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G = C5xD4:S3order 240 = 24·3·5

Direct product of C5 and D4:S3

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

Aliases: C5xD4:S3, C15:9D8, D12:2C10, C20.36D6, C30.47D4, C60.43C22, D4:(C5xS3), C3:2(C5xD8), C3:C8:1C10, (C5xD4):4S3, C6.7(C5xD4), (C3xD4):1C10, (C5xD12):8C2, (D4xC15):7C2, C4.1(S3xC10), C12.1(C2xC10), C10.23(C3:D4), (C5xC3:C8):8C2, C2.4(C5xC3:D4), SmallGroup(240,60)

Series: Derived Chief Lower central Upper central

C1C12 — C5xD4:S3
C1C3C6C12C60C5xD12 — C5xD4:S3
C3C6C12 — C5xD4:S3
C1C10C20C5xD4

Generators and relations for C5xD4: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 >

Subgroups: 112 in 44 conjugacy classes, 22 normal (all characteristic)
Quotients: C1, C2, C22, C5, S3, D4, C10, D6, D8, C2xC10, C3:D4, C5xS3, C5xD4, D4:S3, S3xC10, C5xD8, C5xC3:D4, C5xD4:S3
4C2
12C2
2C22
6C22
4C6
4S3
4C10
12C10
3D4
3C8
2C2xC6
2D6
2C2xC10
6C2xC10
4C30
4C5xS3
3D8
3C5xD4
3C40
2S3xC10
2C2xC30
3C5xD8

Smallest permutation representation of C5xD4: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)]])

C5xD4:S3 is a maximal subgroup of
Dic10:3D6  D15:D8  D30.8D4  D12:10D10  D12.24D10  D30.11D4  D12:5D10  C5xS3xD8

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:D4C5xS3C5xD4S3xC10C5xD8C5xC3:D4D4:S3C5xD4:S3
kernelC5xD4:S3C5xC3:C8C5xD12D4xC15D4:S3C3:C8D12C3xD4C5xD4C30C20C15C10D4C6C4C3C2C5C1
# reps11114444111224448814

Matrix representation of C5xD4: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] >;

C5xD4: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 C5xD4:S3 in TeX

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