direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×Q8⋊2S3, C30.49D4, C20.38D6, C15⋊15SD16, D12.2C10, C60.45C22, C3⋊C8⋊3C10, Q8⋊2(C5×S3), (C5×Q8)⋊6S3, C6.9(C5×D4), C3⋊3(C5×SD16), C4.3(S3×C10), (C3×Q8)⋊1C10, (Q8×C15)⋊7C2, C12.3(C2×C10), (C5×D12).4C2, C10.25(C3⋊D4), (C5×C3⋊C8)⋊10C2, C2.6(C5×C3⋊D4), SmallGroup(240,62)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Q8⋊2S3
G = < a,b,c,d,e | a5=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ece=b-1c, ede=d-1 >
Smallest permutation representation of C5×Q8⋊2S3
►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 26 58 78)(2 27 59 79)(3 28 60 80)(4 29 56 76)(5 30 57 77)(6 34 63 54)(7 35 64 55)(8 31 65 51)(9 32 61 52)(10 33 62 53)(11 117 102 97)(12 118 103 98)(13 119 104 99)(14 120 105 100)(15 116 101 96)(16 91 41 90)(17 92 42 86)(18 93 43 87)(19 94 44 88)(20 95 45 89)(21 72 46 114)(22 73 47 115)(23 74 48 111)(24 75 49 112)(25 71 50 113)(36 85 67 110)(37 81 68 106)(38 82 69 107)(39 83 70 108)(40 84 66 109)
(1 118 58 98)(2 119 59 99)(3 120 60 100)(4 116 56 96)(5 117 57 97)(6 38 63 69)(7 39 64 70)(8 40 65 66)(9 36 61 67)(10 37 62 68)(11 77 102 30)(12 78 103 26)(13 79 104 27)(14 80 105 28)(15 76 101 29)(16 75 41 112)(17 71 42 113)(18 72 43 114)(19 73 44 115)(20 74 45 111)(21 87 46 93)(22 88 47 94)(23 89 48 95)(24 90 49 91)(25 86 50 92)(31 109 51 84)(32 110 52 85)(33 106 53 81)(34 107 54 82)(35 108 55 83)
(1 20 61)(2 16 62)(3 17 63)(4 18 64)(5 19 65)(6 60 42)(7 56 43)(8 57 44)(9 58 45)(10 59 41)(11 22 84)(12 23 85)(13 24 81)(14 25 82)(15 21 83)(26 95 52)(27 91 53)(28 92 54)(29 93 55)(30 94 51)(31 77 88)(32 78 89)(33 79 90)(34 80 86)(35 76 87)(36 98 111)(37 99 112)(38 100 113)(39 96 114)(40 97 115)(46 108 101)(47 109 102)(48 110 103)(49 106 104)(50 107 105)(66 117 73)(67 118 74)(68 119 75)(69 120 71)(70 116 72)
(6 42)(7 43)(8 44)(9 45)(10 41)(11 97)(12 98)(13 99)(14 100)(15 96)(16 62)(17 63)(18 64)(19 65)(20 61)(21 39)(22 40)(23 36)(24 37)(25 38)(26 78)(27 79)(28 80)(29 76)(30 77)(31 94)(32 95)(33 91)(34 92)(35 93)(46 70)(47 66)(48 67)(49 68)(50 69)(51 88)(52 89)(53 90)(54 86)(55 87)(71 107)(72 108)(73 109)(74 110)(75 106)(81 112)(82 113)(83 114)(84 115)(85 111)(101 116)(102 117)(103 118)(104 119)(105 120)
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), (1,26,58,78)(2,27,59,79)(3,28,60,80)(4,29,56,76)(5,30,57,77)(6,34,63,54)(7,35,64,55)(8,31,65,51)(9,32,61,52)(10,33,62,53)(11,117,102,97)(12,118,103,98)(13,119,104,99)(14,120,105,100)(15,116,101,96)(16,91,41,90)(17,92,42,86)(18,93,43,87)(19,94,44,88)(20,95,45,89)(21,72,46,114)(22,73,47,115)(23,74,48,111)(24,75,49,112)(25,71,50,113)(36,85,67,110)(37,81,68,106)(38,82,69,107)(39,83,70,108)(40,84,66,109), (1,118,58,98)(2,119,59,99)(3,120,60,100)(4,116,56,96)(5,117,57,97)(6,38,63,69)(7,39,64,70)(8,40,65,66)(9,36,61,67)(10,37,62,68)(11,77,102,30)(12,78,103,26)(13,79,104,27)(14,80,105,28)(15,76,101,29)(16,75,41,112)(17,71,42,113)(18,72,43,114)(19,73,44,115)(20,74,45,111)(21,87,46,93)(22,88,47,94)(23,89,48,95)(24,90,49,91)(25,86,50,92)(31,109,51,84)(32,110,52,85)(33,106,53,81)(34,107,54,82)(35,108,55,83), (1,20,61)(2,16,62)(3,17,63)(4,18,64)(5,19,65)(6,60,42)(7,56,43)(8,57,44)(9,58,45)(10,59,41)(11,22,84)(12,23,85)(13,24,81)(14,25,82)(15,21,83)(26,95,52)(27,91,53)(28,92,54)(29,93,55)(30,94,51)(31,77,88)(32,78,89)(33,79,90)(34,80,86)(35,76,87)(36,98,111)(37,99,112)(38,100,113)(39,96,114)(40,97,115)(46,108,101)(47,109,102)(48,110,103)(49,106,104)(50,107,105)(66,117,73)(67,118,74)(68,119,75)(69,120,71)(70,116,72), (6,42)(7,43)(8,44)(9,45)(10,41)(11,97)(12,98)(13,99)(14,100)(15,96)(16,62)(17,63)(18,64)(19,65)(20,61)(21,39)(22,40)(23,36)(24,37)(25,38)(26,78)(27,79)(28,80)(29,76)(30,77)(31,94)(32,95)(33,91)(34,92)(35,93)(46,70)(47,66)(48,67)(49,68)(50,69)(51,88)(52,89)(53,90)(54,86)(55,87)(71,107)(72,108)(73,109)(74,110)(75,106)(81,112)(82,113)(83,114)(84,115)(85,111)(101,116)(102,117)(103,118)(104,119)(105,120)>;
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), (1,26,58,78)(2,27,59,79)(3,28,60,80)(4,29,56,76)(5,30,57,77)(6,34,63,54)(7,35,64,55)(8,31,65,51)(9,32,61,52)(10,33,62,53)(11,117,102,97)(12,118,103,98)(13,119,104,99)(14,120,105,100)(15,116,101,96)(16,91,41,90)(17,92,42,86)(18,93,43,87)(19,94,44,88)(20,95,45,89)(21,72,46,114)(22,73,47,115)(23,74,48,111)(24,75,49,112)(25,71,50,113)(36,85,67,110)(37,81,68,106)(38,82,69,107)(39,83,70,108)(40,84,66,109), (1,118,58,98)(2,119,59,99)(3,120,60,100)(4,116,56,96)(5,117,57,97)(6,38,63,69)(7,39,64,70)(8,40,65,66)(9,36,61,67)(10,37,62,68)(11,77,102,30)(12,78,103,26)(13,79,104,27)(14,80,105,28)(15,76,101,29)(16,75,41,112)(17,71,42,113)(18,72,43,114)(19,73,44,115)(20,74,45,111)(21,87,46,93)(22,88,47,94)(23,89,48,95)(24,90,49,91)(25,86,50,92)(31,109,51,84)(32,110,52,85)(33,106,53,81)(34,107,54,82)(35,108,55,83), (1,20,61)(2,16,62)(3,17,63)(4,18,64)(5,19,65)(6,60,42)(7,56,43)(8,57,44)(9,58,45)(10,59,41)(11,22,84)(12,23,85)(13,24,81)(14,25,82)(15,21,83)(26,95,52)(27,91,53)(28,92,54)(29,93,55)(30,94,51)(31,77,88)(32,78,89)(33,79,90)(34,80,86)(35,76,87)(36,98,111)(37,99,112)(38,100,113)(39,96,114)(40,97,115)(46,108,101)(47,109,102)(48,110,103)(49,106,104)(50,107,105)(66,117,73)(67,118,74)(68,119,75)(69,120,71)(70,116,72), (6,42)(7,43)(8,44)(9,45)(10,41)(11,97)(12,98)(13,99)(14,100)(15,96)(16,62)(17,63)(18,64)(19,65)(20,61)(21,39)(22,40)(23,36)(24,37)(25,38)(26,78)(27,79)(28,80)(29,76)(30,77)(31,94)(32,95)(33,91)(34,92)(35,93)(46,70)(47,66)(48,67)(49,68)(50,69)(51,88)(52,89)(53,90)(54,86)(55,87)(71,107)(72,108)(73,109)(74,110)(75,106)(81,112)(82,113)(83,114)(84,115)(85,111)(101,116)(102,117)(103,118)(104,119)(105,120) );
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)], [(1,26,58,78),(2,27,59,79),(3,28,60,80),(4,29,56,76),(5,30,57,77),(6,34,63,54),(7,35,64,55),(8,31,65,51),(9,32,61,52),(10,33,62,53),(11,117,102,97),(12,118,103,98),(13,119,104,99),(14,120,105,100),(15,116,101,96),(16,91,41,90),(17,92,42,86),(18,93,43,87),(19,94,44,88),(20,95,45,89),(21,72,46,114),(22,73,47,115),(23,74,48,111),(24,75,49,112),(25,71,50,113),(36,85,67,110),(37,81,68,106),(38,82,69,107),(39,83,70,108),(40,84,66,109)], [(1,118,58,98),(2,119,59,99),(3,120,60,100),(4,116,56,96),(5,117,57,97),(6,38,63,69),(7,39,64,70),(8,40,65,66),(9,36,61,67),(10,37,62,68),(11,77,102,30),(12,78,103,26),(13,79,104,27),(14,80,105,28),(15,76,101,29),(16,75,41,112),(17,71,42,113),(18,72,43,114),(19,73,44,115),(20,74,45,111),(21,87,46,93),(22,88,47,94),(23,89,48,95),(24,90,49,91),(25,86,50,92),(31,109,51,84),(32,110,52,85),(33,106,53,81),(34,107,54,82),(35,108,55,83)], [(1,20,61),(2,16,62),(3,17,63),(4,18,64),(5,19,65),(6,60,42),(7,56,43),(8,57,44),(9,58,45),(10,59,41),(11,22,84),(12,23,85),(13,24,81),(14,25,82),(15,21,83),(26,95,52),(27,91,53),(28,92,54),(29,93,55),(30,94,51),(31,77,88),(32,78,89),(33,79,90),(34,80,86),(35,76,87),(36,98,111),(37,99,112),(38,100,113),(39,96,114),(40,97,115),(46,108,101),(47,109,102),(48,110,103),(49,106,104),(50,107,105),(66,117,73),(67,118,74),(68,119,75),(69,120,71),(70,116,72)], [(6,42),(7,43),(8,44),(9,45),(10,41),(11,97),(12,98),(13,99),(14,100),(15,96),(16,62),(17,63),(18,64),(19,65),(20,61),(21,39),(22,40),(23,36),(24,37),(25,38),(26,78),(27,79),(28,80),(29,76),(30,77),(31,94),(32,95),(33,91),(34,92),(35,93),(46,70),(47,66),(48,67),(49,68),(50,69),(51,88),(52,89),(53,90),(54,86),(55,87),(71,107),(72,108),(73,109),(74,110),(75,106),(81,112),(82,113),(83,114),(84,115),(85,111),(101,116),(102,117),(103,118),(104,119),(105,120)]])
C5×Q8⋊2S3 is a maximal subgroup of
D20⋊D6 D15⋊SD16 D60⋊C22 D12.27D10 D20.14D6 D12.D10 D30.44D4 C5×S3×SD16
60 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6 | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 60A | ··· | 60L |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 12 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 6 | 6 | 1 | 1 | 1 | 1 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | S3 | D4 | D6 | SD16 | C3⋊D4 | C5×S3 | C5×D4 | S3×C10 | C5×SD16 | C5×C3⋊D4 | Q8⋊2S3 | C5×Q8⋊2S3 |
| kernel | C5×Q8⋊2S3 | C5×C3⋊C8 | C5×D12 | Q8×C15 | Q8⋊2S3 | C3⋊C8 | D12 | C3×Q8 | C5×Q8 | C30 | C20 | C15 | C10 | Q8 | C6 | C4 | C3 | C2 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 1 | 4 |
Matrix representation of C5×Q8⋊2S3 ►in GL4(𝔽241) generated by
| 91 | 0 | 0 | 0 |
| 0 | 91 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 239 |
| 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 38 |
| 0 | 0 | 19 | 0 |
| 240 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 240 | 240 |
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,240,1,0,0,239,1],[1,0,0,0,0,1,0,0,0,0,0,19,0,0,38,0],[240,240,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,240,0,0,0,240] >;
C5×Q8⋊2S3 in GAP, Magma, Sage, TeX
C_5\times Q_8\rtimes_2S_3
% in TeX
G:=Group("C5xQ8:2S3"); // GroupNames label
G:=SmallGroup(240,62);
// by ID
G=gap.SmallGroup(240,62);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-3,265,247,1443,729,69,5765]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=d^3=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations
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