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## G = C5×Q8⋊2S3order 240 = 24·3·5

### Direct product of C5 and Q8⋊2S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×Q8⋊2S3
 Chief series C1 — C3 — C6 — C12 — C60 — C5×D12 — C5×Q8⋊2S3
 Lower central C3 — C6 — C12 — C5×Q8⋊2S3
 Upper central C1 — C10 — C20 — C5×Q8

Generators and relations for C5×Q82S3
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×Q82S3
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 66)(2 27 59 67)(3 28 60 68)(4 29 56 69)(5 30 57 70)(6 20 90 52)(7 16 86 53)(8 17 87 54)(9 18 88 55)(10 19 89 51)(11 98 48 111)(12 99 49 112)(13 100 50 113)(14 96 46 114)(15 97 47 115)(21 75 93 107)(22 71 94 108)(23 72 95 109)(24 73 91 110)(25 74 92 106)(31 76 43 63)(32 77 44 64)(33 78 45 65)(34 79 41 61)(35 80 42 62)(36 118 103 81)(37 119 104 82)(38 120 105 83)(39 116 101 84)(40 117 102 85)
(1 118 58 81)(2 119 59 82)(3 120 60 83)(4 116 56 84)(5 117 57 85)(6 24 90 91)(7 25 86 92)(8 21 87 93)(9 22 88 94)(10 23 89 95)(11 65 48 78)(12 61 49 79)(13 62 50 80)(14 63 46 76)(15 64 47 77)(16 106 53 74)(17 107 54 75)(18 108 55 71)(19 109 51 72)(20 110 52 73)(26 36 66 103)(27 37 67 104)(28 38 68 105)(29 39 69 101)(30 40 70 102)(31 96 43 114)(32 97 44 115)(33 98 45 111)(34 99 41 112)(35 100 42 113)
(1 33 90)(2 34 86)(3 35 87)(4 31 88)(5 32 89)(6 58 45)(7 59 41)(8 60 42)(9 56 43)(10 57 44)(11 73 36)(12 74 37)(13 75 38)(14 71 39)(15 72 40)(16 67 61)(17 68 62)(18 69 63)(19 70 64)(20 66 65)(21 83 113)(22 84 114)(23 85 115)(24 81 111)(25 82 112)(26 78 52)(27 79 53)(28 80 54)(29 76 55)(30 77 51)(46 108 101)(47 109 102)(48 110 103)(49 106 104)(50 107 105)(91 118 98)(92 119 99)(93 120 100)(94 116 96)(95 117 97)
(6 45)(7 41)(8 42)(9 43)(10 44)(11 24)(12 25)(13 21)(14 22)(15 23)(16 79)(17 80)(18 76)(19 77)(20 78)(26 66)(27 67)(28 68)(29 69)(30 70)(31 88)(32 89)(33 90)(34 86)(35 87)(36 81)(37 82)(38 83)(39 84)(40 85)(46 94)(47 95)(48 91)(49 92)(50 93)(51 64)(52 65)(53 61)(54 62)(55 63)(71 114)(72 115)(73 111)(74 112)(75 113)(96 108)(97 109)(98 110)(99 106)(100 107)(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), (1,26,58,66)(2,27,59,67)(3,28,60,68)(4,29,56,69)(5,30,57,70)(6,20,90,52)(7,16,86,53)(8,17,87,54)(9,18,88,55)(10,19,89,51)(11,98,48,111)(12,99,49,112)(13,100,50,113)(14,96,46,114)(15,97,47,115)(21,75,93,107)(22,71,94,108)(23,72,95,109)(24,73,91,110)(25,74,92,106)(31,76,43,63)(32,77,44,64)(33,78,45,65)(34,79,41,61)(35,80,42,62)(36,118,103,81)(37,119,104,82)(38,120,105,83)(39,116,101,84)(40,117,102,85), (1,118,58,81)(2,119,59,82)(3,120,60,83)(4,116,56,84)(5,117,57,85)(6,24,90,91)(7,25,86,92)(8,21,87,93)(9,22,88,94)(10,23,89,95)(11,65,48,78)(12,61,49,79)(13,62,50,80)(14,63,46,76)(15,64,47,77)(16,106,53,74)(17,107,54,75)(18,108,55,71)(19,109,51,72)(20,110,52,73)(26,36,66,103)(27,37,67,104)(28,38,68,105)(29,39,69,101)(30,40,70,102)(31,96,43,114)(32,97,44,115)(33,98,45,111)(34,99,41,112)(35,100,42,113), (1,33,90)(2,34,86)(3,35,87)(4,31,88)(5,32,89)(6,58,45)(7,59,41)(8,60,42)(9,56,43)(10,57,44)(11,73,36)(12,74,37)(13,75,38)(14,71,39)(15,72,40)(16,67,61)(17,68,62)(18,69,63)(19,70,64)(20,66,65)(21,83,113)(22,84,114)(23,85,115)(24,81,111)(25,82,112)(26,78,52)(27,79,53)(28,80,54)(29,76,55)(30,77,51)(46,108,101)(47,109,102)(48,110,103)(49,106,104)(50,107,105)(91,118,98)(92,119,99)(93,120,100)(94,116,96)(95,117,97), (6,45)(7,41)(8,42)(9,43)(10,44)(11,24)(12,25)(13,21)(14,22)(15,23)(16,79)(17,80)(18,76)(19,77)(20,78)(26,66)(27,67)(28,68)(29,69)(30,70)(31,88)(32,89)(33,90)(34,86)(35,87)(36,81)(37,82)(38,83)(39,84)(40,85)(46,94)(47,95)(48,91)(49,92)(50,93)(51,64)(52,65)(53,61)(54,62)(55,63)(71,114)(72,115)(73,111)(74,112)(75,113)(96,108)(97,109)(98,110)(99,106)(100,107)(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), (1,26,58,66)(2,27,59,67)(3,28,60,68)(4,29,56,69)(5,30,57,70)(6,20,90,52)(7,16,86,53)(8,17,87,54)(9,18,88,55)(10,19,89,51)(11,98,48,111)(12,99,49,112)(13,100,50,113)(14,96,46,114)(15,97,47,115)(21,75,93,107)(22,71,94,108)(23,72,95,109)(24,73,91,110)(25,74,92,106)(31,76,43,63)(32,77,44,64)(33,78,45,65)(34,79,41,61)(35,80,42,62)(36,118,103,81)(37,119,104,82)(38,120,105,83)(39,116,101,84)(40,117,102,85), (1,118,58,81)(2,119,59,82)(3,120,60,83)(4,116,56,84)(5,117,57,85)(6,24,90,91)(7,25,86,92)(8,21,87,93)(9,22,88,94)(10,23,89,95)(11,65,48,78)(12,61,49,79)(13,62,50,80)(14,63,46,76)(15,64,47,77)(16,106,53,74)(17,107,54,75)(18,108,55,71)(19,109,51,72)(20,110,52,73)(26,36,66,103)(27,37,67,104)(28,38,68,105)(29,39,69,101)(30,40,70,102)(31,96,43,114)(32,97,44,115)(33,98,45,111)(34,99,41,112)(35,100,42,113), (1,33,90)(2,34,86)(3,35,87)(4,31,88)(5,32,89)(6,58,45)(7,59,41)(8,60,42)(9,56,43)(10,57,44)(11,73,36)(12,74,37)(13,75,38)(14,71,39)(15,72,40)(16,67,61)(17,68,62)(18,69,63)(19,70,64)(20,66,65)(21,83,113)(22,84,114)(23,85,115)(24,81,111)(25,82,112)(26,78,52)(27,79,53)(28,80,54)(29,76,55)(30,77,51)(46,108,101)(47,109,102)(48,110,103)(49,106,104)(50,107,105)(91,118,98)(92,119,99)(93,120,100)(94,116,96)(95,117,97), (6,45)(7,41)(8,42)(9,43)(10,44)(11,24)(12,25)(13,21)(14,22)(15,23)(16,79)(17,80)(18,76)(19,77)(20,78)(26,66)(27,67)(28,68)(29,69)(30,70)(31,88)(32,89)(33,90)(34,86)(35,87)(36,81)(37,82)(38,83)(39,84)(40,85)(46,94)(47,95)(48,91)(49,92)(50,93)(51,64)(52,65)(53,61)(54,62)(55,63)(71,114)(72,115)(73,111)(74,112)(75,113)(96,108)(97,109)(98,110)(99,106)(100,107)(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)], [(1,26,58,66),(2,27,59,67),(3,28,60,68),(4,29,56,69),(5,30,57,70),(6,20,90,52),(7,16,86,53),(8,17,87,54),(9,18,88,55),(10,19,89,51),(11,98,48,111),(12,99,49,112),(13,100,50,113),(14,96,46,114),(15,97,47,115),(21,75,93,107),(22,71,94,108),(23,72,95,109),(24,73,91,110),(25,74,92,106),(31,76,43,63),(32,77,44,64),(33,78,45,65),(34,79,41,61),(35,80,42,62),(36,118,103,81),(37,119,104,82),(38,120,105,83),(39,116,101,84),(40,117,102,85)], [(1,118,58,81),(2,119,59,82),(3,120,60,83),(4,116,56,84),(5,117,57,85),(6,24,90,91),(7,25,86,92),(8,21,87,93),(9,22,88,94),(10,23,89,95),(11,65,48,78),(12,61,49,79),(13,62,50,80),(14,63,46,76),(15,64,47,77),(16,106,53,74),(17,107,54,75),(18,108,55,71),(19,109,51,72),(20,110,52,73),(26,36,66,103),(27,37,67,104),(28,38,68,105),(29,39,69,101),(30,40,70,102),(31,96,43,114),(32,97,44,115),(33,98,45,111),(34,99,41,112),(35,100,42,113)], [(1,33,90),(2,34,86),(3,35,87),(4,31,88),(5,32,89),(6,58,45),(7,59,41),(8,60,42),(9,56,43),(10,57,44),(11,73,36),(12,74,37),(13,75,38),(14,71,39),(15,72,40),(16,67,61),(17,68,62),(18,69,63),(19,70,64),(20,66,65),(21,83,113),(22,84,114),(23,85,115),(24,81,111),(25,82,112),(26,78,52),(27,79,53),(28,80,54),(29,76,55),(30,77,51),(46,108,101),(47,109,102),(48,110,103),(49,106,104),(50,107,105),(91,118,98),(92,119,99),(93,120,100),(94,116,96),(95,117,97)], [(6,45),(7,41),(8,42),(9,43),(10,44),(11,24),(12,25),(13,21),(14,22),(15,23),(16,79),(17,80),(18,76),(19,77),(20,78),(26,66),(27,67),(28,68),(29,69),(30,70),(31,88),(32,89),(33,90),(34,86),(35,87),(36,81),(37,82),(38,83),(39,84),(40,85),(46,94),(47,95),(48,91),(49,92),(50,93),(51,64),(52,65),(53,61),(54,62),(55,63),(71,114),(72,115),(73,111),(74,112),(75,113),(96,108),(97,109),(98,110),(99,106),(100,107),(101,116),(102,117),(103,118),(104,119),(105,120)])

C5×Q82S3 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×Q82S3 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×Q82S3 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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