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, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, D4, Q8, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, C4○D4, Dic5, C20, C20, D10, D10, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D5, D15, C30, C4×D5, C4×D5, D20, C5×Q8, C4○D12, C5×Dic3, C3×Dic5, C60, C6×D5, D30, Q8⋊2D5, D30.C2, C3⋊D20, D5×C12, C5×Dic6, D60, C12.28D10
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, C22×D5, C4○D12, S3×D5, Q8⋊2D5, C2×S3×D5, C12.28D10
►On 120 points
(1 28 99 105 44 63 11 38 89 115 54 73)(2 74 55 116 90 39 12 64 45 106 100 29)(3 30 81 107 46 65 13 40 91 117 56 75)(4 76 57 118 92 21 14 66 47 108 82 31)(5 32 83 109 48 67 15 22 93 119 58 77)(6 78 59 120 94 23 16 68 49 110 84 33)(7 34 85 111 50 69 17 24 95 101 60 79)(8 80 41 102 96 25 18 70 51 112 86 35)(9 36 87 113 52 71 19 26 97 103 42 61)(10 62 43 104 98 27 20 72 53 114 88 37)
(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 64)(22 63)(23 62)(24 61)(25 80)(26 79)(27 78)(28 77)(29 76)(30 75)(31 74)(32 73)(33 72)(34 71)(35 70)(36 69)(37 68)(38 67)(39 66)(40 65)(41 96)(42 95)(43 94)(44 93)(45 92)(46 91)(47 90)(48 89)(49 88)(50 87)(51 86)(52 85)(53 84)(54 83)(55 82)(56 81)(57 100)(58 99)(59 98)(60 97)(101 103)(104 120)(105 119)(106 118)(107 117)(108 116)(109 115)(110 114)(111 113)
G:=sub<Sym(120)| (1,28,99,105,44,63,11,38,89,115,54,73)(2,74,55,116,90,39,12,64,45,106,100,29)(3,30,81,107,46,65,13,40,91,117,56,75)(4,76,57,118,92,21,14,66,47,108,82,31)(5,32,83,109,48,67,15,22,93,119,58,77)(6,78,59,120,94,23,16,68,49,110,84,33)(7,34,85,111,50,69,17,24,95,101,60,79)(8,80,41,102,96,25,18,70,51,112,86,35)(9,36,87,113,52,71,19,26,97,103,42,61)(10,62,43,104,98,27,20,72,53,114,88,37), (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)(21,64)(22,63)(23,62)(24,61)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,100)(58,99)(59,98)(60,97)(101,103)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)>;
G:=Group( (1,28,99,105,44,63,11,38,89,115,54,73)(2,74,55,116,90,39,12,64,45,106,100,29)(3,30,81,107,46,65,13,40,91,117,56,75)(4,76,57,118,92,21,14,66,47,108,82,31)(5,32,83,109,48,67,15,22,93,119,58,77)(6,78,59,120,94,23,16,68,49,110,84,33)(7,34,85,111,50,69,17,24,95,101,60,79)(8,80,41,102,96,25,18,70,51,112,86,35)(9,36,87,113,52,71,19,26,97,103,42,61)(10,62,43,104,98,27,20,72,53,114,88,37), (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)(21,64)(22,63)(23,62)(24,61)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,100)(58,99)(59,98)(60,97)(101,103)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113) );
G=PermutationGroup([[(1,28,99,105,44,63,11,38,89,115,54,73),(2,74,55,116,90,39,12,64,45,106,100,29),(3,30,81,107,46,65,13,40,91,117,56,75),(4,76,57,118,92,21,14,66,47,108,82,31),(5,32,83,109,48,67,15,22,93,119,58,77),(6,78,59,120,94,23,16,68,49,110,84,33),(7,34,85,111,50,69,17,24,95,101,60,79),(8,80,41,102,96,25,18,70,51,112,86,35),(9,36,87,113,52,71,19,26,97,103,42,61),(10,62,43,104,98,27,20,72,53,114,88,37)], [(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),(21,64),(22,63),(23,62),(24,61),(25,80),(26,79),(27,78),(28,77),(29,76),(30,75),(31,74),(32,73),(33,72),(34,71),(35,70),(36,69),(37,68),(38,67),(39,66),(40,65),(41,96),(42,95),(43,94),(44,93),(45,92),(46,91),(47,90),(48,89),(49,88),(50,87),(51,86),(52,85),(53,84),(54,83),(55,82),(56,81),(57,100),(58,99),(59,98),(60,97),(101,103),(104,120),(105,119),(106,118),(107,117),(108,116),(109,115),(110,114),(111,113)]])
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