direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×D4○D12, C30.95C24, C60.242C23, C15⋊142+ (1+4), (C5×D4)⋊30D6, (S3×D4)⋊5C10, D4⋊8(S3×C10), (C2×C20)⋊23D6, (C5×Q8)⋊29D6, Q8⋊8(S3×C10), C4○D12⋊8C10, (C2×D12)⋊13C10, D12⋊11(C2×C10), (C10×D12)⋊29C2, (C2×C60)⋊30C22, Q8⋊3S3⋊5C10, (S3×C20)⋊15C22, Dic6⋊12(C2×C10), C3⋊2(C5×2+ (1+4)), (D4×C15)⋊40C22, (C5×D12)⋊41C22, C6.12(C23×C10), C10.80(S3×C23), (Q8×C15)⋊35C22, D6.6(C22×C10), (S3×C10).42C23, C12.26(C22×C10), (C2×C30).260C23, C20.239(C22×S3), (C5×Dic6)⋊39C22, (C5×Dic3).44C23, Dic3.8(C22×C10), (C5×S3×D4)⋊12C2, (C2×C4)⋊4(S3×C10), C4○D4⋊5(C5×S3), C4.26(S3×C2×C10), (C4×S3)⋊2(C2×C10), (C2×C12)⋊5(C2×C10), (C5×C4○D4)⋊12S3, (C3×C4○D4)⋊4C10, (C3×D4)⋊9(C2×C10), C3⋊D4⋊5(C2×C10), (C3×Q8)⋊8(C2×C10), C22.4(S3×C2×C10), (C5×C4○D12)⋊18C2, (C15×C4○D4)⋊14C2, (S3×C2×C10)⋊16C22, C2.13(S3×C22×C10), (C22×S3)⋊4(C2×C10), (C5×Q8⋊3S3)⋊12C2, (C5×C3⋊D4)⋊21C22, (C2×C6).4(C22×C10), (C2×C10).23(C22×S3), SmallGroup(480,1161)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 788 in 332 conjugacy classes, 170 normal (24 characteristic)
C1, C2, C2 [×9], C3, C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], C5, S3 [×6], C6, C6 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×3], D4 [×15], Q8, Q8, C23 [×6], C10, C10 [×9], Dic3 [×2], C12, C12 [×3], D6 [×6], D6 [×6], C2×C6 [×3], C15, C2×D4 [×9], C4○D4, C4○D4 [×5], C20, C20 [×3], C20 [×2], C2×C10 [×3], C2×C10 [×12], Dic6, C4×S3 [×6], D12 [×9], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×6], C5×S3 [×6], C30, C30 [×3], 2+ (1+4), C2×C20 [×3], C2×C20 [×6], C5×D4 [×3], C5×D4 [×15], C5×Q8, C5×Q8, C22×C10 [×6], C2×D12 [×3], C4○D12 [×3], S3×D4 [×6], Q8⋊3S3 [×2], C3×C4○D4, C5×Dic3 [×2], C60, C60 [×3], S3×C10 [×6], S3×C10 [×6], C2×C30 [×3], D4×C10 [×9], C5×C4○D4, C5×C4○D4 [×5], D4○D12, C5×Dic6, S3×C20 [×6], C5×D12 [×9], C5×C3⋊D4 [×6], C2×C60 [×3], D4×C15 [×3], Q8×C15, S3×C2×C10 [×6], C5×2+ (1+4), C10×D12 [×3], C5×C4○D12 [×3], C5×S3×D4 [×6], C5×Q8⋊3S3 [×2], C15×C4○D4, C5×D4○D12
Quotients:
C1, C2 [×15], C22 [×35], C5, S3, C23 [×15], C10 [×15], D6 [×7], C24, C2×C10 [×35], C22×S3 [×7], C5×S3, 2+ (1+4), C22×C10 [×15], S3×C23, S3×C10 [×7], C23×C10, D4○D12, S3×C2×C10 [×7], C5×2+ (1+4), S3×C22×C10, C5×D4○D12
Generators and relations
G = < a,b,c,d,e | a5=b4=c2=e2=1, d6=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d5 >
►On 120 points
(1 14 103 36 39)(2 15 104 25 40)(3 16 105 26 41)(4 17 106 27 42)(5 18 107 28 43)(6 19 108 29 44)(7 20 97 30 45)(8 21 98 31 46)(9 22 99 32 47)(10 23 100 33 48)(11 24 101 34 37)(12 13 102 35 38)(49 69 76 109 92)(50 70 77 110 93)(51 71 78 111 94)(52 72 79 112 95)(53 61 80 113 96)(54 62 81 114 85)(55 63 82 115 86)(56 64 83 116 87)(57 65 84 117 88)(58 66 73 118 89)(59 67 74 119 90)(60 68 75 120 91)
(1 56 7 50)(2 57 8 51)(3 58 9 52)(4 59 10 53)(5 60 11 54)(6 49 12 55)(13 63 19 69)(14 64 20 70)(15 65 21 71)(16 66 22 72)(17 67 23 61)(18 68 24 62)(25 117 31 111)(26 118 32 112)(27 119 33 113)(28 120 34 114)(29 109 35 115)(30 110 36 116)(37 85 43 91)(38 86 44 92)(39 87 45 93)(40 88 46 94)(41 89 47 95)(42 90 48 96)(73 99 79 105)(74 100 80 106)(75 101 81 107)(76 102 82 108)(77 103 83 97)(78 104 84 98)
(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 14)(15 24)(16 23)(17 22)(18 21)(19 20)(25 34)(26 33)(27 32)(28 31)(29 30)(35 36)(37 40)(38 39)(41 48)(42 47)(43 46)(44 45)(49 50)(51 60)(52 59)(53 58)(54 57)(55 56)(61 66)(62 65)(63 64)(67 72)(68 71)(69 70)(73 80)(74 79)(75 78)(76 77)(81 84)(82 83)(85 88)(86 87)(89 96)(90 95)(91 94)(92 93)(97 108)(98 107)(99 106)(100 105)(101 104)(102 103)(109 110)(111 120)(112 119)(113 118)(114 117)(115 116)
G:=sub<Sym(120)| (1,14,103,36,39)(2,15,104,25,40)(3,16,105,26,41)(4,17,106,27,42)(5,18,107,28,43)(6,19,108,29,44)(7,20,97,30,45)(8,21,98,31,46)(9,22,99,32,47)(10,23,100,33,48)(11,24,101,34,37)(12,13,102,35,38)(49,69,76,109,92)(50,70,77,110,93)(51,71,78,111,94)(52,72,79,112,95)(53,61,80,113,96)(54,62,81,114,85)(55,63,82,115,86)(56,64,83,116,87)(57,65,84,117,88)(58,66,73,118,89)(59,67,74,119,90)(60,68,75,120,91), (1,56,7,50)(2,57,8,51)(3,58,9,52)(4,59,10,53)(5,60,11,54)(6,49,12,55)(13,63,19,69)(14,64,20,70)(15,65,21,71)(16,66,22,72)(17,67,23,61)(18,68,24,62)(25,117,31,111)(26,118,32,112)(27,119,33,113)(28,120,34,114)(29,109,35,115)(30,110,36,116)(37,85,43,91)(38,86,44,92)(39,87,45,93)(40,88,46,94)(41,89,47,95)(42,90,48,96)(73,99,79,105)(74,100,80,106)(75,101,81,107)(76,102,82,108)(77,103,83,97)(78,104,84,98), (49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(109,115)(110,116)(111,117)(112,118)(113,119)(114,120), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,14)(15,24)(16,23)(17,22)(18,21)(19,20)(25,34)(26,33)(27,32)(28,31)(29,30)(35,36)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)(49,50)(51,60)(52,59)(53,58)(54,57)(55,56)(61,66)(62,65)(63,64)(67,72)(68,71)(69,70)(73,80)(74,79)(75,78)(76,77)(81,84)(82,83)(85,88)(86,87)(89,96)(90,95)(91,94)(92,93)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)(109,110)(111,120)(112,119)(113,118)(114,117)(115,116)>;
G:=Group( (1,14,103,36,39)(2,15,104,25,40)(3,16,105,26,41)(4,17,106,27,42)(5,18,107,28,43)(6,19,108,29,44)(7,20,97,30,45)(8,21,98,31,46)(9,22,99,32,47)(10,23,100,33,48)(11,24,101,34,37)(12,13,102,35,38)(49,69,76,109,92)(50,70,77,110,93)(51,71,78,111,94)(52,72,79,112,95)(53,61,80,113,96)(54,62,81,114,85)(55,63,82,115,86)(56,64,83,116,87)(57,65,84,117,88)(58,66,73,118,89)(59,67,74,119,90)(60,68,75,120,91), (1,56,7,50)(2,57,8,51)(3,58,9,52)(4,59,10,53)(5,60,11,54)(6,49,12,55)(13,63,19,69)(14,64,20,70)(15,65,21,71)(16,66,22,72)(17,67,23,61)(18,68,24,62)(25,117,31,111)(26,118,32,112)(27,119,33,113)(28,120,34,114)(29,109,35,115)(30,110,36,116)(37,85,43,91)(38,86,44,92)(39,87,45,93)(40,88,46,94)(41,89,47,95)(42,90,48,96)(73,99,79,105)(74,100,80,106)(75,101,81,107)(76,102,82,108)(77,103,83,97)(78,104,84,98), (49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(109,115)(110,116)(111,117)(112,118)(113,119)(114,120), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,14)(15,24)(16,23)(17,22)(18,21)(19,20)(25,34)(26,33)(27,32)(28,31)(29,30)(35,36)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)(49,50)(51,60)(52,59)(53,58)(54,57)(55,56)(61,66)(62,65)(63,64)(67,72)(68,71)(69,70)(73,80)(74,79)(75,78)(76,77)(81,84)(82,83)(85,88)(86,87)(89,96)(90,95)(91,94)(92,93)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)(109,110)(111,120)(112,119)(113,118)(114,117)(115,116) );
G=PermutationGroup([(1,14,103,36,39),(2,15,104,25,40),(3,16,105,26,41),(4,17,106,27,42),(5,18,107,28,43),(6,19,108,29,44),(7,20,97,30,45),(8,21,98,31,46),(9,22,99,32,47),(10,23,100,33,48),(11,24,101,34,37),(12,13,102,35,38),(49,69,76,109,92),(50,70,77,110,93),(51,71,78,111,94),(52,72,79,112,95),(53,61,80,113,96),(54,62,81,114,85),(55,63,82,115,86),(56,64,83,116,87),(57,65,84,117,88),(58,66,73,118,89),(59,67,74,119,90),(60,68,75,120,91)], [(1,56,7,50),(2,57,8,51),(3,58,9,52),(4,59,10,53),(5,60,11,54),(6,49,12,55),(13,63,19,69),(14,64,20,70),(15,65,21,71),(16,66,22,72),(17,67,23,61),(18,68,24,62),(25,117,31,111),(26,118,32,112),(27,119,33,113),(28,120,34,114),(29,109,35,115),(30,110,36,116),(37,85,43,91),(38,86,44,92),(39,87,45,93),(40,88,46,94),(41,89,47,95),(42,90,48,96),(73,99,79,105),(74,100,80,106),(75,101,81,107),(76,102,82,108),(77,103,83,97),(78,104,84,98)], [(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96),(109,115),(110,116),(111,117),(112,118),(113,119),(114,120)], [(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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14),(15,24),(16,23),(17,22),(18,21),(19,20),(25,34),(26,33),(27,32),(28,31),(29,30),(35,36),(37,40),(38,39),(41,48),(42,47),(43,46),(44,45),(49,50),(51,60),(52,59),(53,58),(54,57),(55,56),(61,66),(62,65),(63,64),(67,72),(68,71),(69,70),(73,80),(74,79),(75,78),(76,77),(81,84),(82,83),(85,88),(86,87),(89,96),(90,95),(91,94),(92,93),(97,108),(98,107),(99,106),(100,105),(101,104),(102,103),(109,110),(111,120),(112,119),(113,118),(114,117),(115,116)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 34 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 |
| 0 | 0 | 34 | 0 |
| 0 | 0 | 0 | 34 |
| 1 | 0 | 2 | 0 |
| 0 | 1 | 0 | 2 |
| 60 | 0 | 60 | 0 |
| 0 | 60 | 0 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 60 | 0 | 60 | 0 |
| 0 | 60 | 0 | 60 |
| 46 | 23 | 0 | 0 |
| 38 | 23 | 0 | 0 |
| 0 | 0 | 46 | 23 |
| 0 | 0 | 38 | 23 |
| 46 | 23 | 0 | 0 |
| 38 | 15 | 0 | 0 |
| 0 | 0 | 46 | 23 |
| 0 | 0 | 38 | 15 |
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,34,0,0,0,0,34],[1,0,60,0,0,1,0,60,2,0,60,0,0,2,0,60],[1,0,60,0,0,1,0,60,0,0,60,0,0,0,0,60],[46,38,0,0,23,23,0,0,0,0,46,38,0,0,23,23],[46,38,0,0,23,15,0,0,0,0,46,38,0,0,23,15] >;
135 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | ··· | 2J | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 10A | 10B | 10C | 10D | 10E | ··· | 10P | 10Q | ··· | 10AN | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 20Q | ··· | 20X | 30A | 30B | 30C | 30D | 30E | ··· | 30P | 60A | ··· | 60H | 60I | ··· | 60T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | S3 | D6 | D6 | D6 | C5×S3 | S3×C10 | S3×C10 | S3×C10 | 2+ (1+4) | D4○D12 | C5×2+ (1+4) | C5×D4○D12 |
| kernel | C5×D4○D12 | C10×D12 | C5×C4○D12 | C5×S3×D4 | C5×Q8⋊3S3 | C15×C4○D4 | D4○D12 | C2×D12 | C4○D12 | S3×D4 | Q8⋊3S3 | C3×C4○D4 | C5×C4○D4 | C2×C20 | C5×D4 | C5×Q8 | C4○D4 | C2×C4 | D4 | Q8 | C15 | C5 | C3 | C1 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 4 | 12 | 12 | 24 | 8 | 4 | 1 | 3 | 3 | 1 | 4 | 12 | 12 | 4 | 1 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_5\times D_4\circ D_{12} % in TeX
G:=Group("C5xD4oD12"); // GroupNames label
G:=SmallGroup(480,1161);
// by ID
G=gap.SmallGroup(480,1161);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,891,2467,304,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^2=e^2=1,d^6=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d^5>;
// generators/relations