direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C15⋊7D4, C30⋊7D4, C23⋊2D15, C22⋊3D30, D30⋊7C22, C30.37C23, Dic15⋊4C22, (C2×C6)⋊8D10, C15⋊16(C2×D4), (C2×C10)⋊11D6, C6⋊3(C5⋊D4), (C22×C6)⋊2D5, C10⋊3(C3⋊D4), (C2×C30)⋊9C22, (C22×C10)⋊4S3, (C22×C30)⋊2C2, (C2×Dic15)⋊4C2, (C22×D15)⋊3C2, C6.37(C22×D5), C10.37(C22×S3), C2.10(C22×D15), C5⋊4(C2×C3⋊D4), C3⋊4(C2×C5⋊D4), SmallGroup(240,184)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C15⋊7D4
G = < a,b,c,d | a2=b15=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 520 in 108 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, C10, Dic3, D6, C2×C6, C2×C6, C2×C6, C15, C2×D4, Dic5, D10, C2×C10, C2×C10, C2×C10, C2×Dic3, C3⋊D4, C22×S3, C22×C6, D15, C30, C30, C30, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C2×C3⋊D4, Dic15, D30, D30, C2×C30, C2×C30, C2×C30, C2×C5⋊D4, C2×Dic15, C15⋊7D4, C22×D15, C22×C30, C2×C15⋊7D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, D15, C5⋊D4, C22×D5, C2×C3⋊D4, D30, C2×C5⋊D4, C15⋊7D4, C22×D15, C2×C15⋊7D4
►On 120 points
(1 74)(2 75)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 81)(17 82)(18 83)(19 84)(20 85)(21 86)(22 87)(23 88)(24 89)(25 90)(26 76)(27 77)(28 78)(29 79)(30 80)(31 100)(32 101)(33 102)(34 103)(35 104)(36 105)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 97)(44 98)(45 99)(46 107)(47 108)(48 109)(49 110)(50 111)(51 112)(52 113)(53 114)(54 115)(55 116)(56 117)(57 118)(58 119)(59 120)(60 106)
(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 40 29 59)(2 39 30 58)(3 38 16 57)(4 37 17 56)(5 36 18 55)(6 35 19 54)(7 34 20 53)(8 33 21 52)(9 32 22 51)(10 31 23 50)(11 45 24 49)(12 44 25 48)(13 43 26 47)(14 42 27 46)(15 41 28 60)(61 92 81 118)(62 91 82 117)(63 105 83 116)(64 104 84 115)(65 103 85 114)(66 102 86 113)(67 101 87 112)(68 100 88 111)(69 99 89 110)(70 98 90 109)(71 97 76 108)(72 96 77 107)(73 95 78 106)(74 94 79 120)(75 93 80 119)
(1 74)(2 73)(3 72)(4 71)(5 70)(6 69)(7 68)(8 67)(9 66)(10 65)(11 64)(12 63)(13 62)(14 61)(15 75)(16 77)(17 76)(18 90)(19 89)(20 88)(21 87)(22 86)(23 85)(24 84)(25 83)(26 82)(27 81)(28 80)(29 79)(30 78)(31 114)(32 113)(33 112)(34 111)(35 110)(36 109)(37 108)(38 107)(39 106)(40 120)(41 119)(42 118)(43 117)(44 116)(45 115)(46 92)(47 91)(48 105)(49 104)(50 103)(51 102)(52 101)(53 100)(54 99)(55 98)(56 97)(57 96)(58 95)(59 94)(60 93)
G:=sub<Sym(120)| (1,74)(2,75)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,81)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,89)(25,90)(26,76)(27,77)(28,78)(29,79)(30,80)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(46,107)(47,108)(48,109)(49,110)(50,111)(51,112)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,106), (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), (1,40,29,59)(2,39,30,58)(3,38,16,57)(4,37,17,56)(5,36,18,55)(6,35,19,54)(7,34,20,53)(8,33,21,52)(9,32,22,51)(10,31,23,50)(11,45,24,49)(12,44,25,48)(13,43,26,47)(14,42,27,46)(15,41,28,60)(61,92,81,118)(62,91,82,117)(63,105,83,116)(64,104,84,115)(65,103,85,114)(66,102,86,113)(67,101,87,112)(68,100,88,111)(69,99,89,110)(70,98,90,109)(71,97,76,108)(72,96,77,107)(73,95,78,106)(74,94,79,120)(75,93,80,119), (1,74)(2,73)(3,72)(4,71)(5,70)(6,69)(7,68)(8,67)(9,66)(10,65)(11,64)(12,63)(13,62)(14,61)(15,75)(16,77)(17,76)(18,90)(19,89)(20,88)(21,87)(22,86)(23,85)(24,84)(25,83)(26,82)(27,81)(28,80)(29,79)(30,78)(31,114)(32,113)(33,112)(34,111)(35,110)(36,109)(37,108)(38,107)(39,106)(40,120)(41,119)(42,118)(43,117)(44,116)(45,115)(46,92)(47,91)(48,105)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,96)(58,95)(59,94)(60,93)>;
G:=Group( (1,74)(2,75)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,81)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,89)(25,90)(26,76)(27,77)(28,78)(29,79)(30,80)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(46,107)(47,108)(48,109)(49,110)(50,111)(51,112)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,106), (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), (1,40,29,59)(2,39,30,58)(3,38,16,57)(4,37,17,56)(5,36,18,55)(6,35,19,54)(7,34,20,53)(8,33,21,52)(9,32,22,51)(10,31,23,50)(11,45,24,49)(12,44,25,48)(13,43,26,47)(14,42,27,46)(15,41,28,60)(61,92,81,118)(62,91,82,117)(63,105,83,116)(64,104,84,115)(65,103,85,114)(66,102,86,113)(67,101,87,112)(68,100,88,111)(69,99,89,110)(70,98,90,109)(71,97,76,108)(72,96,77,107)(73,95,78,106)(74,94,79,120)(75,93,80,119), (1,74)(2,73)(3,72)(4,71)(5,70)(6,69)(7,68)(8,67)(9,66)(10,65)(11,64)(12,63)(13,62)(14,61)(15,75)(16,77)(17,76)(18,90)(19,89)(20,88)(21,87)(22,86)(23,85)(24,84)(25,83)(26,82)(27,81)(28,80)(29,79)(30,78)(31,114)(32,113)(33,112)(34,111)(35,110)(36,109)(37,108)(38,107)(39,106)(40,120)(41,119)(42,118)(43,117)(44,116)(45,115)(46,92)(47,91)(48,105)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,96)(58,95)(59,94)(60,93) );
G=PermutationGroup([[(1,74),(2,75),(3,61),(4,62),(5,63),(6,64),(7,65),(8,66),(9,67),(10,68),(11,69),(12,70),(13,71),(14,72),(15,73),(16,81),(17,82),(18,83),(19,84),(20,85),(21,86),(22,87),(23,88),(24,89),(25,90),(26,76),(27,77),(28,78),(29,79),(30,80),(31,100),(32,101),(33,102),(34,103),(35,104),(36,105),(37,91),(38,92),(39,93),(40,94),(41,95),(42,96),(43,97),(44,98),(45,99),(46,107),(47,108),(48,109),(49,110),(50,111),(51,112),(52,113),(53,114),(54,115),(55,116),(56,117),(57,118),(58,119),(59,120),(60,106)], [(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)], [(1,40,29,59),(2,39,30,58),(3,38,16,57),(4,37,17,56),(5,36,18,55),(6,35,19,54),(7,34,20,53),(8,33,21,52),(9,32,22,51),(10,31,23,50),(11,45,24,49),(12,44,25,48),(13,43,26,47),(14,42,27,46),(15,41,28,60),(61,92,81,118),(62,91,82,117),(63,105,83,116),(64,104,84,115),(65,103,85,114),(66,102,86,113),(67,101,87,112),(68,100,88,111),(69,99,89,110),(70,98,90,109),(71,97,76,108),(72,96,77,107),(73,95,78,106),(74,94,79,120),(75,93,80,119)], [(1,74),(2,73),(3,72),(4,71),(5,70),(6,69),(7,68),(8,67),(9,66),(10,65),(11,64),(12,63),(13,62),(14,61),(15,75),(16,77),(17,76),(18,90),(19,89),(20,88),(21,87),(22,86),(23,85),(24,84),(25,83),(26,82),(27,81),(28,80),(29,79),(30,78),(31,114),(32,113),(33,112),(34,111),(35,110),(36,109),(37,108),(38,107),(39,106),(40,120),(41,119),(42,118),(43,117),(44,116),(45,115),(46,92),(47,91),(48,105),(49,104),(50,103),(51,102),(52,101),(53,100),(54,99),(55,98),(56,97),(57,96),(58,95),(59,94),(60,93)]])
C2×C15⋊7D4 is a maximal subgroup of
C15⋊9(C23⋊C4) C23.6D30 Dic15.19D4 D30⋊6D4 C10.(C2×D12) C6.D4⋊D5 Dic15⋊3D4 C15⋊26(C4×D4) C15⋊28(C4×D4) D30⋊7D4 Dic15⋊4D4 D30.16D4 (C2×C6)⋊8D20 (C2×C10)⋊4D12 Dic15⋊5D4 (C2×C6)⋊D20 (C2×C10)⋊11D12 D30⋊8D4 Dic15⋊19D4 D30⋊16D4 D30.28D4 D30⋊9D4 C23.11D30 C22.D60 C23.28D30 C60⋊29D4 D30⋊17D4 C60⋊2D4 Dic15⋊12D4 C60⋊3D4 C24⋊5D15 C2×D5×C3⋊D4 C2×S3×C5⋊D4 C15⋊2+ 1+4 C2×D4×D15 D4⋊6D30
C2×C15⋊7D4 is a maximal quotient of
C60.205D4 C23.28D30 C60⋊29D4 D4.D30 C23.22D30 C60.17D4 D30⋊17D4 C60⋊2D4 Dic15⋊12D4 C60⋊3D4 Q8.11D30 Dic15⋊4Q8 D30⋊7Q8 C60.23D4 D4⋊D30 D4.8D30 D4.9D30 C24⋊5D15
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 5A | 5B | 6A | ··· | 6G | 10A | ··· | 10N | 15A | 15B | 15C | 15D | 30A | ··· | 30AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 30 | 30 | 2 | 30 | 30 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D10 | C3⋊D4 | D15 | C5⋊D4 | D30 | C15⋊7D4 |
| kernel | C2×C15⋊7D4 | C2×Dic15 | C15⋊7D4 | C22×D15 | C22×C30 | C22×C10 | C30 | C22×C6 | C2×C10 | C2×C6 | C10 | C23 | C6 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 3 | 6 | 4 | 4 | 8 | 12 | 16 |
Matrix representation of C2×C15⋊7D4 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 18 | 18 | 0 | 0 |
| 43 | 60 | 0 | 0 |
| 0 | 0 | 47 | 31 |
| 0 | 0 | 30 | 25 |
| 1 | 0 | 0 | 0 |
| 43 | 60 | 0 | 0 |
| 0 | 0 | 37 | 8 |
| 0 | 0 | 27 | 24 |
| 60 | 0 | 0 | 0 |
| 18 | 1 | 0 | 0 |
| 0 | 0 | 25 | 33 |
| 0 | 0 | 31 | 36 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[18,43,0,0,18,60,0,0,0,0,47,30,0,0,31,25],[1,43,0,0,0,60,0,0,0,0,37,27,0,0,8,24],[60,18,0,0,0,1,0,0,0,0,25,31,0,0,33,36] >;
C2×C15⋊7D4 in GAP, Magma, Sage, TeX
C_2\times C_{15}\rtimes_7D_4 % in TeX
G:=Group("C2xC15:7D4"); // GroupNames label
G:=SmallGroup(240,184);
// by ID
G=gap.SmallGroup(240,184);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,218,964,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^15=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations