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## G = C5×D4⋊S3order 240 = 24·3·5

### Direct product of C5 and D4⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×D4⋊S3
 Chief series C1 — C3 — C6 — C12 — C60 — C5×D12 — C5×D4⋊S3
 Lower central C3 — C6 — C12 — C5×D4⋊S3
 Upper central C1 — C10 — C20 — C5×D4

Generators and relations for C5×D4⋊S3
G = < a,b,c,d,e | a5=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >

Smallest permutation representation of C5×D4⋊S3
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 16 58 66)(2 17 59 67)(3 18 60 68)(4 19 56 69)(5 20 57 70)(6 93 43 63)(7 94 44 64)(8 95 45 65)(9 91 41 61)(10 92 42 62)(11 81 48 111)(12 82 49 112)(13 83 50 113)(14 84 46 114)(15 85 47 115)(21 120 105 100)(22 116 101 96)(23 117 102 97)(24 118 103 98)(25 119 104 99)(26 31 89 51)(27 32 90 52)(28 33 86 53)(29 34 87 54)(30 35 88 55)(36 73 79 110)(37 74 80 106)(38 75 76 107)(39 71 77 108)(40 72 78 109)
(1 103)(2 104)(3 105)(4 101)(5 102)(6 46)(7 47)(8 48)(9 49)(10 50)(11 45)(12 41)(13 42)(14 43)(15 44)(16 118)(17 119)(18 120)(19 116)(20 117)(21 60)(22 56)(23 57)(24 58)(25 59)(26 72)(27 73)(28 74)(29 75)(30 71)(31 40)(32 36)(33 37)(34 38)(35 39)(51 78)(52 79)(53 80)(54 76)(55 77)(61 112)(62 113)(63 114)(64 115)(65 111)(66 98)(67 99)(68 100)(69 96)(70 97)(81 95)(82 91)(83 92)(84 93)(85 94)(86 106)(87 107)(88 108)(89 109)(90 110)
(1 8 90)(2 9 86)(3 10 87)(4 6 88)(5 7 89)(11 73 24)(12 74 25)(13 75 21)(14 71 22)(15 72 23)(16 95 52)(17 91 53)(18 92 54)(19 93 55)(20 94 51)(26 57 44)(27 58 45)(28 59 41)(29 60 42)(30 56 43)(31 70 64)(32 66 65)(33 67 61)(34 68 62)(35 69 63)(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)(76 120 83)(77 116 84)(78 117 85)(79 118 81)(80 119 82)
(6 88)(7 89)(8 90)(9 86)(10 87)(11 36)(12 37)(13 38)(14 39)(15 40)(16 66)(17 67)(18 68)(19 69)(20 70)(21 100)(22 96)(23 97)(24 98)(25 99)(26 44)(27 45)(28 41)(29 42)(30 43)(31 94)(32 95)(33 91)(34 92)(35 93)(46 77)(47 78)(48 79)(49 80)(50 76)(51 64)(52 65)(53 61)(54 62)(55 63)(71 114)(72 115)(73 111)(74 112)(75 113)(81 110)(82 106)(83 107)(84 108)(85 109)(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), (1,16,58,66)(2,17,59,67)(3,18,60,68)(4,19,56,69)(5,20,57,70)(6,93,43,63)(7,94,44,64)(8,95,45,65)(9,91,41,61)(10,92,42,62)(11,81,48,111)(12,82,49,112)(13,83,50,113)(14,84,46,114)(15,85,47,115)(21,120,105,100)(22,116,101,96)(23,117,102,97)(24,118,103,98)(25,119,104,99)(26,31,89,51)(27,32,90,52)(28,33,86,53)(29,34,87,54)(30,35,88,55)(36,73,79,110)(37,74,80,106)(38,75,76,107)(39,71,77,108)(40,72,78,109), (1,103)(2,104)(3,105)(4,101)(5,102)(6,46)(7,47)(8,48)(9,49)(10,50)(11,45)(12,41)(13,42)(14,43)(15,44)(16,118)(17,119)(18,120)(19,116)(20,117)(21,60)(22,56)(23,57)(24,58)(25,59)(26,72)(27,73)(28,74)(29,75)(30,71)(31,40)(32,36)(33,37)(34,38)(35,39)(51,78)(52,79)(53,80)(54,76)(55,77)(61,112)(62,113)(63,114)(64,115)(65,111)(66,98)(67,99)(68,100)(69,96)(70,97)(81,95)(82,91)(83,92)(84,93)(85,94)(86,106)(87,107)(88,108)(89,109)(90,110), (1,8,90)(2,9,86)(3,10,87)(4,6,88)(5,7,89)(11,73,24)(12,74,25)(13,75,21)(14,71,22)(15,72,23)(16,95,52)(17,91,53)(18,92,54)(19,93,55)(20,94,51)(26,57,44)(27,58,45)(28,59,41)(29,60,42)(30,56,43)(31,70,64)(32,66,65)(33,67,61)(34,68,62)(35,69,63)(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)(76,120,83)(77,116,84)(78,117,85)(79,118,81)(80,119,82), (6,88)(7,89)(8,90)(9,86)(10,87)(11,36)(12,37)(13,38)(14,39)(15,40)(16,66)(17,67)(18,68)(19,69)(20,70)(21,100)(22,96)(23,97)(24,98)(25,99)(26,44)(27,45)(28,41)(29,42)(30,43)(31,94)(32,95)(33,91)(34,92)(35,93)(46,77)(47,78)(48,79)(49,80)(50,76)(51,64)(52,65)(53,61)(54,62)(55,63)(71,114)(72,115)(73,111)(74,112)(75,113)(81,110)(82,106)(83,107)(84,108)(85,109)(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), (1,16,58,66)(2,17,59,67)(3,18,60,68)(4,19,56,69)(5,20,57,70)(6,93,43,63)(7,94,44,64)(8,95,45,65)(9,91,41,61)(10,92,42,62)(11,81,48,111)(12,82,49,112)(13,83,50,113)(14,84,46,114)(15,85,47,115)(21,120,105,100)(22,116,101,96)(23,117,102,97)(24,118,103,98)(25,119,104,99)(26,31,89,51)(27,32,90,52)(28,33,86,53)(29,34,87,54)(30,35,88,55)(36,73,79,110)(37,74,80,106)(38,75,76,107)(39,71,77,108)(40,72,78,109), (1,103)(2,104)(3,105)(4,101)(5,102)(6,46)(7,47)(8,48)(9,49)(10,50)(11,45)(12,41)(13,42)(14,43)(15,44)(16,118)(17,119)(18,120)(19,116)(20,117)(21,60)(22,56)(23,57)(24,58)(25,59)(26,72)(27,73)(28,74)(29,75)(30,71)(31,40)(32,36)(33,37)(34,38)(35,39)(51,78)(52,79)(53,80)(54,76)(55,77)(61,112)(62,113)(63,114)(64,115)(65,111)(66,98)(67,99)(68,100)(69,96)(70,97)(81,95)(82,91)(83,92)(84,93)(85,94)(86,106)(87,107)(88,108)(89,109)(90,110), (1,8,90)(2,9,86)(3,10,87)(4,6,88)(5,7,89)(11,73,24)(12,74,25)(13,75,21)(14,71,22)(15,72,23)(16,95,52)(17,91,53)(18,92,54)(19,93,55)(20,94,51)(26,57,44)(27,58,45)(28,59,41)(29,60,42)(30,56,43)(31,70,64)(32,66,65)(33,67,61)(34,68,62)(35,69,63)(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)(76,120,83)(77,116,84)(78,117,85)(79,118,81)(80,119,82), (6,88)(7,89)(8,90)(9,86)(10,87)(11,36)(12,37)(13,38)(14,39)(15,40)(16,66)(17,67)(18,68)(19,69)(20,70)(21,100)(22,96)(23,97)(24,98)(25,99)(26,44)(27,45)(28,41)(29,42)(30,43)(31,94)(32,95)(33,91)(34,92)(35,93)(46,77)(47,78)(48,79)(49,80)(50,76)(51,64)(52,65)(53,61)(54,62)(55,63)(71,114)(72,115)(73,111)(74,112)(75,113)(81,110)(82,106)(83,107)(84,108)(85,109)(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)], [(1,16,58,66),(2,17,59,67),(3,18,60,68),(4,19,56,69),(5,20,57,70),(6,93,43,63),(7,94,44,64),(8,95,45,65),(9,91,41,61),(10,92,42,62),(11,81,48,111),(12,82,49,112),(13,83,50,113),(14,84,46,114),(15,85,47,115),(21,120,105,100),(22,116,101,96),(23,117,102,97),(24,118,103,98),(25,119,104,99),(26,31,89,51),(27,32,90,52),(28,33,86,53),(29,34,87,54),(30,35,88,55),(36,73,79,110),(37,74,80,106),(38,75,76,107),(39,71,77,108),(40,72,78,109)], [(1,103),(2,104),(3,105),(4,101),(5,102),(6,46),(7,47),(8,48),(9,49),(10,50),(11,45),(12,41),(13,42),(14,43),(15,44),(16,118),(17,119),(18,120),(19,116),(20,117),(21,60),(22,56),(23,57),(24,58),(25,59),(26,72),(27,73),(28,74),(29,75),(30,71),(31,40),(32,36),(33,37),(34,38),(35,39),(51,78),(52,79),(53,80),(54,76),(55,77),(61,112),(62,113),(63,114),(64,115),(65,111),(66,98),(67,99),(68,100),(69,96),(70,97),(81,95),(82,91),(83,92),(84,93),(85,94),(86,106),(87,107),(88,108),(89,109),(90,110)], [(1,8,90),(2,9,86),(3,10,87),(4,6,88),(5,7,89),(11,73,24),(12,74,25),(13,75,21),(14,71,22),(15,72,23),(16,95,52),(17,91,53),(18,92,54),(19,93,55),(20,94,51),(26,57,44),(27,58,45),(28,59,41),(29,60,42),(30,56,43),(31,70,64),(32,66,65),(33,67,61),(34,68,62),(35,69,63),(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),(76,120,83),(77,116,84),(78,117,85),(79,118,81),(80,119,82)], [(6,88),(7,89),(8,90),(9,86),(10,87),(11,36),(12,37),(13,38),(14,39),(15,40),(16,66),(17,67),(18,68),(19,69),(20,70),(21,100),(22,96),(23,97),(24,98),(25,99),(26,44),(27,45),(28,41),(29,42),(30,43),(31,94),(32,95),(33,91),(34,92),(35,93),(46,77),(47,78),(48,79),(49,80),(50,76),(51,64),(52,65),(53,61),(54,62),(55,63),(71,114),(72,115),(73,111),(74,112),(75,113),(81,110),(82,106),(83,107),(84,108),(85,109),(101,116),(102,117),(103,118),(104,119),(105,120)]])

C5×D4⋊S3 is a maximal subgroup of
Dic103D6  D15⋊D8  D30.8D4  D1210D10  D12.24D10  D30.11D4  D125D10  C5×S3×D8

60 conjugacy classes

 class 1 2A 2B 2C 3 4 5A 5B 5C 5D 6A 6B 6C 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 12 15A 15B 15C 15D 20A 20B 20C 20D 30A 30B 30C 30D 30E ··· 30L 40A ··· 40H 60A 60B 60C 60D order 1 2 2 2 3 4 5 5 5 5 6 6 6 8 8 10 10 10 10 10 10 10 10 10 10 10 10 12 15 15 15 15 20 20 20 20 30 30 30 30 30 ··· 30 40 ··· 40 60 60 60 60 size 1 1 4 12 2 2 1 1 1 1 2 4 4 6 6 1 1 1 1 4 4 4 4 12 12 12 12 4 2 2 2 2 2 2 2 2 2 2 2 2 4 ··· 4 6 ··· 6 4 4 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 D8 C3⋊D4 C5×S3 C5×D4 S3×C10 C5×D8 C5×C3⋊D4 D4⋊S3 C5×D4⋊S3 kernel C5×D4⋊S3 C5×C3⋊C8 C5×D12 D4×C15 D4⋊S3 C3⋊C8 D12 C3×D4 C5×D4 C30 C20 C15 C10 D4 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×D4⋊S3 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 0 1 0 0 240 0
,
 240 0 0 0 0 240 0 0 0 0 11 230 0 0 230 230
,
 0 1 0 0 240 240 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 240 240 0 0 0 0 1 0 0 0 0 240
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,240,0,0,1,0],[240,0,0,0,0,240,0,0,0,0,11,230,0,0,230,230],[0,240,0,0,1,240,0,0,0,0,1,0,0,0,0,1],[1,240,0,0,0,240,0,0,0,0,1,0,0,0,0,240] >;

C5×D4⋊S3 in GAP, Magma, Sage, TeX

C_5\times D_4\rtimes S_3
% in TeX

G:=Group("C5xD4:S3");
// GroupNames label

G:=SmallGroup(240,60);
// by ID

G=gap.SmallGroup(240,60);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-3,265,1443,729,69,5765]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations

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