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G = C5×D4⋊S3order 240 = 24·3·5

Direct product of C5 and D4⋊S3

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5×D4⋊S3, C159D8, D122C10, C20.36D6, C30.47D4, C60.43C22, D4⋊(C5×S3), C32(C5×D8), C3⋊C81C10, (C5×D4)⋊4S3, C6.7(C5×D4), (C3×D4)⋊1C10, (C5×D12)⋊8C2, (D4×C15)⋊7C2, C4.1(S3×C10), C12.1(C2×C10), C10.23(C3⋊D4), (C5×C3⋊C8)⋊8C2, C2.4(C5×C3⋊D4), SmallGroup(240,60)

Series: Derived Chief Lower central Upper central

C1C12 — C5×D4⋊S3
C1C3C6C12C60C5×D12 — C5×D4⋊S3
C3C6C12 — C5×D4⋊S3
C1C10C20C5×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 >

4C2
12C2
2C22
6C22
4C6
4S3
4C10
12C10
3D4
3C8
2C2×C6
2D6
2C2×C10
6C2×C10
4C30
4C5×S3
3D8
3C5×D4
3C40
2S3×C10
2C2×C30
3C5×D8

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)
(1 103)(2 104)(3 105)(4 101)(5 102)(6 73)(7 74)(8 75)(9 71)(10 72)(11 45)(12 41)(13 42)(14 43)(15 44)(16 25)(17 21)(18 22)(19 23)(20 24)(26 118)(27 119)(28 120)(29 116)(30 117)(31 46)(32 47)(33 48)(34 49)(35 50)(36 58)(37 59)(38 60)(39 56)(40 57)(51 95)(52 91)(53 92)(54 93)(55 94)(61 112)(62 113)(63 114)(64 115)(65 111)(66 81)(67 82)(68 83)(69 84)(70 85)(76 96)(77 97)(78 98)(79 99)(80 100)(86 106)(87 107)(88 108)(89 109)(90 110)
(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), (1,103)(2,104)(3,105)(4,101)(5,102)(6,73)(7,74)(8,75)(9,71)(10,72)(11,45)(12,41)(13,42)(14,43)(15,44)(16,25)(17,21)(18,22)(19,23)(20,24)(26,118)(27,119)(28,120)(29,116)(30,117)(31,46)(32,47)(33,48)(34,49)(35,50)(36,58)(37,59)(38,60)(39,56)(40,57)(51,95)(52,91)(53,92)(54,93)(55,94)(61,112)(62,113)(63,114)(64,115)(65,111)(66,81)(67,82)(68,83)(69,84)(70,85)(76,96)(77,97)(78,98)(79,99)(80,100)(86,106)(87,107)(88,108)(89,109)(90,110), (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), (1,103)(2,104)(3,105)(4,101)(5,102)(6,73)(7,74)(8,75)(9,71)(10,72)(11,45)(12,41)(13,42)(14,43)(15,44)(16,25)(17,21)(18,22)(19,23)(20,24)(26,118)(27,119)(28,120)(29,116)(30,117)(31,46)(32,47)(33,48)(34,49)(35,50)(36,58)(37,59)(38,60)(39,56)(40,57)(51,95)(52,91)(53,92)(54,93)(55,94)(61,112)(62,113)(63,114)(64,115)(65,111)(66,81)(67,82)(68,83)(69,84)(70,85)(76,96)(77,97)(78,98)(79,99)(80,100)(86,106)(87,107)(88,108)(89,109)(90,110), (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)], [(1,103),(2,104),(3,105),(4,101),(5,102),(6,73),(7,74),(8,75),(9,71),(10,72),(11,45),(12,41),(13,42),(14,43),(15,44),(16,25),(17,21),(18,22),(19,23),(20,24),(26,118),(27,119),(28,120),(29,116),(30,117),(31,46),(32,47),(33,48),(34,49),(35,50),(36,58),(37,59),(38,60),(39,56),(40,57),(51,95),(52,91),(53,92),(54,93),(55,94),(61,112),(62,113),(63,114),(64,115),(65,111),(66,81),(67,82),(68,83),(69,84),(70,85),(76,96),(77,97),(78,98),(79,99),(80,100),(86,106),(87,107),(88,108),(89,109),(90,110)], [(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×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 2A2B2C 3  4 5A5B5C5D6A6B6C8A8B10A10B10C10D10E10F10G10H10I10J10K10L 12 15A15B15C15D20A20B20C20D30A30B30C30D30E···30L40A···40H60A60B60C60D
order1222345555666881010101010101010101010101215151515202020203030303030···3040···4060606060
size1141222111124466111144441212121242222222222224···46···64444

60 irreducible representations

dim11111111222222222244
type+++++++++
imageC1C2C2C2C5C10C10C10S3D4D6D8C3⋊D4C5×S3C5×D4S3×C10C5×D8C5×C3⋊D4D4⋊S3C5×D4⋊S3
kernelC5×D4⋊S3C5×C3⋊C8C5×D12D4×C15D4⋊S3C3⋊C8D12C3×D4C5×D4C30C20C15C10D4C6C4C3C2C5C1
# reps11114444111224448814

Matrix representation of C5×D4⋊S3 in GL4(𝔽241) generated by

91000
09100
0010
0001
,
1000
0100
0001
002400
,
240000
024000
0011230
00230230
,
0100
24024000
0010
0001
,
1000
24024000
0010
000240
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

Export

Subgroup lattice of C5×D4⋊S3 in TeX

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