direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C22×C6, C15⋊3C24, C30⋊3C23, C5⋊(C23×C6), C10⋊(C22×C6), (C22×C10)⋊5C6, (C22×C30)⋊5C2, (C2×C30)⋊12C22, (C2×C10)⋊6(C2×C6), SmallGroup(240,205)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C22×C6 |
Generators and relations for D5×C22×C6
G = < a,b,c,d,e | a2=b2=c6=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 676 in 268 conjugacy classes, 166 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, C5, C6, C6, C23, C23, D5, C10, C2×C6, C2×C6, C15, C24, D10, C2×C10, C22×C6, C22×C6, C3×D5, C30, C22×D5, C22×C10, C23×C6, C6×D5, C2×C30, C23×D5, D5×C2×C6, C22×C30, D5×C22×C6
Quotients: C1, C2, C3, C22, C6, C23, D5, C2×C6, C24, D10, C22×C6, C3×D5, C22×D5, C23×C6, C6×D5, C23×D5, D5×C2×C6, D5×C22×C6
►On 120 points
Generators in S120
(1 86)(2 87)(3 88)(4 89)(5 90)(6 85)(7 82)(8 83)(9 84)(10 79)(11 80)(12 81)(13 78)(14 73)(15 74)(16 75)(17 76)(18 77)(19 63)(20 64)(21 65)(22 66)(23 61)(24 62)(25 69)(26 70)(27 71)(28 72)(29 67)(30 68)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)
(1 59)(2 60)(3 55)(4 56)(5 57)(6 58)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 105)(14 106)(15 107)(16 108)(17 103)(18 104)(19 96)(20 91)(21 92)(22 93)(23 94)(24 95)(25 102)(26 97)(27 98)(28 99)(29 100)(30 101)(31 64)(32 65)(33 66)(34 61)(35 62)(36 63)(37 70)(38 71)(39 72)(40 67)(41 68)(42 69)(43 76)(44 77)(45 78)(46 73)(47 74)(48 75)(49 82)(50 83)(51 84)(52 79)(53 80)(54 81)(85 118)(86 119)(87 120)(88 115)(89 116)(90 117)
(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 36 49 37 43)(2 31 50 38 44)(3 32 51 39 45)(4 33 52 40 46)(5 34 53 41 47)(6 35 54 42 48)(7 26 17 119 19)(8 27 18 120 20)(9 28 13 115 21)(10 29 14 116 22)(11 30 15 117 23)(12 25 16 118 24)(55 65 84 72 78)(56 66 79 67 73)(57 61 80 68 74)(58 62 81 69 75)(59 63 82 70 76)(60 64 83 71 77)(85 95 114 102 108)(86 96 109 97 103)(87 91 110 98 104)(88 92 111 99 105)(89 93 112 100 106)(90 94 113 101 107)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 112)(8 113)(9 114)(10 109)(11 110)(12 111)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 100)(20 101)(21 102)(22 97)(23 98)(24 99)(25 92)(26 93)(27 94)(28 95)(29 96)(30 91)(31 68)(32 69)(33 70)(34 71)(35 72)(36 67)(37 66)(38 61)(39 62)(40 63)(41 64)(42 65)(43 56)(44 57)(45 58)(46 59)(47 60)(48 55)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(103 116)(104 117)(105 118)(106 119)(107 120)(108 115)
G:=sub<Sym(120)| (1,86)(2,87)(3,88)(4,89)(5,90)(6,85)(7,82)(8,83)(9,84)(10,79)(11,80)(12,81)(13,78)(14,73)(15,74)(16,75)(17,76)(18,77)(19,63)(20,64)(21,65)(22,66)(23,61)(24,62)(25,69)(26,70)(27,71)(28,72)(29,67)(30,68)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120), (1,59)(2,60)(3,55)(4,56)(5,57)(6,58)(7,109)(8,110)(9,111)(10,112)(11,113)(12,114)(13,105)(14,106)(15,107)(16,108)(17,103)(18,104)(19,96)(20,91)(21,92)(22,93)(23,94)(24,95)(25,102)(26,97)(27,98)(28,99)(29,100)(30,101)(31,64)(32,65)(33,66)(34,61)(35,62)(36,63)(37,70)(38,71)(39,72)(40,67)(41,68)(42,69)(43,76)(44,77)(45,78)(46,73)(47,74)(48,75)(49,82)(50,83)(51,84)(52,79)(53,80)(54,81)(85,118)(86,119)(87,120)(88,115)(89,116)(90,117), (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), (1,36,49,37,43)(2,31,50,38,44)(3,32,51,39,45)(4,33,52,40,46)(5,34,53,41,47)(6,35,54,42,48)(7,26,17,119,19)(8,27,18,120,20)(9,28,13,115,21)(10,29,14,116,22)(11,30,15,117,23)(12,25,16,118,24)(55,65,84,72,78)(56,66,79,67,73)(57,61,80,68,74)(58,62,81,69,75)(59,63,82,70,76)(60,64,83,71,77)(85,95,114,102,108)(86,96,109,97,103)(87,91,110,98,104)(88,92,111,99,105)(89,93,112,100,106)(90,94,113,101,107), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,112)(8,113)(9,114)(10,109)(11,110)(12,111)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,100)(20,101)(21,102)(22,97)(23,98)(24,99)(25,92)(26,93)(27,94)(28,95)(29,96)(30,91)(31,68)(32,69)(33,70)(34,71)(35,72)(36,67)(37,66)(38,61)(39,62)(40,63)(41,64)(42,65)(43,56)(44,57)(45,58)(46,59)(47,60)(48,55)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(103,116)(104,117)(105,118)(106,119)(107,120)(108,115)>;
G:=Group( (1,86)(2,87)(3,88)(4,89)(5,90)(6,85)(7,82)(8,83)(9,84)(10,79)(11,80)(12,81)(13,78)(14,73)(15,74)(16,75)(17,76)(18,77)(19,63)(20,64)(21,65)(22,66)(23,61)(24,62)(25,69)(26,70)(27,71)(28,72)(29,67)(30,68)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120), (1,59)(2,60)(3,55)(4,56)(5,57)(6,58)(7,109)(8,110)(9,111)(10,112)(11,113)(12,114)(13,105)(14,106)(15,107)(16,108)(17,103)(18,104)(19,96)(20,91)(21,92)(22,93)(23,94)(24,95)(25,102)(26,97)(27,98)(28,99)(29,100)(30,101)(31,64)(32,65)(33,66)(34,61)(35,62)(36,63)(37,70)(38,71)(39,72)(40,67)(41,68)(42,69)(43,76)(44,77)(45,78)(46,73)(47,74)(48,75)(49,82)(50,83)(51,84)(52,79)(53,80)(54,81)(85,118)(86,119)(87,120)(88,115)(89,116)(90,117), (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), (1,36,49,37,43)(2,31,50,38,44)(3,32,51,39,45)(4,33,52,40,46)(5,34,53,41,47)(6,35,54,42,48)(7,26,17,119,19)(8,27,18,120,20)(9,28,13,115,21)(10,29,14,116,22)(11,30,15,117,23)(12,25,16,118,24)(55,65,84,72,78)(56,66,79,67,73)(57,61,80,68,74)(58,62,81,69,75)(59,63,82,70,76)(60,64,83,71,77)(85,95,114,102,108)(86,96,109,97,103)(87,91,110,98,104)(88,92,111,99,105)(89,93,112,100,106)(90,94,113,101,107), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,112)(8,113)(9,114)(10,109)(11,110)(12,111)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,100)(20,101)(21,102)(22,97)(23,98)(24,99)(25,92)(26,93)(27,94)(28,95)(29,96)(30,91)(31,68)(32,69)(33,70)(34,71)(35,72)(36,67)(37,66)(38,61)(39,62)(40,63)(41,64)(42,65)(43,56)(44,57)(45,58)(46,59)(47,60)(48,55)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(103,116)(104,117)(105,118)(106,119)(107,120)(108,115) );
G=PermutationGroup([[(1,86),(2,87),(3,88),(4,89),(5,90),(6,85),(7,82),(8,83),(9,84),(10,79),(11,80),(12,81),(13,78),(14,73),(15,74),(16,75),(17,76),(18,77),(19,63),(20,64),(21,65),(22,66),(23,61),(24,62),(25,69),(26,70),(27,71),(28,72),(29,67),(30,68),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,97),(38,98),(39,99),(40,100),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120)], [(1,59),(2,60),(3,55),(4,56),(5,57),(6,58),(7,109),(8,110),(9,111),(10,112),(11,113),(12,114),(13,105),(14,106),(15,107),(16,108),(17,103),(18,104),(19,96),(20,91),(21,92),(22,93),(23,94),(24,95),(25,102),(26,97),(27,98),(28,99),(29,100),(30,101),(31,64),(32,65),(33,66),(34,61),(35,62),(36,63),(37,70),(38,71),(39,72),(40,67),(41,68),(42,69),(43,76),(44,77),(45,78),(46,73),(47,74),(48,75),(49,82),(50,83),(51,84),(52,79),(53,80),(54,81),(85,118),(86,119),(87,120),(88,115),(89,116),(90,117)], [(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)], [(1,36,49,37,43),(2,31,50,38,44),(3,32,51,39,45),(4,33,52,40,46),(5,34,53,41,47),(6,35,54,42,48),(7,26,17,119,19),(8,27,18,120,20),(9,28,13,115,21),(10,29,14,116,22),(11,30,15,117,23),(12,25,16,118,24),(55,65,84,72,78),(56,66,79,67,73),(57,61,80,68,74),(58,62,81,69,75),(59,63,82,70,76),(60,64,83,71,77),(85,95,114,102,108),(86,96,109,97,103),(87,91,110,98,104),(88,92,111,99,105),(89,93,112,100,106),(90,94,113,101,107)], [(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,112),(8,113),(9,114),(10,109),(11,110),(12,111),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,100),(20,101),(21,102),(22,97),(23,98),(24,99),(25,92),(26,93),(27,94),(28,95),(29,96),(30,91),(31,68),(32,69),(33,70),(34,71),(35,72),(36,67),(37,66),(38,61),(39,62),(40,63),(41,64),(42,65),(43,56),(44,57),(45,58),(46,59),(47,60),(48,55),(49,79),(50,80),(51,81),(52,82),(53,83),(54,84),(103,116),(104,117),(105,118),(106,119),(107,120),(108,115)]])
D5×C22×C6 is a maximal subgroup of
(C2×C30)⋊D4 (C2×C6)⋊8D20
96 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 3A | 3B | 5A | 5B | 6A | ··· | 6N | 6O | ··· | 6AD | 10A | ··· | 10N | 15A | 15B | 15C | 15D | 30A | ··· | 30AB |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 3 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | ··· | 1 | 5 | ··· | 5 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D5 | D10 | C3×D5 | C6×D5 |
| kernel | D5×C22×C6 | D5×C2×C6 | C22×C30 | C23×D5 | C22×D5 | C22×C10 | C22×C6 | C2×C6 | C23 | C22 |
| # reps | 1 | 14 | 1 | 2 | 28 | 2 | 2 | 14 | 4 | 28 |
Matrix representation of D5×C22×C6 ►in GL4(𝔽31) generated by
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 30 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 26 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 25 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 18 | 1 |
| 0 | 0 | 30 | 0 |
| 30 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 30 |
| 0 | 0 | 30 | 0 |
G:=sub<GL(4,GF(31))| [30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[30,0,0,0,0,1,0,0,0,0,30,0,0,0,0,30],[26,0,0,0,0,6,0,0,0,0,25,0,0,0,0,25],[1,0,0,0,0,1,0,0,0,0,18,30,0,0,1,0],[30,0,0,0,0,1,0,0,0,0,0,30,0,0,30,0] >;
D5×C22×C6 in GAP, Magma, Sage, TeX
D_5\times C_2^2\times C_6
% in TeX
G:=Group("D5xC2^2xC6"); // GroupNames label
G:=SmallGroup(240,205);
// by ID
G=gap.SmallGroup(240,205);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,6917]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^6=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations