metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊2F5, Dic30⋊5C4, Dic5.20D12, C15⋊2C4≀C2, (C4×F5)⋊1S3, C4.3(S3×F5), (C5×D12)⋊5C4, C20.6(C4×S3), (C12×F5)⋊1C2, C60.10(C2×C4), C3⋊2(D4⋊F5), (C6×D5).22D4, (C4×D5).24D6, C2.9(D6⋊F5), C12.24(C2×F5), C12.F5⋊1C2, C5⋊2(C42⋊4S3), C10.6(D6⋊C4), C6.6(C22⋊F5), D12⋊5D5.5C2, D10.1(C3⋊D4), C30.6(C22⋊C4), (C3×Dic5).25D4, (D5×C12).40C22, SmallGroup(480,232)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊2F5
G = < a,b,c,d | a12=b2=c5=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a9b, dcd-1=c3 >
Subgroups: 500 in 88 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C42, M4(2), C4○D4, Dic5, Dic5, C20, F5, D10, C2×C10, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C5×S3, C3×D5, C30, C4≀C2, C5⋊C8, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C2×F5, C4.Dic3, C4×C12, C4○D12, C3×Dic5, Dic15, C60, C3×F5, C6×D5, S3×C10, C4.F5, C4×F5, D4⋊2D5, C42⋊4S3, C15⋊C8, S3×Dic5, C15⋊D4, D5×C12, C5×D12, Dic30, C6×F5, D4⋊F5, C12×F5, C12.F5, D12⋊5D5, D12⋊2F5
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, F5, C4×S3, D12, C3⋊D4, C4≀C2, C2×F5, D6⋊C4, C22⋊F5, C42⋊4S3, S3×F5, D4⋊F5, D6⋊F5, D12⋊2F5
►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)
(1 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 107)(14 106)(15 105)(16 104)(17 103)(18 102)(19 101)(20 100)(21 99)(22 98)(23 97)(24 108)(37 90)(38 89)(39 88)(40 87)(41 86)(42 85)(43 96)(44 95)(45 94)(46 93)(47 92)(48 91)(49 71)(50 70)(51 69)(52 68)(53 67)(54 66)(55 65)(56 64)(57 63)(58 62)(59 61)(60 72)(73 111)(74 110)(75 109)(76 120)(77 119)(78 118)(79 117)(80 116)(81 115)(82 114)(83 113)(84 112)
(1 81 108 96 65)(2 82 97 85 66)(3 83 98 86 67)(4 84 99 87 68)(5 73 100 88 69)(6 74 101 89 70)(7 75 102 90 71)(8 76 103 91 72)(9 77 104 92 61)(10 78 105 93 62)(11 79 106 94 63)(12 80 107 95 64)(13 44 56 31 116)(14 45 57 32 117)(15 46 58 33 118)(16 47 59 34 119)(17 48 60 35 120)(18 37 49 36 109)(19 38 50 25 110)(20 39 51 26 111)(21 40 52 27 112)(22 41 53 28 113)(23 42 54 29 114)(24 43 55 30 115)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 59 38 113)(14 60 39 114)(15 49 40 115)(16 50 41 116)(17 51 42 117)(18 52 43 118)(19 53 44 119)(20 54 45 120)(21 55 46 109)(22 56 47 110)(23 57 48 111)(24 58 37 112)(25 28 31 34)(26 29 32 35)(27 30 33 36)(61 86 77 98)(62 87 78 99)(63 88 79 100)(64 89 80 101)(65 90 81 102)(66 91 82 103)(67 92 83 104)(68 93 84 105)(69 94 73 106)(70 95 74 107)(71 96 75 108)(72 85 76 97)
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,107)(14,106)(15,105)(16,104)(17,103)(18,102)(19,101)(20,100)(21,99)(22,98)(23,97)(24,108)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(60,72)(73,111)(74,110)(75,109)(76,120)(77,119)(78,118)(79,117)(80,116)(81,115)(82,114)(83,113)(84,112), (1,81,108,96,65)(2,82,97,85,66)(3,83,98,86,67)(4,84,99,87,68)(5,73,100,88,69)(6,74,101,89,70)(7,75,102,90,71)(8,76,103,91,72)(9,77,104,92,61)(10,78,105,93,62)(11,79,106,94,63)(12,80,107,95,64)(13,44,56,31,116)(14,45,57,32,117)(15,46,58,33,118)(16,47,59,34,119)(17,48,60,35,120)(18,37,49,36,109)(19,38,50,25,110)(20,39,51,26,111)(21,40,52,27,112)(22,41,53,28,113)(23,42,54,29,114)(24,43,55,30,115), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,59,38,113)(14,60,39,114)(15,49,40,115)(16,50,41,116)(17,51,42,117)(18,52,43,118)(19,53,44,119)(20,54,45,120)(21,55,46,109)(22,56,47,110)(23,57,48,111)(24,58,37,112)(25,28,31,34)(26,29,32,35)(27,30,33,36)(61,86,77,98)(62,87,78,99)(63,88,79,100)(64,89,80,101)(65,90,81,102)(66,91,82,103)(67,92,83,104)(68,93,84,105)(69,94,73,106)(70,95,74,107)(71,96,75,108)(72,85,76,97)>;
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,107)(14,106)(15,105)(16,104)(17,103)(18,102)(19,101)(20,100)(21,99)(22,98)(23,97)(24,108)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(60,72)(73,111)(74,110)(75,109)(76,120)(77,119)(78,118)(79,117)(80,116)(81,115)(82,114)(83,113)(84,112), (1,81,108,96,65)(2,82,97,85,66)(3,83,98,86,67)(4,84,99,87,68)(5,73,100,88,69)(6,74,101,89,70)(7,75,102,90,71)(8,76,103,91,72)(9,77,104,92,61)(10,78,105,93,62)(11,79,106,94,63)(12,80,107,95,64)(13,44,56,31,116)(14,45,57,32,117)(15,46,58,33,118)(16,47,59,34,119)(17,48,60,35,120)(18,37,49,36,109)(19,38,50,25,110)(20,39,51,26,111)(21,40,52,27,112)(22,41,53,28,113)(23,42,54,29,114)(24,43,55,30,115), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,59,38,113)(14,60,39,114)(15,49,40,115)(16,50,41,116)(17,51,42,117)(18,52,43,118)(19,53,44,119)(20,54,45,120)(21,55,46,109)(22,56,47,110)(23,57,48,111)(24,58,37,112)(25,28,31,34)(26,29,32,35)(27,30,33,36)(61,86,77,98)(62,87,78,99)(63,88,79,100)(64,89,80,101)(65,90,81,102)(66,91,82,103)(67,92,83,104)(68,93,84,105)(69,94,73,106)(70,95,74,107)(71,96,75,108)(72,85,76,97) );
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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,36),(8,35),(9,34),(10,33),(11,32),(12,31),(13,107),(14,106),(15,105),(16,104),(17,103),(18,102),(19,101),(20,100),(21,99),(22,98),(23,97),(24,108),(37,90),(38,89),(39,88),(40,87),(41,86),(42,85),(43,96),(44,95),(45,94),(46,93),(47,92),(48,91),(49,71),(50,70),(51,69),(52,68),(53,67),(54,66),(55,65),(56,64),(57,63),(58,62),(59,61),(60,72),(73,111),(74,110),(75,109),(76,120),(77,119),(78,118),(79,117),(80,116),(81,115),(82,114),(83,113),(84,112)], [(1,81,108,96,65),(2,82,97,85,66),(3,83,98,86,67),(4,84,99,87,68),(5,73,100,88,69),(6,74,101,89,70),(7,75,102,90,71),(8,76,103,91,72),(9,77,104,92,61),(10,78,105,93,62),(11,79,106,94,63),(12,80,107,95,64),(13,44,56,31,116),(14,45,57,32,117),(15,46,58,33,118),(16,47,59,34,119),(17,48,60,35,120),(18,37,49,36,109),(19,38,50,25,110),(20,39,51,26,111),(21,40,52,27,112),(22,41,53,28,113),(23,42,54,29,114),(24,43,55,30,115)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,59,38,113),(14,60,39,114),(15,49,40,115),(16,50,41,116),(17,51,42,117),(18,52,43,118),(19,53,44,119),(20,54,45,120),(21,55,46,109),(22,56,47,110),(23,57,48,111),(24,58,37,112),(25,28,31,34),(26,29,32,35),(27,30,33,36),(61,86,77,98),(62,87,78,99),(63,88,79,100),(64,89,80,101),(65,90,81,102),(66,91,82,103),(67,92,83,104),(68,93,84,105),(69,94,73,106),(70,95,74,107),(71,96,75,108),(72,85,76,97)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 12A | 12B | 12C | ··· | 12L | 15 | 20 | 30 | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | 12 | ··· | 12 | 15 | 20 | 30 | 60 | 60 |
| size | 1 | 1 | 10 | 12 | 2 | 2 | 5 | 5 | 10 | 10 | 10 | 10 | 60 | 4 | 2 | 10 | 10 | 60 | 60 | 4 | 24 | 24 | 2 | 2 | 10 | ··· | 10 | 8 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D6 | D12 | C4×S3 | C3⋊D4 | C4≀C2 | C42⋊4S3 | F5 | C2×F5 | C22⋊F5 | S3×F5 | D4⋊F5 | D6⋊F5 | D12⋊2F5 |
| kernel | D12⋊2F5 | C12×F5 | C12.F5 | D12⋊5D5 | C5×D12 | Dic30 | C4×F5 | C3×Dic5 | C6×D5 | C4×D5 | Dic5 | C20 | D10 | C15 | C5 | D12 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 2 | 1 | 1 | 1 | 2 |
Matrix representation of D12⋊2F5 ►in GL8(𝔽241)
| 64 | 177 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
| 177 | 64 | 0 | 0 | 0 | 0 | 0 | 0 |
| 113 | 64 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 124 | 234 | 0 | 7 |
| 0 | 0 | 0 | 0 | 0 | 117 | 234 | 7 |
| 0 | 0 | 0 | 0 | 7 | 234 | 117 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 | 234 | 124 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 240 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 240 |
| 240 | 153 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(8,GF(241))| [64,0,0,0,0,0,0,0,177,177,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[177,113,0,0,0,0,0,0,64,64,0,0,0,0,0,0,0,0,240,240,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,124,0,7,7,0,0,0,0,234,117,234,0,0,0,0,0,0,234,117,234,0,0,0,0,7,7,0,124],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,240,240,240,240],[240,0,0,0,0,0,0,0,153,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;
D12⋊2F5 in GAP, Magma, Sage, TeX
D_{12}\rtimes_2F_5 % in TeX
G:=Group("D12:2F5"); // GroupNames label
G:=SmallGroup(480,232);
// by ID
G=gap.SmallGroup(480,232);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,219,346,80,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^5=d^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^9*b,d*c*d^-1=c^3>;
// generators/relations