Copied to
clipboard

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, C30.48D4, C20.37D6, C1512SD16, Dic62C10, C60.44C22, C3⋊C82C10, D4.(C5×S3), C6.8(C5×D4), C32(C5×SD16), C4.2(S3×C10), (C5×D4).2S3, C12.2(C2×C10), (C5×Dic6)⋊8C2, (D4×C15).3C2, (C3×D4).1C10, C10.24(C3⋊D4), (C5×C3⋊C8)⋊9C2, C2.5(C5×C3⋊D4), SmallGroup(240,61)

Series: Derived Chief Lower central Upper central

C1C12 — C5×D4.S3
C1C3C6C12C60C5×Dic6 — 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=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 >

4C2
2C22
6C4
4C6
4C10
3Q8
3C8
2Dic3
2C2×C6
2C2×C10
6C20
4C30
3SD16
3C5×Q8
3C40
2C2×C30
2C5×Dic3
3C5×SD16

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 2A2B 3 4A4B5A5B5C5D6A6B6C8A8B10A10B10C10D10E10F10G10H 12 15A15B15C15D20A20B20C20D20E20F20G20H30A30B30C30D30E···30L40A···40H60A60B60C60D
order1223445555666881010101010101010121515151520202020202020203030303030···3040···4060606060
size1142212111124466111144444222222221212121222224···46···64444

60 irreducible representations

dim11111111222222222244
type+++++++-
imageC1C2C2C2C5C10C10C10S3D4D6SD16C3⋊D4C5×S3C5×D4S3×C10C5×SD16C5×C3⋊D4D4.S3C5×D4.S3
kernelC5×D4.S3C5×C3⋊C8C5×Dic6D4×C15D4.S3C3⋊C8Dic6C3×D4C5×D4C30C20C15C10D4C6C4C3C2C5C1
# reps11114444111224448814

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

1000
0100
00910
00091
,
122700
6924000
0010
0001
,
122700
024000
0010
0001
,
1000
0100
0001
00240240
,
02500
106000
00109221
00112132
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

Export

Subgroup lattice of C5×D4.S3 in TeX

׿
×
𝔽