metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C30.38D4, C23.2D15, C22.7D30, C22⋊2Dic15, (C2×C30)⋊6C4, (C2×C6)⋊2Dic5, C30.53(C2×C4), (C2×C10)⋊7Dic3, (C2×C6).25D10, (C2×C10).25D6, C3⋊2(C23.D5), (C22×C6).2D5, C15⋊10(C22⋊C4), (C2×Dic15)⋊2C2, C6.20(C5⋊D4), C5⋊3(C6.D4), C2.3(C15⋊7D4), (C22×C10).4S3, (C22×C30).2C2, C6.10(C2×Dic5), C2.5(C2×Dic15), C10.20(C3⋊D4), (C2×C30).26C22, C10.17(C2×Dic3), SmallGroup(240,80)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C30.38D4
G = < a,b,c | a30=b4=1, c2=a15, bab-1=cac-1=a-1, cbc-1=a15b-1 >
Subgroups: 232 in 68 conjugacy classes, 35 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C23, C10, C10, C10, Dic3, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, Dic5, C2×C10, C2×C10, C2×C10, C2×Dic3, C22×C6, C30, C30, C30, C2×Dic5, C22×C10, C6.D4, Dic15, C2×C30, C2×C30, C2×C30, C23.D5, C2×Dic15, C22×C30, C30.38D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, Dic3, D6, C22⋊C4, Dic5, D10, C2×Dic3, C3⋊D4, D15, C2×Dic5, C5⋊D4, C6.D4, Dic15, D30, C23.D5, C2×Dic15, C15⋊7D4, C30.38D4
►On 120 points
(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 35 86 105)(2 34 87 104)(3 33 88 103)(4 32 89 102)(5 31 90 101)(6 60 61 100)(7 59 62 99)(8 58 63 98)(9 57 64 97)(10 56 65 96)(11 55 66 95)(12 54 67 94)(13 53 68 93)(14 52 69 92)(15 51 70 91)(16 50 71 120)(17 49 72 119)(18 48 73 118)(19 47 74 117)(20 46 75 116)(21 45 76 115)(22 44 77 114)(23 43 78 113)(24 42 79 112)(25 41 80 111)(26 40 81 110)(27 39 82 109)(28 38 83 108)(29 37 84 107)(30 36 85 106)
(1 120 16 105)(2 119 17 104)(3 118 18 103)(4 117 19 102)(5 116 20 101)(6 115 21 100)(7 114 22 99)(8 113 23 98)(9 112 24 97)(10 111 25 96)(11 110 26 95)(12 109 27 94)(13 108 28 93)(14 107 29 92)(15 106 30 91)(31 90 46 75)(32 89 47 74)(33 88 48 73)(34 87 49 72)(35 86 50 71)(36 85 51 70)(37 84 52 69)(38 83 53 68)(39 82 54 67)(40 81 55 66)(41 80 56 65)(42 79 57 64)(43 78 58 63)(44 77 59 62)(45 76 60 61)
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), (1,35,86,105)(2,34,87,104)(3,33,88,103)(4,32,89,102)(5,31,90,101)(6,60,61,100)(7,59,62,99)(8,58,63,98)(9,57,64,97)(10,56,65,96)(11,55,66,95)(12,54,67,94)(13,53,68,93)(14,52,69,92)(15,51,70,91)(16,50,71,120)(17,49,72,119)(18,48,73,118)(19,47,74,117)(20,46,75,116)(21,45,76,115)(22,44,77,114)(23,43,78,113)(24,42,79,112)(25,41,80,111)(26,40,81,110)(27,39,82,109)(28,38,83,108)(29,37,84,107)(30,36,85,106), (1,120,16,105)(2,119,17,104)(3,118,18,103)(4,117,19,102)(5,116,20,101)(6,115,21,100)(7,114,22,99)(8,113,23,98)(9,112,24,97)(10,111,25,96)(11,110,26,95)(12,109,27,94)(13,108,28,93)(14,107,29,92)(15,106,30,91)(31,90,46,75)(32,89,47,74)(33,88,48,73)(34,87,49,72)(35,86,50,71)(36,85,51,70)(37,84,52,69)(38,83,53,68)(39,82,54,67)(40,81,55,66)(41,80,56,65)(42,79,57,64)(43,78,58,63)(44,77,59,62)(45,76,60,61)>;
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), (1,35,86,105)(2,34,87,104)(3,33,88,103)(4,32,89,102)(5,31,90,101)(6,60,61,100)(7,59,62,99)(8,58,63,98)(9,57,64,97)(10,56,65,96)(11,55,66,95)(12,54,67,94)(13,53,68,93)(14,52,69,92)(15,51,70,91)(16,50,71,120)(17,49,72,119)(18,48,73,118)(19,47,74,117)(20,46,75,116)(21,45,76,115)(22,44,77,114)(23,43,78,113)(24,42,79,112)(25,41,80,111)(26,40,81,110)(27,39,82,109)(28,38,83,108)(29,37,84,107)(30,36,85,106), (1,120,16,105)(2,119,17,104)(3,118,18,103)(4,117,19,102)(5,116,20,101)(6,115,21,100)(7,114,22,99)(8,113,23,98)(9,112,24,97)(10,111,25,96)(11,110,26,95)(12,109,27,94)(13,108,28,93)(14,107,29,92)(15,106,30,91)(31,90,46,75)(32,89,47,74)(33,88,48,73)(34,87,49,72)(35,86,50,71)(36,85,51,70)(37,84,52,69)(38,83,53,68)(39,82,54,67)(40,81,55,66)(41,80,56,65)(42,79,57,64)(43,78,58,63)(44,77,59,62)(45,76,60,61) );
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)], [(1,35,86,105),(2,34,87,104),(3,33,88,103),(4,32,89,102),(5,31,90,101),(6,60,61,100),(7,59,62,99),(8,58,63,98),(9,57,64,97),(10,56,65,96),(11,55,66,95),(12,54,67,94),(13,53,68,93),(14,52,69,92),(15,51,70,91),(16,50,71,120),(17,49,72,119),(18,48,73,118),(19,47,74,117),(20,46,75,116),(21,45,76,115),(22,44,77,114),(23,43,78,113),(24,42,79,112),(25,41,80,111),(26,40,81,110),(27,39,82,109),(28,38,83,108),(29,37,84,107),(30,36,85,106)], [(1,120,16,105),(2,119,17,104),(3,118,18,103),(4,117,19,102),(5,116,20,101),(6,115,21,100),(7,114,22,99),(8,113,23,98),(9,112,24,97),(10,111,25,96),(11,110,26,95),(12,109,27,94),(13,108,28,93),(14,107,29,92),(15,106,30,91),(31,90,46,75),(32,89,47,74),(33,88,48,73),(34,87,49,72),(35,86,50,71),(36,85,51,70),(37,84,52,69),(38,83,53,68),(39,82,54,67),(40,81,55,66),(41,80,56,65),(42,79,57,64),(43,78,58,63),(44,77,59,62),(45,76,60,61)]])
C30.38D4 is a maximal subgroup of
(C2×C6).D20 C15⋊8(C23⋊C4) C23.6D30 C23.7D30 C23.D5⋊S3 (C6×Dic5)⋊7C4 C23.26(S3×D5) C23.13(S3×D5) C23.14(S3×D5) C6.(D4×D5) (C2×C30).D4 C30.(C2×D4) D5×C6.D4 C23.17(S3×D5) (C6×D5)⋊D4 Dic5×C3⋊D4 Dic3×C5⋊D4 S3×C23.D5 (S3×C10).D4 (C2×C30)⋊D4 (S3×C10)⋊D4 C15⋊C22≀C2 (C2×C30)⋊Q8 (C2×C10)⋊8Dic6 C23.15D30 C22⋊2Dic30 C23.8D30 C22⋊C4×D15 D30.28D4 C23.11D30 C60.205D4 C23.26D30 C4×C15⋊7D4 C23.28D30 D4×Dic15 C23.22D30 C60.17D4 D30⋊17D4 C60⋊2D4 Dic15⋊12D4 C24⋊5D15
C30.38D4 is a maximal quotient of
C60.212D4 C30.29C42 D4⋊Dic15 C60.8D4 C23.7D30 Q8⋊2Dic15 C60.10D4 Q8⋊3Dic15
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | ··· | 6G | 10A | ··· | 10N | 15A | 15B | 15C | 15D | 30A | ··· | 30AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | + | + | - | + | ||||
| image | C1 | C2 | C2 | C4 | S3 | D4 | D5 | Dic3 | D6 | Dic5 | D10 | C3⋊D4 | D15 | C5⋊D4 | Dic15 | D30 | C15⋊7D4 |
| kernel | C30.38D4 | C2×Dic15 | C22×C30 | C2×C30 | C22×C10 | C30 | C22×C6 | C2×C10 | C2×C10 | C2×C6 | C2×C6 | C10 | C23 | C6 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 4 | 8 | 8 | 4 | 16 |
Matrix representation of C30.38D4 ►in GL3(𝔽61) generated by
| 60 | 0 | 0 |
| 0 | 49 | 0 |
| 0 | 0 | 5 |
| 11 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 50 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 60 | 0 |
G:=sub<GL(3,GF(61))| [60,0,0,0,49,0,0,0,5],[11,0,0,0,0,1,0,1,0],[50,0,0,0,0,60,0,1,0] >;
C30.38D4 in GAP, Magma, Sage, TeX
C_{30}._{38}D_4 % in TeX
G:=Group("C30.38D4"); // GroupNames label
G:=SmallGroup(240,80);
// by ID
G=gap.SmallGroup(240,80);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,964,6917]);
// Polycyclic
G:=Group<a,b,c|a^30=b^4=1,c^2=a^15,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^15*b^-1>;
// generators/relations