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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×Dic6 — 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=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=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 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 66)(2 67)(3 68)(4 69)(5 70)(6 52)(7 53)(8 54)(9 55)(10 51)(16 86)(17 87)(18 88)(19 89)(20 90)(21 93)(22 94)(23 95)(24 91)(25 92)(26 58)(27 59)(28 60)(29 56)(30 57)(31 63)(32 64)(33 65)(34 61)(35 62)(41 79)(42 80)(43 76)(44 77)(45 78)(81 118)(82 119)(83 120)(84 116)(85 117)(96 114)(97 115)(98 111)(99 112)(100 113)
(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 36 73)(12 37 74)(13 38 75)(14 39 71)(15 40 72)(16 67 61)(17 68 62)(18 69 63)(19 70 64)(20 66 65)(21 113 83)(22 114 84)(23 115 85)(24 111 81)(25 112 82)(26 78 52)(27 79 53)(28 80 54)(29 76 55)(30 77 51)(46 101 108)(47 102 109)(48 103 110)(49 104 106)(50 105 107)(91 98 118)(92 99 119)(93 100 120)(94 96 116)(95 97 117)
(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)

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

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

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

C5×D4.S3 is a maximal subgroup of
C60.8C23  Dic10⋊D6  D30.9D4  D20.9D6  C60.16C23  D20.10D6  Dic6⋊D10  C5×S3×SD16

60 conjugacy classes

 class 1 2A 2B 3 4A 4B 5A 5B 5C 5D 6A 6B 6C 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 12 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 20G 20H 30A 30B 30C 30D 30E ··· 30L 40A ··· 40H 60A 60B 60C 60D order 1 2 2 3 4 4 5 5 5 5 6 6 6 8 8 10 10 10 10 10 10 10 10 12 15 15 15 15 20 20 20 20 20 20 20 20 30 30 30 30 30 ··· 30 40 ··· 40 60 60 60 60 size 1 1 4 2 2 12 1 1 1 1 2 4 4 6 6 1 1 1 1 4 4 4 4 4 2 2 2 2 2 2 2 2 12 12 12 12 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 SD16 C3⋊D4 C5×S3 C5×D4 S3×C10 C5×SD16 C5×C3⋊D4 D4.S3 C5×D4.S3 kernel C5×D4.S3 C5×C3⋊C8 C5×Dic6 D4×C15 D4.S3 C3⋊C8 Dic6 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

 1 0 0 0 0 1 0 0 0 0 91 0 0 0 0 91
,
 1 227 0 0 69 240 0 0 0 0 1 0 0 0 0 1
,
 1 227 0 0 0 240 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 240 240
,
 0 25 0 0 106 0 0 0 0 0 109 221 0 0 112 132
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,91,0,0,0,0,91],[1,69,0,0,227,240,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,227,240,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,240,0,0,1,240],[0,106,0,0,25,0,0,0,0,0,109,112,0,0,221,132] >;

C5×D4.S3 in GAP, Magma, Sage, TeX

C_5\times D_4.S_3
% in TeX

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

G:=SmallGroup(240,61);
// by ID

G=gap.SmallGroup(240,61);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-3,240,265,1443,729,69,5765]);
// Polycyclic

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

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