metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊C8⋊5D6, D5⋊C8⋊4S3, D5⋊(C8⋊S3), (C3×D5)⋊M4(2), D6.F5⋊7C2, D6.8(C2×F5), (C4×S3).4F5, C4.32(S3×F5), (S3×C20).6C4, C20.32(C4×S3), C60.32(C2×C4), (C4×D15).6C4, (C4×D5).74D6, C15⋊3(C2×M4(2)), C12.39(C2×F5), C60.C4⋊5C2, C15⋊C8⋊6C22, D10.23(C4×S3), D30.10(C2×C4), C3⋊1(D5⋊M4(2)), Dic3.F5⋊7C2, Dic3.9(C2×F5), (D5×Dic3).7C4, C6.12(C22×F5), C30.12(C22×C4), Dic15.12(C2×C4), (D5×C12).66C22, D30.C2.15C22, (C3×Dic5).30C23, (S3×Dic5).15C22, Dic5.32(C22×S3), C5⋊1(C2×C8⋊S3), (C2×S3×D5).7C4, (C3×D5⋊C8)⋊5C2, C2.16(C2×S3×F5), (C3×C5⋊C8)⋊6C22, C10.12(S3×C2×C4), (C4×S3×D5).11C2, (C6×D5).21(C2×C4), (S3×C10).10(C2×C4), (C5×Dic3).12(C2×C4), SmallGroup(480,993)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊C8⋊D6
G = < a,b,c,d | a5=b8=c6=d2=1, bab-1=a3, cac-1=dad=a-1, bc=cb, dbd=b5, dcd=c-1 >
Subgroups: 676 in 136 conjugacy classes, 50 normal (36 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, C23, D5, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8, M4(2), C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C3⋊C8, C24, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C2×M4(2), C5⋊C8, C5⋊C8, C4×D5, C4×D5, C2×Dic5, C2×C20, C22×D5, C8⋊S3, C2×C3⋊C8, C2×C24, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, D5⋊C8, D5⋊C8, C4.F5, C22.F5, C2×C4×D5, C2×C8⋊S3, C3×C5⋊C8, C15⋊C8, D5×Dic3, S3×Dic5, D30.C2, D5×C12, S3×C20, C4×D15, C2×S3×D5, D5⋊M4(2), D6.F5, Dic3.F5, C3×D5⋊C8, C60.C4, C4×S3×D5, C5⋊C8⋊D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, M4(2), C22×C4, F5, C4×S3, C22×S3, C2×M4(2), C2×F5, C8⋊S3, S3×C2×C4, C22×F5, C2×C8⋊S3, S3×F5, D5⋊M4(2), C2×S3×F5, C5⋊C8⋊D6
►On 120 points
Generators in S120
(1 45 115 38 88)(2 39 46 81 116)(3 82 40 117 47)(4 118 83 48 33)(5 41 119 34 84)(6 35 42 85 120)(7 86 36 113 43)(8 114 87 44 37)(9 58 22 72 53)(10 65 59 54 23)(11 55 66 24 60)(12 17 56 61 67)(13 62 18 68 49)(14 69 63 50 19)(15 51 70 20 64)(16 21 52 57 71)(25 100 106 74 95)(26 75 101 96 107)(27 89 76 108 102)(28 109 90 103 77)(29 104 110 78 91)(30 79 97 92 111)(31 93 80 112 98)(32 105 94 99 73)
(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 58 99 5 62 103)(2 59 100 6 63 104)(3 60 101 7 64 97)(4 61 102 8 57 98)(9 73 84 18 90 45)(10 74 85 19 91 46)(11 75 86 20 92 47)(12 76 87 21 93 48)(13 77 88 22 94 41)(14 78 81 23 95 42)(15 79 82 24 96 43)(16 80 83 17 89 44)(25 35 69 110 116 54)(26 36 70 111 117 55)(27 37 71 112 118 56)(28 38 72 105 119 49)(29 39 65 106 120 50)(30 40 66 107 113 51)(31 33 67 108 114 52)(32 34 68 109 115 53)
(1 103)(2 100)(3 97)(4 102)(5 99)(6 104)(7 101)(8 98)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)(25 39)(26 36)(27 33)(28 38)(29 35)(30 40)(31 37)(32 34)(41 94)(42 91)(43 96)(44 93)(45 90)(46 95)(47 92)(48 89)(49 72)(50 69)(51 66)(52 71)(53 68)(54 65)(55 70)(56 67)(58 62)(60 64)(73 84)(74 81)(75 86)(76 83)(77 88)(78 85)(79 82)(80 87)(105 119)(106 116)(107 113)(108 118)(109 115)(110 120)(111 117)(112 114)
G:=sub<Sym(120)| (1,45,115,38,88)(2,39,46,81,116)(3,82,40,117,47)(4,118,83,48,33)(5,41,119,34,84)(6,35,42,85,120)(7,86,36,113,43)(8,114,87,44,37)(9,58,22,72,53)(10,65,59,54,23)(11,55,66,24,60)(12,17,56,61,67)(13,62,18,68,49)(14,69,63,50,19)(15,51,70,20,64)(16,21,52,57,71)(25,100,106,74,95)(26,75,101,96,107)(27,89,76,108,102)(28,109,90,103,77)(29,104,110,78,91)(30,79,97,92,111)(31,93,80,112,98)(32,105,94,99,73), (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), (1,58,99,5,62,103)(2,59,100,6,63,104)(3,60,101,7,64,97)(4,61,102,8,57,98)(9,73,84,18,90,45)(10,74,85,19,91,46)(11,75,86,20,92,47)(12,76,87,21,93,48)(13,77,88,22,94,41)(14,78,81,23,95,42)(15,79,82,24,96,43)(16,80,83,17,89,44)(25,35,69,110,116,54)(26,36,70,111,117,55)(27,37,71,112,118,56)(28,38,72,105,119,49)(29,39,65,106,120,50)(30,40,66,107,113,51)(31,33,67,108,114,52)(32,34,68,109,115,53), (1,103)(2,100)(3,97)(4,102)(5,99)(6,104)(7,101)(8,98)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21)(25,39)(26,36)(27,33)(28,38)(29,35)(30,40)(31,37)(32,34)(41,94)(42,91)(43,96)(44,93)(45,90)(46,95)(47,92)(48,89)(49,72)(50,69)(51,66)(52,71)(53,68)(54,65)(55,70)(56,67)(58,62)(60,64)(73,84)(74,81)(75,86)(76,83)(77,88)(78,85)(79,82)(80,87)(105,119)(106,116)(107,113)(108,118)(109,115)(110,120)(111,117)(112,114)>;
G:=Group( (1,45,115,38,88)(2,39,46,81,116)(3,82,40,117,47)(4,118,83,48,33)(5,41,119,34,84)(6,35,42,85,120)(7,86,36,113,43)(8,114,87,44,37)(9,58,22,72,53)(10,65,59,54,23)(11,55,66,24,60)(12,17,56,61,67)(13,62,18,68,49)(14,69,63,50,19)(15,51,70,20,64)(16,21,52,57,71)(25,100,106,74,95)(26,75,101,96,107)(27,89,76,108,102)(28,109,90,103,77)(29,104,110,78,91)(30,79,97,92,111)(31,93,80,112,98)(32,105,94,99,73), (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), (1,58,99,5,62,103)(2,59,100,6,63,104)(3,60,101,7,64,97)(4,61,102,8,57,98)(9,73,84,18,90,45)(10,74,85,19,91,46)(11,75,86,20,92,47)(12,76,87,21,93,48)(13,77,88,22,94,41)(14,78,81,23,95,42)(15,79,82,24,96,43)(16,80,83,17,89,44)(25,35,69,110,116,54)(26,36,70,111,117,55)(27,37,71,112,118,56)(28,38,72,105,119,49)(29,39,65,106,120,50)(30,40,66,107,113,51)(31,33,67,108,114,52)(32,34,68,109,115,53), (1,103)(2,100)(3,97)(4,102)(5,99)(6,104)(7,101)(8,98)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21)(25,39)(26,36)(27,33)(28,38)(29,35)(30,40)(31,37)(32,34)(41,94)(42,91)(43,96)(44,93)(45,90)(46,95)(47,92)(48,89)(49,72)(50,69)(51,66)(52,71)(53,68)(54,65)(55,70)(56,67)(58,62)(60,64)(73,84)(74,81)(75,86)(76,83)(77,88)(78,85)(79,82)(80,87)(105,119)(106,116)(107,113)(108,118)(109,115)(110,120)(111,117)(112,114) );
G=PermutationGroup([[(1,45,115,38,88),(2,39,46,81,116),(3,82,40,117,47),(4,118,83,48,33),(5,41,119,34,84),(6,35,42,85,120),(7,86,36,113,43),(8,114,87,44,37),(9,58,22,72,53),(10,65,59,54,23),(11,55,66,24,60),(12,17,56,61,67),(13,62,18,68,49),(14,69,63,50,19),(15,51,70,20,64),(16,21,52,57,71),(25,100,106,74,95),(26,75,101,96,107),(27,89,76,108,102),(28,109,90,103,77),(29,104,110,78,91),(30,79,97,92,111),(31,93,80,112,98),(32,105,94,99,73)], [(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)], [(1,58,99,5,62,103),(2,59,100,6,63,104),(3,60,101,7,64,97),(4,61,102,8,57,98),(9,73,84,18,90,45),(10,74,85,19,91,46),(11,75,86,20,92,47),(12,76,87,21,93,48),(13,77,88,22,94,41),(14,78,81,23,95,42),(15,79,82,24,96,43),(16,80,83,17,89,44),(25,35,69,110,116,54),(26,36,70,111,117,55),(27,37,71,112,118,56),(28,38,72,105,119,49),(29,39,65,106,120,50),(30,40,66,107,113,51),(31,33,67,108,114,52),(32,34,68,109,115,53)], [(1,103),(2,100),(3,97),(4,102),(5,99),(6,104),(7,101),(8,98),(9,18),(10,23),(11,20),(12,17),(13,22),(14,19),(15,24),(16,21),(25,39),(26,36),(27,33),(28,38),(29,35),(30,40),(31,37),(32,34),(41,94),(42,91),(43,96),(44,93),(45,90),(46,95),(47,92),(48,89),(49,72),(50,69),(51,66),(52,71),(53,68),(54,65),(55,70),(56,67),(58,62),(60,64),(73,84),(74,81),(75,86),(76,83),(77,88),(78,85),(79,82),(80,87),(105,119),(106,116),(107,113),(108,118),(109,115),(110,120),(111,117),(112,114)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 15 | 20A | 20B | 20C | 20D | 24A | ··· | 24H | 30 | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | 60 | 60 |
| size | 1 | 1 | 5 | 5 | 6 | 30 | 2 | 1 | 1 | 5 | 5 | 6 | 30 | 4 | 2 | 10 | 10 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 4 | 12 | 12 | 2 | 2 | 10 | 10 | 8 | 4 | 4 | 12 | 12 | 10 | ··· | 10 | 8 | 8 | 8 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D6 | D6 | M4(2) | C4×S3 | C4×S3 | C8⋊S3 | F5 | C2×F5 | C2×F5 | C2×F5 | D5⋊M4(2) | S3×F5 | C2×S3×F5 | C5⋊C8⋊D6 |
| kernel | C5⋊C8⋊D6 | D6.F5 | Dic3.F5 | C3×D5⋊C8 | C60.C4 | C4×S3×D5 | D5×Dic3 | S3×C20 | C4×D15 | C2×S3×D5 | D5⋊C8 | C5⋊C8 | C4×D5 | C3×D5 | C20 | D10 | D5 | C4×S3 | Dic3 | C12 | D6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 4 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 |
Matrix representation of C5⋊C8⋊D6 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 190 | 240 | 0 | 0 |
| 0 | 0 | 191 | 240 | 0 | 0 |
| 0 | 0 | 195 | 195 | 52 | 52 |
| 0 | 0 | 177 | 0 | 189 | 240 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 177 | 64 |
| 0 | 0 | 52 | 1 | 113 | 46 |
| 0 | 0 | 100 | 39 | 240 | 0 |
| 0 | 0 | 49 | 38 | 240 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 240 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 190 | 240 | 0 | 0 |
| 0 | 0 | 190 | 51 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 195 | 64 | 52 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 190 | 240 | 0 | 0 |
| 0 | 0 | 190 | 51 | 0 | 0 |
| 0 | 0 | 34 | 47 | 1 | 0 |
| 0 | 0 | 80 | 224 | 189 | 240 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,190,191,195,177,0,0,240,240,195,0,0,0,0,0,52,189,0,0,0,0,52,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,52,100,49,0,0,0,1,39,38,0,0,177,113,240,240,0,0,64,46,0,0],[0,240,0,0,0,0,1,1,0,0,0,0,0,0,190,190,0,195,0,0,240,51,0,64,0,0,0,0,240,52,0,0,0,0,0,1],[1,1,0,0,0,0,0,240,0,0,0,0,0,0,190,190,34,80,0,0,240,51,47,224,0,0,0,0,1,189,0,0,0,0,0,240] >;
C5⋊C8⋊D6 in GAP, Magma, Sage, TeX
C_5\rtimes C_8\rtimes D_6
% in TeX
G:=Group("C5:C8:D6"); // GroupNames label
G:=SmallGroup(480,993);
// by ID
G=gap.SmallGroup(480,993);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,100,80,1356,9414,2379]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^6=d^2=1,b*a*b^-1=a^3,c*a*c^-1=d*a*d=a^-1,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations