metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D60⋊9C2, Dic6⋊5D5, C20.18D6, D10.11D6, C12.28D10, C60.21C22, C30.10C23, Dic5.20D6, Dic3.4D10, D30.2C22, (C4×D5)⋊2S3, (D5×C12)⋊2C2, C15⋊7(C4○D4), C5⋊1(C4○D12), C3⋊D20⋊2C2, C4.14(S3×D5), D30.C2⋊2C2, (C5×Dic6)⋊3C2, C3⋊1(Q8⋊2D5), C6.10(C22×D5), C10.10(C22×S3), (C6×D5).12C22, (C5×Dic3).4C22, (C3×Dic5).14C22, C2.14(C2×S3×D5), SmallGroup(240,134)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.28D10
G = < a,b,c | a12=c2=1, b10=a6, bab-1=cac=a-1, cbc=a6b9 >
Subgroups: 384 in 80 conjugacy classes, 32 normal (22 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3 [×2], C6, C6, C2×C4 [×3], D4 [×3], Q8, D5 [×3], C10, Dic3 [×2], C12, C12, D6 [×2], C2×C6, C15, C4○D4, Dic5, C20, C20 [×2], D10, D10 [×2], Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12, C3×D5, D15 [×2], C30, C4×D5, C4×D5 [×2], D20 [×3], C5×Q8, C4○D12, C5×Dic3 [×2], C3×Dic5, C60, C6×D5, D30 [×2], Q8⋊2D5, D30.C2 [×2], C3⋊D20 [×2], D5×C12, C5×Dic6, D60, C12.28D10
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4, D10 [×3], C22×S3, C22×D5, C4○D12, S3×D5, Q8⋊2D5, C2×S3×D5, C12.28D10
►On 120 points
(1 59 97 38 80 119 11 49 87 28 70 109)(2 110 71 29 88 50 12 120 61 39 98 60)(3 41 99 40 62 101 13 51 89 30 72 111)(4 112 73 31 90 52 14 102 63 21 100 42)(5 43 81 22 64 103 15 53 91 32 74 113)(6 114 75 33 92 54 16 104 65 23 82 44)(7 45 83 24 66 105 17 55 93 34 76 115)(8 116 77 35 94 56 18 106 67 25 84 46)(9 47 85 26 68 107 19 57 95 36 78 117)(10 118 79 37 96 58 20 108 69 27 86 48)
(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 5)(2 4)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(21 29)(22 28)(23 27)(24 26)(30 40)(31 39)(32 38)(33 37)(34 36)(41 111)(42 110)(43 109)(44 108)(45 107)(46 106)(47 105)(48 104)(49 103)(50 102)(51 101)(52 120)(53 119)(54 118)(55 117)(56 116)(57 115)(58 114)(59 113)(60 112)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 100)(72 99)(73 98)(74 97)(75 96)(76 95)(77 94)(78 93)(79 92)(80 91)
G:=sub<Sym(120)| (1,59,97,38,80,119,11,49,87,28,70,109)(2,110,71,29,88,50,12,120,61,39,98,60)(3,41,99,40,62,101,13,51,89,30,72,111)(4,112,73,31,90,52,14,102,63,21,100,42)(5,43,81,22,64,103,15,53,91,32,74,113)(6,114,75,33,92,54,16,104,65,23,82,44)(7,45,83,24,66,105,17,55,93,34,76,115)(8,116,77,35,94,56,18,106,67,25,84,46)(9,47,85,26,68,107,19,57,95,36,78,117)(10,118,79,37,96,58,20,108,69,27,86,48), (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), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(21,29)(22,28)(23,27)(24,26)(30,40)(31,39)(32,38)(33,37)(34,36)(41,111)(42,110)(43,109)(44,108)(45,107)(46,106)(47,105)(48,104)(49,103)(50,102)(51,101)(52,120)(53,119)(54,118)(55,117)(56,116)(57,115)(58,114)(59,113)(60,112)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)>;
G:=Group( (1,59,97,38,80,119,11,49,87,28,70,109)(2,110,71,29,88,50,12,120,61,39,98,60)(3,41,99,40,62,101,13,51,89,30,72,111)(4,112,73,31,90,52,14,102,63,21,100,42)(5,43,81,22,64,103,15,53,91,32,74,113)(6,114,75,33,92,54,16,104,65,23,82,44)(7,45,83,24,66,105,17,55,93,34,76,115)(8,116,77,35,94,56,18,106,67,25,84,46)(9,47,85,26,68,107,19,57,95,36,78,117)(10,118,79,37,96,58,20,108,69,27,86,48), (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), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(21,29)(22,28)(23,27)(24,26)(30,40)(31,39)(32,38)(33,37)(34,36)(41,111)(42,110)(43,109)(44,108)(45,107)(46,106)(47,105)(48,104)(49,103)(50,102)(51,101)(52,120)(53,119)(54,118)(55,117)(56,116)(57,115)(58,114)(59,113)(60,112)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91) );
G=PermutationGroup([(1,59,97,38,80,119,11,49,87,28,70,109),(2,110,71,29,88,50,12,120,61,39,98,60),(3,41,99,40,62,101,13,51,89,30,72,111),(4,112,73,31,90,52,14,102,63,21,100,42),(5,43,81,22,64,103,15,53,91,32,74,113),(6,114,75,33,92,54,16,104,65,23,82,44),(7,45,83,24,66,105,17,55,93,34,76,115),(8,116,77,35,94,56,18,106,67,25,84,46),(9,47,85,26,68,107,19,57,95,36,78,117),(10,118,79,37,96,58,20,108,69,27,86,48)], [(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)], [(1,5),(2,4),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(21,29),(22,28),(23,27),(24,26),(30,40),(31,39),(32,38),(33,37),(34,36),(41,111),(42,110),(43,109),(44,108),(45,107),(46,106),(47,105),(48,104),(49,103),(50,102),(51,101),(52,120),(53,119),(54,118),(55,117),(56,116),(57,115),(58,114),(59,113),(60,112),(61,90),(62,89),(63,88),(64,87),(65,86),(66,85),(67,84),(68,83),(69,82),(70,81),(71,100),(72,99),(73,98),(74,97),(75,96),(76,95),(77,94),(78,93),(79,92),(80,91)])
C12.28D10 is a maximal subgroup of
D60⋊2C4 D60⋊5C4 C24⋊D10 C24.2D10 C40.31D6 D120⋊C2 D20.9D6 C60.16C23 C60.39C23 D20.D6 D60.C4 Dic6.F5 D20.38D6 D5×C4○D12 D20⋊29D6 D30.C23 D20⋊14D6 C30.33C24 S3×Q8⋊2D5
C12.28D10 is a maximal quotient of
Dic5×Dic6 C4⋊Dic3⋊D5 Dic3.Dic10 Dic3⋊C4⋊D5 D30.D4 (C4×D5)⋊Dic3 C60.67D4 (C2×C60).C22 C60.70D4 (C4×Dic5)⋊S3 C20.Dic6 D30.C2⋊C4 D60⋊17C4 D30⋊3Q8 D30.6D4 C15⋊20(C4×D4) C12⋊7D20 (C2×Dic6)⋊D5 D30⋊2D4
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 30A | 30B | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 10 | 30 | 30 | 2 | 2 | 5 | 5 | 6 | 6 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 10 | 10 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | C4○D12 | S3×D5 | Q8⋊2D5 | C2×S3×D5 | C12.28D10 |
| kernel | C12.28D10 | D30.C2 | C3⋊D20 | D5×C12 | C5×Dic6 | D60 | C4×D5 | Dic6 | Dic5 | C20 | D10 | C15 | Dic3 | C12 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 4 |
Matrix representation of C12.28D10 ►in GL6(𝔽61)
| 60 | 15 | 0 | 0 | 0 | 0 |
| 12 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 3 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 49 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 44 | 0 | 0 |
| 0 | 0 | 17 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 33 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 49 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 44 | 0 | 0 |
| 0 | 0 | 60 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 21 | 60 |
G:=sub<GL(6,GF(61))| [60,12,0,0,0,0,15,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,40,0,0,0,0,3,1],[1,49,0,0,0,0,0,60,0,0,0,0,0,0,44,17,0,0,0,0,44,60,0,0,0,0,0,0,50,0,0,0,0,0,33,11],[1,49,0,0,0,0,0,60,0,0,0,0,0,0,44,60,0,0,0,0,44,17,0,0,0,0,0,0,1,21,0,0,0,0,0,60] >;
C12.28D10 in GAP, Magma, Sage, TeX
C_{12}._{28}D_{10} % in TeX
G:=Group("C12.28D10"); // GroupNames label
G:=SmallGroup(240,134);
// by ID
G=gap.SmallGroup(240,134);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,121,55,116,50,490,6917]);
// Polycyclic
G:=Group<a,b,c|a^12=c^2=1,b^10=a^6,b*a*b^-1=c*a*c=a^-1,c*b*c=a^6*b^9>;
// generators/relations