metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.4D6, D6.3D10, Dic5.6D6, C30.17C23, Dic3.5D10, Dic15.14C22, C15⋊Q8⋊5C2, C5⋊D4⋊1S3, C3⋊D4⋊1D5, C15⋊9(C4○D4), C15⋊D4⋊4C2, (C2×C6).2D10, (C2×C10).1D6, C3⋊4(D4⋊2D5), C5⋊4(D4⋊2S3), (D5×Dic3)⋊4C2, (S3×Dic5)⋊3C2, C22.3(S3×D5), (C2×Dic15)⋊8C2, (C6×D5).4C22, C6.17(C22×D5), (S3×C10).3C22, (C2×C30).11C22, C10.17(C22×S3), (C3×Dic5).6C22, (C5×Dic3).5C22, C2.19(C2×S3×D5), (C3×C5⋊D4)⋊2C2, (C5×C3⋊D4)⋊1C2, SmallGroup(240,141)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C30.C23
G = < a,b,c,d | a30=b2=d2=1, c2=a15, bab=a19, cac-1=a11, ad=da, bc=cb, dbd=a15b, dcd=a15c >
Subgroups: 312 in 80 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, Q8, D5, C10, C10, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C15, C4○D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3⋊D4, C3×D4, C5×S3, C3×D5, C30, C30, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5⋊D4, C5×D4, D4⋊2S3, C5×Dic3, C3×Dic5, Dic15, C6×D5, S3×C10, C2×C30, D4⋊2D5, D5×Dic3, S3×Dic5, C15⋊D4, C15⋊Q8, C3×C5⋊D4, C5×C3⋊D4, C2×Dic15, C30.C23
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, C22×D5, D4⋊2S3, S3×D5, D4⋊2D5, C2×S3×D5, C30.C23
►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)
(2 20)(3 9)(4 28)(5 17)(7 25)(8 14)(10 22)(12 30)(13 19)(15 27)(18 24)(23 29)(31 58)(32 47)(33 36)(34 55)(35 44)(37 52)(38 41)(39 60)(40 49)(42 57)(43 46)(45 54)(48 51)(50 59)(53 56)(61 79)(62 68)(63 87)(64 76)(66 84)(67 73)(69 81)(71 89)(72 78)(74 86)(77 83)(82 88)(91 118)(92 107)(93 96)(94 115)(95 104)(97 112)(98 101)(99 120)(100 109)(102 117)(103 106)(105 114)(108 111)(110 119)(113 116)
(1 90 16 75)(2 71 17 86)(3 82 18 67)(4 63 19 78)(5 74 20 89)(6 85 21 70)(7 66 22 81)(8 77 23 62)(9 88 24 73)(10 69 25 84)(11 80 26 65)(12 61 27 76)(13 72 28 87)(14 83 29 68)(15 64 30 79)(31 106 46 91)(32 117 47 102)(33 98 48 113)(34 109 49 94)(35 120 50 105)(36 101 51 116)(37 112 52 97)(38 93 53 108)(39 104 54 119)(40 115 55 100)(41 96 56 111)(42 107 57 92)(43 118 58 103)(44 99 59 114)(45 110 60 95)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 50)(30 51)(61 113)(62 114)(63 115)(64 116)(65 117)(66 118)(67 119)(68 120)(69 91)(70 92)(71 93)(72 94)(73 95)(74 96)(75 97)(76 98)(77 99)(78 100)(79 101)(80 102)(81 103)(82 104)(83 105)(84 106)(85 107)(86 108)(87 109)(88 110)(89 111)(90 112)
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), (2,20)(3,9)(4,28)(5,17)(7,25)(8,14)(10,22)(12,30)(13,19)(15,27)(18,24)(23,29)(31,58)(32,47)(33,36)(34,55)(35,44)(37,52)(38,41)(39,60)(40,49)(42,57)(43,46)(45,54)(48,51)(50,59)(53,56)(61,79)(62,68)(63,87)(64,76)(66,84)(67,73)(69,81)(71,89)(72,78)(74,86)(77,83)(82,88)(91,118)(92,107)(93,96)(94,115)(95,104)(97,112)(98,101)(99,120)(100,109)(102,117)(103,106)(105,114)(108,111)(110,119)(113,116), (1,90,16,75)(2,71,17,86)(3,82,18,67)(4,63,19,78)(5,74,20,89)(6,85,21,70)(7,66,22,81)(8,77,23,62)(9,88,24,73)(10,69,25,84)(11,80,26,65)(12,61,27,76)(13,72,28,87)(14,83,29,68)(15,64,30,79)(31,106,46,91)(32,117,47,102)(33,98,48,113)(34,109,49,94)(35,120,50,105)(36,101,51,116)(37,112,52,97)(38,93,53,108)(39,104,54,119)(40,115,55,100)(41,96,56,111)(42,107,57,92)(43,118,58,103)(44,99,59,114)(45,110,60,95), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(61,113)(62,114)(63,115)(64,116)(65,117)(66,118)(67,119)(68,120)(69,91)(70,92)(71,93)(72,94)(73,95)(74,96)(75,97)(76,98)(77,99)(78,100)(79,101)(80,102)(81,103)(82,104)(83,105)(84,106)(85,107)(86,108)(87,109)(88,110)(89,111)(90,112)>;
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), (2,20)(3,9)(4,28)(5,17)(7,25)(8,14)(10,22)(12,30)(13,19)(15,27)(18,24)(23,29)(31,58)(32,47)(33,36)(34,55)(35,44)(37,52)(38,41)(39,60)(40,49)(42,57)(43,46)(45,54)(48,51)(50,59)(53,56)(61,79)(62,68)(63,87)(64,76)(66,84)(67,73)(69,81)(71,89)(72,78)(74,86)(77,83)(82,88)(91,118)(92,107)(93,96)(94,115)(95,104)(97,112)(98,101)(99,120)(100,109)(102,117)(103,106)(105,114)(108,111)(110,119)(113,116), (1,90,16,75)(2,71,17,86)(3,82,18,67)(4,63,19,78)(5,74,20,89)(6,85,21,70)(7,66,22,81)(8,77,23,62)(9,88,24,73)(10,69,25,84)(11,80,26,65)(12,61,27,76)(13,72,28,87)(14,83,29,68)(15,64,30,79)(31,106,46,91)(32,117,47,102)(33,98,48,113)(34,109,49,94)(35,120,50,105)(36,101,51,116)(37,112,52,97)(38,93,53,108)(39,104,54,119)(40,115,55,100)(41,96,56,111)(42,107,57,92)(43,118,58,103)(44,99,59,114)(45,110,60,95), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(61,113)(62,114)(63,115)(64,116)(65,117)(66,118)(67,119)(68,120)(69,91)(70,92)(71,93)(72,94)(73,95)(74,96)(75,97)(76,98)(77,99)(78,100)(79,101)(80,102)(81,103)(82,104)(83,105)(84,106)(85,107)(86,108)(87,109)(88,110)(89,111)(90,112) );
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)], [(2,20),(3,9),(4,28),(5,17),(7,25),(8,14),(10,22),(12,30),(13,19),(15,27),(18,24),(23,29),(31,58),(32,47),(33,36),(34,55),(35,44),(37,52),(38,41),(39,60),(40,49),(42,57),(43,46),(45,54),(48,51),(50,59),(53,56),(61,79),(62,68),(63,87),(64,76),(66,84),(67,73),(69,81),(71,89),(72,78),(74,86),(77,83),(82,88),(91,118),(92,107),(93,96),(94,115),(95,104),(97,112),(98,101),(99,120),(100,109),(102,117),(103,106),(105,114),(108,111),(110,119),(113,116)], [(1,90,16,75),(2,71,17,86),(3,82,18,67),(4,63,19,78),(5,74,20,89),(6,85,21,70),(7,66,22,81),(8,77,23,62),(9,88,24,73),(10,69,25,84),(11,80,26,65),(12,61,27,76),(13,72,28,87),(14,83,29,68),(15,64,30,79),(31,106,46,91),(32,117,47,102),(33,98,48,113),(34,109,49,94),(35,120,50,105),(36,101,51,116),(37,112,52,97),(38,93,53,108),(39,104,54,119),(40,115,55,100),(41,96,56,111),(42,107,57,92),(43,118,58,103),(44,99,59,114),(45,110,60,95)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(29,50),(30,51),(61,113),(62,114),(63,115),(64,116),(65,117),(66,118),(67,119),(68,120),(69,91),(70,92),(71,93),(72,94),(73,95),(74,96),(75,97),(76,98),(77,99),(78,100),(79,101),(80,102),(81,103),(82,104),(83,105),(84,106),(85,107),(86,108),(87,109),(88,110),(89,111),(90,112)]])
C30.C23 is a maximal subgroup of
D20.39D6 D20⋊24D6 C15⋊2- 1+4 D5×D4⋊2S3 S3×D4⋊2D5 D20⋊13D6 C15⋊2+ 1+4
C30.C23 is a maximal quotient of
Dic15⋊5Q8 Dic5.1Dic6 C60⋊5C4⋊C2 Dic3.Dic10 (C2×C60).C22 (C4×Dic15)⋊C2 D6⋊Dic5.C2 Dic15.4Q8 D10.19(C4×S3) (S3×Dic5)⋊C4 D10.17D12 D6⋊2Dic10 D10⋊2Dic6 Dic15⋊9D4 D6.9D20 Dic15.31D4 C23.D5⋊S3 C23.13(S3×D5) C23.14(S3×D5) C23.48(S3×D5) C6.(D4×D5) C6.(C2×D20) (C2×C10).D12 C23.17(S3×D5) (C6×D5)⋊D4 (S3×C10).D4 Dic15⋊16D4 Dic15⋊17D4 (S3×C10)⋊D4 Dic15⋊18D4 Dic15.48D4
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 10C | 10D | 10E | 10F | 12 | 15A | 15B | 20A | 20B | 30A | ··· | 30F |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 2 | 6 | 10 | 2 | 6 | 10 | 15 | 15 | 30 | 2 | 2 | 2 | 4 | 20 | 2 | 2 | 4 | 4 | 12 | 12 | 20 | 4 | 4 | 12 | 12 | 4 | ··· | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | D4⋊2S3 | S3×D5 | D4⋊2D5 | C2×S3×D5 | C30.C23 |
| kernel | C30.C23 | D5×Dic3 | S3×Dic5 | C15⋊D4 | C15⋊Q8 | C3×C5⋊D4 | C5×C3⋊D4 | C2×Dic15 | C5⋊D4 | C3⋊D4 | Dic5 | D10 | C2×C10 | C15 | Dic3 | D6 | C2×C6 | C5 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C30.C23 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 1 | 0 | 0 |
| 0 | 0 | 16 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 18 | 0 | 0 |
| 0 | 0 | 45 | 44 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 49 | 31 |
| 0 | 0 | 0 | 0 | 19 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,44,16,0,0,0,0,1,60,0,0,0,0,0,0,0,1,0,0,0,0,60,60],[1,0,0,0,0,0,0,60,0,0,0,0,0,0,17,45,0,0,0,0,18,44,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,0,0,0,0,0,0,50,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,49,19,0,0,0,0,31,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;
C30.C23 in GAP, Magma, Sage, TeX
C_{30}.C_2^3 % in TeX
G:=Group("C30.C2^3"); // GroupNames label
G:=SmallGroup(240,141);
// by ID
G=gap.SmallGroup(240,141);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,55,218,116,490,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=d^2=1,c^2=a^15,b*a*b=a^19,c*a*c^-1=a^11,a*d=d*a,b*c=c*b,d*b*d=a^15*b,d*c*d=a^15*c>;
// generators/relations