metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.6D30, C22.2D60, (C2×C6).3D20, C22⋊C4⋊1D15, (C2×C10).3D12, C15⋊10(C23⋊C4), (C2×Dic15)⋊1C4, (C2×C30).134D4, (C22×D15)⋊1C4, C22.3(C4×D15), C10.28(D6⋊C4), C30.38D4⋊1C2, (C22×C10).30D6, (C22×C6).15D10, C5⋊4(C23.6D6), C2.4(D30⋊3C4), C30.70(C22⋊C4), C3⋊2(C23.1D10), C22.8(C15⋊7D4), (C22×C30).6C22, C6.13(D10⋊C4), (C2×C6).4(C4×D5), (C5×C22⋊C4)⋊2S3, (C3×C22⋊C4)⋊2D5, (C15×C22⋊C4)⋊3C2, (C2×C10).27(C4×S3), (C2×C30).64(C2×C4), (C2×C15⋊7D4).1C2, (C2×C6).66(C5⋊D4), (C2×C10).66(C3⋊D4), SmallGroup(480,166)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.6D30
G = < a,b,c,d,e | a2=b2=c2=1, d30=a, e2=abc, ab=ba, eae-1=ac=ca, ad=da, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=bcd29 >
Subgroups: 740 in 104 conjugacy classes, 33 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C22⋊C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, D15, C30, C30, C23⋊C4, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C6.D4, C3×C22⋊C4, C2×C3⋊D4, Dic15, C60, D30, C2×C30, C2×C30, C23.D5, C5×C22⋊C4, C2×C5⋊D4, C23.6D6, C2×Dic15, C2×Dic15, C15⋊7D4, C2×C60, C22×D15, C22×C30, C23.1D10, C30.38D4, C15×C22⋊C4, C2×C15⋊7D4, C23.6D30
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, D15, C23⋊C4, C4×D5, D20, C5⋊D4, D6⋊C4, D30, D10⋊C4, C23.6D6, C4×D15, D60, C15⋊7D4, C23.1D10, D30⋊3C4, C23.6D30
►On 120 points
Generators in S120
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)
(1 117)(2 32)(3 119)(4 34)(5 61)(6 36)(7 63)(8 38)(9 65)(10 40)(11 67)(12 42)(13 69)(14 44)(15 71)(16 46)(17 73)(18 48)(19 75)(20 50)(21 77)(22 52)(23 79)(24 54)(25 81)(26 56)(27 83)(28 58)(29 85)(30 60)(31 87)(33 89)(35 91)(37 93)(39 95)(41 97)(43 99)(45 101)(47 103)(49 105)(51 107)(53 109)(55 111)(57 113)(59 115)(62 92)(64 94)(66 96)(68 98)(70 100)(72 102)(74 104)(76 106)(78 108)(80 110)(82 112)(84 114)(86 116)(88 118)(90 120)
(1 87)(2 88)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 98)(13 99)(14 100)(15 101)(16 102)(17 103)(18 104)(19 105)(20 106)(21 107)(22 108)(23 109)(24 110)(25 111)(26 112)(27 113)(28 114)(29 115)(30 116)(31 117)(32 118)(33 119)(34 120)(35 61)(36 62)(37 63)(38 64)(39 65)(40 66)(41 67)(42 68)(43 69)(44 70)(45 71)(46 72)(47 73)(48 74)(49 75)(50 76)(51 77)(52 78)(53 79)(54 80)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)
(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 45)(2 70 88 44)(3 69)(4 42 90 68)(5 41)(6 66 92 40)(7 65)(8 38 94 64)(9 37)(10 62 96 36)(11 61)(12 34 98 120)(13 33)(14 118 100 32)(15 117)(16 30 102 116)(17 29)(18 114 104 28)(19 113)(20 26 106 112)(21 25)(22 110 108 24)(23 109)(27 105)(31 101)(35 97)(39 93)(43 89)(46 86 72 60)(47 85)(48 58 74 84)(49 57)(50 82 76 56)(51 81)(52 54 78 80)(55 77)(59 73)(63 95)(67 91)(71 87)(75 83)(99 119)(103 115)(107 111)
G:=sub<Sym(120)| (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120), (1,117)(2,32)(3,119)(4,34)(5,61)(6,36)(7,63)(8,38)(9,65)(10,40)(11,67)(12,42)(13,69)(14,44)(15,71)(16,46)(17,73)(18,48)(19,75)(20,50)(21,77)(22,52)(23,79)(24,54)(25,81)(26,56)(27,83)(28,58)(29,85)(30,60)(31,87)(33,89)(35,91)(37,93)(39,95)(41,97)(43,99)(45,101)(47,103)(49,105)(51,107)(53,109)(55,111)(57,113)(59,115)(62,92)(64,94)(66,96)(68,98)(70,100)(72,102)(74,104)(76,106)(78,108)(80,110)(82,112)(84,114)(86,116)(88,118)(90,120), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,113)(28,114)(29,115)(30,116)(31,117)(32,118)(33,119)(34,120)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66)(41,67)(42,68)(43,69)(44,70)(45,71)(46,72)(47,73)(48,74)(49,75)(50,76)(51,77)(52,78)(53,79)(54,80)(55,81)(56,82)(57,83)(58,84)(59,85)(60,86), (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), (1,45)(2,70,88,44)(3,69)(4,42,90,68)(5,41)(6,66,92,40)(7,65)(8,38,94,64)(9,37)(10,62,96,36)(11,61)(12,34,98,120)(13,33)(14,118,100,32)(15,117)(16,30,102,116)(17,29)(18,114,104,28)(19,113)(20,26,106,112)(21,25)(22,110,108,24)(23,109)(27,105)(31,101)(35,97)(39,93)(43,89)(46,86,72,60)(47,85)(48,58,74,84)(49,57)(50,82,76,56)(51,81)(52,54,78,80)(55,77)(59,73)(63,95)(67,91)(71,87)(75,83)(99,119)(103,115)(107,111)>;
G:=Group( (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120), (1,117)(2,32)(3,119)(4,34)(5,61)(6,36)(7,63)(8,38)(9,65)(10,40)(11,67)(12,42)(13,69)(14,44)(15,71)(16,46)(17,73)(18,48)(19,75)(20,50)(21,77)(22,52)(23,79)(24,54)(25,81)(26,56)(27,83)(28,58)(29,85)(30,60)(31,87)(33,89)(35,91)(37,93)(39,95)(41,97)(43,99)(45,101)(47,103)(49,105)(51,107)(53,109)(55,111)(57,113)(59,115)(62,92)(64,94)(66,96)(68,98)(70,100)(72,102)(74,104)(76,106)(78,108)(80,110)(82,112)(84,114)(86,116)(88,118)(90,120), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,113)(28,114)(29,115)(30,116)(31,117)(32,118)(33,119)(34,120)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66)(41,67)(42,68)(43,69)(44,70)(45,71)(46,72)(47,73)(48,74)(49,75)(50,76)(51,77)(52,78)(53,79)(54,80)(55,81)(56,82)(57,83)(58,84)(59,85)(60,86), (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), (1,45)(2,70,88,44)(3,69)(4,42,90,68)(5,41)(6,66,92,40)(7,65)(8,38,94,64)(9,37)(10,62,96,36)(11,61)(12,34,98,120)(13,33)(14,118,100,32)(15,117)(16,30,102,116)(17,29)(18,114,104,28)(19,113)(20,26,106,112)(21,25)(22,110,108,24)(23,109)(27,105)(31,101)(35,97)(39,93)(43,89)(46,86,72,60)(47,85)(48,58,74,84)(49,57)(50,82,76,56)(51,81)(52,54,78,80)(55,77)(59,73)(63,95)(67,91)(71,87)(75,83)(99,119)(103,115)(107,111) );
G=PermutationGroup([[(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,97),(68,98),(69,99),(70,100),(71,101),(72,102),(73,103),(74,104),(75,105),(76,106),(77,107),(78,108),(79,109),(80,110),(81,111),(82,112),(83,113),(84,114),(85,115),(86,116),(87,117),(88,118),(89,119),(90,120)], [(1,117),(2,32),(3,119),(4,34),(5,61),(6,36),(7,63),(8,38),(9,65),(10,40),(11,67),(12,42),(13,69),(14,44),(15,71),(16,46),(17,73),(18,48),(19,75),(20,50),(21,77),(22,52),(23,79),(24,54),(25,81),(26,56),(27,83),(28,58),(29,85),(30,60),(31,87),(33,89),(35,91),(37,93),(39,95),(41,97),(43,99),(45,101),(47,103),(49,105),(51,107),(53,109),(55,111),(57,113),(59,115),(62,92),(64,94),(66,96),(68,98),(70,100),(72,102),(74,104),(76,106),(78,108),(80,110),(82,112),(84,114),(86,116),(88,118),(90,120)], [(1,87),(2,88),(3,89),(4,90),(5,91),(6,92),(7,93),(8,94),(9,95),(10,96),(11,97),(12,98),(13,99),(14,100),(15,101),(16,102),(17,103),(18,104),(19,105),(20,106),(21,107),(22,108),(23,109),(24,110),(25,111),(26,112),(27,113),(28,114),(29,115),(30,116),(31,117),(32,118),(33,119),(34,120),(35,61),(36,62),(37,63),(38,64),(39,65),(40,66),(41,67),(42,68),(43,69),(44,70),(45,71),(46,72),(47,73),(48,74),(49,75),(50,76),(51,77),(52,78),(53,79),(54,80),(55,81),(56,82),(57,83),(58,84),(59,85),(60,86)], [(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)], [(1,45),(2,70,88,44),(3,69),(4,42,90,68),(5,41),(6,66,92,40),(7,65),(8,38,94,64),(9,37),(10,62,96,36),(11,61),(12,34,98,120),(13,33),(14,118,100,32),(15,117),(16,30,102,116),(17,29),(18,114,104,28),(19,113),(20,26,106,112),(21,25),(22,110,108,24),(23,109),(27,105),(31,101),(35,97),(39,93),(43,89),(46,86,72,60),(47,85),(48,58,74,84),(49,57),(50,82,76,56),(51,81),(52,54,78,80),(55,77),(59,73),(63,95),(67,91),(71,87),(75,83),(99,119),(103,115),(107,111)]])
81 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 30M | ··· | 30T | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 2 | 2 | 60 | 2 | 4 | 4 | 60 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
81 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D5 | D6 | D10 | C4×S3 | D12 | C3⋊D4 | D15 | C4×D5 | D20 | C5⋊D4 | D30 | C4×D15 | D60 | C15⋊7D4 | C23⋊C4 | C23.6D6 | C23.1D10 | C23.6D30 |
| kernel | C23.6D30 | C30.38D4 | C15×C22⋊C4 | C2×C15⋊7D4 | C2×Dic15 | C22×D15 | C5×C22⋊C4 | C2×C30 | C3×C22⋊C4 | C22×C10 | C22×C6 | C2×C10 | C2×C10 | C2×C10 | C22⋊C4 | C2×C6 | C2×C6 | C2×C6 | C23 | C22 | C22 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C23.6D30 ►in GL4(𝔽61) generated by
| 47 | 44 | 0 | 0 |
| 33 | 14 | 0 | 0 |
| 44 | 0 | 30 | 44 |
| 45 | 17 | 17 | 31 |
| 14 | 17 | 0 | 0 |
| 28 | 47 | 0 | 0 |
| 16 | 44 | 30 | 44 |
| 17 | 0 | 17 | 31 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 21 | 52 | 57 | 7 |
| 38 | 36 | 50 | 60 |
| 19 | 52 | 56 | 9 |
| 24 | 50 | 56 | 9 |
| 28 | 56 | 0 | 0 |
| 59 | 33 | 0 | 0 |
| 35 | 15 | 58 | 2 |
| 28 | 18 | 56 | 3 |
G:=sub<GL(4,GF(61))| [47,33,44,45,44,14,0,17,0,0,30,17,0,0,44,31],[14,28,16,17,17,47,44,0,0,0,30,17,0,0,44,31],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[21,38,19,24,52,36,52,50,57,50,56,56,7,60,9,9],[28,59,35,28,56,33,15,18,0,0,58,56,0,0,2,3] >;
C23.6D30 in GAP, Magma, Sage, TeX
C_2^3._6D_{30} % in TeX
G:=Group("C2^3.6D30"); // GroupNames label
G:=SmallGroup(480,166);
// by ID
G=gap.SmallGroup(480,166);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,422,346,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^30=a,e^2=a*b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^29>;
// generators/relations