,92,71)(63,86,93,72)(64,87,94,73)(65,88,95,74), (1,47,12,28)(2,48,13,29)(3,49,14,30)(4,50,15,26)(5,46,11,27)(6,116,96,20)(7,117,97,16)(8,118,98,17)(9,119,99,18)(10,120,100,19)(21,35,102,111)(22,31,103,112)(23,32,104,113)(24,33,105,114)(25,34,101,115)(36,53,70,41)(37,54,66,42)(38,55,67,43)(39,51,68,44)(40,52,69,45)(56,63,85,93)(57,64,81,94)(58,65,82,95)(59,61,83,91)(60,62,84,92)(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)(6,22,96,103)(7,23,97,104)(8,24,98,105)(9,25,99,101)(10,21,100,102)(16,32,117,113)(17,33,118,114)(18,34,119,115)(19,35,120,111)(20,31,116,112)(26,52,50,45)(27,53,46,41)(28,54,47,42)(29,55,48,43)(30,51,49,44)(56,72,85,86)(57,73,81,87)(58,74,82,88)(59,75,83,89)(60,71,84,90)(61,79,91,108)(62,80,92,109)(63,76,93,110)(64,77,94,106)(65,78,95,107), (1,81,7)(2,82,8)(3,83,9)(4,84,10)(5,85,6)(11,56,96)(12,57,97)(13,58,98)(14,59,99)(15,60,100)(16,66,106)(17,67,107)(18,68,108)(19,69,109)(20,70,110)(21,52,92)(22,53,93)(23,54,94)(24,55,95)(25,51,91)(26,71,111)(27,72,112)(28,73,113)(29,74,114)(30,75,115)(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)(41,63,103)(42,64,104)(43,65,105)(44,61,101)(45,62,102)>;
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), (1,54,12,42)(2,55,13,43)(3,51,14,44)(4,52,15,45)(5,53,11,41)(6,22,96,103)(7,23,97,104)(8,24,98,105)(9,25,99,101)(10,21,100,102)(16,113,117,32)(17,114,118,33)(18,115,119,34)(19,111,120,35)(20,112,116,31)(26,40,50,69)(27,36,46,70)(28,37,47,66)(29,38,48,67)(30,39,49,68)(56,63,85,93)(57,64,81,94)(58,65,82,95)(59,61,83,91)(60,62,84,92)(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)(6,116,96,20)(7,117,97,16)(8,118,98,17)(9,119,99,18)(10,120,100,19)(21,111,102,35)(22,112,103,31)(23,113,104,32)(24,114,105,33)(25,115,101,34)(26,45,50,52)(27,41,46,53)(28,42,47,54)(29,43,48,55)(30,44,49,51)(56,110,85,76)(57,106,81,77)(58,107,82,78)(59,108,83,79)(60,109,84,80)(61,89,91,75)(62,90,92,71)(63,86,93,72)(64,87,94,73)(65,88,95,74), (1,47,12,28)(2,48,13,29)(3,49,14,30)(4,50,15,26)(5,46,11,27)(6,116,96,20)(7,117,97,16)(8,118,98,17)(9,119,99,18)(10,120,100,19)(21,35,102,111)(22,31,103,112)(23,32,104,113)(24,33,105,114)(25,34,101,115)(36,53,70,41)(37,54,66,42)(38,55,67,43)(39,51,68,44)(40,52,69,45)(56,63,85,93)(57,64,81,94)(58,65,82,95)(59,61,83,91)(60,62,84,92)(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)(6,22,96,103)(7,23,97,104)(8,24,98,105)(9,25,99,101)(10,21,100,102)(16,32,117,113)(17,33,118,114)(18,34,119,115)(19,35,120,111)(20,31,116,112)(26,52,50,45)(27,53,46,41)(28,54,47,42)(29,55,48,43)(30,51,49,44)(56,72,85,86)(57,73,81,87)(58,74,82,88)(59,75,83,89)(60,71,84,90)(61,79,91,108)(62,80,92,109)(63,76,93,110)(64,77,94,106)(65,78,95,107), (1,81,7)(2,82,8)(3,83,9)(4,84,10)(5,85,6)(11,56,96)(12,57,97)(13,58,98)(14,59,99)(15,60,100)(16,66,106)(17,67,107)(18,68,108)(19,69,109)(20,70,110)(21,52,92)(22,53,93)(23,54,94)(24,55,95)(25,51,91)(26,71,111)(27,72,112)(28,73,113)(29,74,114)(30,75,115)(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)(41,63,103)(42,64,104)(43,65,105)(44,61,101)(45,62,102) );
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)], [(1,54,12,42),(2,55,13,43),(3,51,14,44),(4,52,15,45),(5,53,11,41),(6,22,96,103),(7,23,97,104),(8,24,98,105),(9,25,99,101),(10,21,100,102),(16,113,117,32),(17,114,118,33),(18,115,119,34),(19,111,120,35),(20,112,116,31),(26,40,50,69),(27,36,46,70),(28,37,47,66),(29,38,48,67),(30,39,49,68),(56,63,85,93),(57,64,81,94),(58,65,82,95),(59,61,83,91),(60,62,84,92),(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),(6,116,96,20),(7,117,97,16),(8,118,98,17),(9,119,99,18),(10,120,100,19),(21,111,102,35),(22,112,103,31),(23,113,104,32),(24,114,105,33),(25,115,101,34),(26,45,50,52),(27,41,46,53),(28,42,47,54),(29,43,48,55),(30,44,49,51),(56,110,85,76),(57,106,81,77),(58,107,82,78),(59,108,83,79),(60,109,84,80),(61,89,91,75),(62,90,92,71),(63,86,93,72),(64,87,94,73),(65,88,95,74)], [(1,47,12,28),(2,48,13,29),(3,49,14,30),(4,50,15,26),(5,46,11,27),(6,116,96,20),(7,117,97,16),(8,118,98,17),(9,119,99,18),(10,120,100,19),(21,35,102,111),(22,31,103,112),(23,32,104,113),(24,33,105,114),(25,34,101,115),(36,53,70,41),(37,54,66,42),(38,55,67,43),(39,51,68,44),(40,52,69,45),(56,63,85,93),(57,64,81,94),(58,65,82,95),(59,61,83,91),(60,62,84,92),(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),(6,22,96,103),(7,23,97,104),(8,24,98,105),(9,25,99,101),(10,21,100,102),(16,32,117,113),(17,33,118,114),(18,34,119,115),(19,35,120,111),(20,31,116,112),(26,52,50,45),(27,53,46,41),(28,54,47,42),(29,55,48,43),(30,51,49,44),(56,72,85,86),(57,73,81,87),(58,74,82,88),(59,75,83,89),(60,71,84,90),(61,79,91,108),(62,80,92,109),(63,76,93,110),(64,77,94,106),(65,78,95,107)], [(1,81,7),(2,82,8),(3,83,9),(4,84,10),(5,85,6),(11,56,96),(12,57,97),(13,58,98),(14,59,99),(15,60,100),(16,66,106),(17,67,107),(18,68,108),(19,69,109),(20,70,110),(21,52,92),(22,53,93),(23,54,94),(24,55,95),(25,51,91),(26,71,111),(27,72,112),(28,73,113),(29,74,114),(30,75,115),(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),(41,63,103),(42,64,104),(43,65,105),(44,61,101),(45,62,102)]])
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 |
Matrix representation of C5×Q8.A4 ►in 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] >;
C5×Q8.A4 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