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G = C3×C8⋊D5order 240 = 24·3·5

Direct product of C3 and C8⋊D5

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

Aliases: C3×C8⋊D5, C404C6, C247D5, C12010C2, D10.1C12, C12.57D10, C1511M4(2), C60.70C22, Dic5.1C12, C83(C3×D5), C52C84C6, (C6×D5).3C4, (C4×D5).2C6, C2.3(D5×C12), C4.13(C6×D5), C6.16(C4×D5), C53(C3×M4(2)), C30.41(C2×C4), C10.9(C2×C12), C20.14(C2×C6), (D5×C12).7C2, (C3×Dic5).3C4, (C3×C52C8)⋊11C2, SmallGroup(240,34)

Series: Derived Chief Lower central Upper central

C1C10 — C3×C8⋊D5
C1C5C10C20C60D5×C12 — C3×C8⋊D5
C5C10 — C3×C8⋊D5
C1C12C24

Generators and relations for C3×C8⋊D5
 G = < a,b,c,d | a3=b8=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

10C2
5C22
5C4
10C6
2D5
5C2×C4
5C8
5C12
5C2×C6
2C3×D5
5M4(2)
5C24
5C2×C12
5C3×M4(2)

Smallest permutation representation of C3×C8⋊D5
On 120 points
Generators in S120
(1 100 60)(2 101 61)(3 102 62)(4 103 63)(5 104 64)(6 97 57)(7 98 58)(8 99 59)(9 86 51)(10 87 52)(11 88 53)(12 81 54)(13 82 55)(14 83 56)(15 84 49)(16 85 50)(17 94 119)(18 95 120)(19 96 113)(20 89 114)(21 90 115)(22 91 116)(23 92 117)(24 93 118)(25 45 66)(26 46 67)(27 47 68)(28 48 69)(29 41 70)(30 42 71)(31 43 72)(32 44 65)(33 108 74)(34 109 75)(35 110 76)(36 111 77)(37 112 78)(38 105 79)(39 106 80)(40 107 73)
(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 53 23 46 109)(2 54 24 47 110)(3 55 17 48 111)(4 56 18 41 112)(5 49 19 42 105)(6 50 20 43 106)(7 51 21 44 107)(8 52 22 45 108)(9 90 65 73 98)(10 91 66 74 99)(11 92 67 75 100)(12 93 68 76 101)(13 94 69 77 102)(14 95 70 78 103)(15 96 71 79 104)(16 89 72 80 97)(25 33 59 87 116)(26 34 60 88 117)(27 35 61 81 118)(28 36 62 82 119)(29 37 63 83 120)(30 38 64 84 113)(31 39 57 85 114)(32 40 58 86 115)
(1 109)(2 106)(3 111)(4 108)(5 105)(6 110)(7 107)(8 112)(9 65)(10 70)(11 67)(12 72)(13 69)(14 66)(15 71)(16 68)(18 22)(20 24)(25 83)(26 88)(27 85)(28 82)(29 87)(30 84)(31 81)(32 86)(33 63)(34 60)(35 57)(36 62)(37 59)(38 64)(39 61)(40 58)(41 52)(42 49)(43 54)(44 51)(45 56)(46 53)(47 50)(48 55)(73 98)(74 103)(75 100)(76 97)(77 102)(78 99)(79 104)(80 101)(89 93)(91 95)(114 118)(116 120)

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

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

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

C3×C8⋊D5 is a maximal subgroup of
C40⋊D6  C24⋊D10  D24⋊D5  Dic60⋊C2  C24.2D10  C40.55D6  C40.35D6  C3×D5×M4(2)

78 conjugacy classes

class 1 2A2B3A3B4A4B4C5A5B6A6B6C6D8A8B8C8D10A10B12A12B12C12D12E12F15A15B15C15D20A20B20C20D24A24B24C24D24E24F24G24H30A30B30C30D40A···40H60A···60H120A···120P
order1223344455666688881010121212121212151515152020202024242424242424243030303040···4060···60120···120
size11101111102211101022101022111110102222222222221010101022222···22···22···2

78 irreducible representations

dim1111111111112222222222
type++++++
imageC1C2C2C2C3C4C4C6C6C6C12C12D5M4(2)D10C3×D5C4×D5C3×M4(2)C6×D5C8⋊D5D5×C12C3×C8⋊D5
kernelC3×C8⋊D5C3×C52C8C120D5×C12C8⋊D5C3×Dic5C6×D5C52C8C40C4×D5Dic5D10C24C15C12C8C6C5C4C3C2C1
# reps11112222224422244448816

Matrix representation of C3×C8⋊D5 in GL3(𝔽241) generated by

22500
010
001
,
100
0216140
010125
,
100
00240
01189
,
24000
052240
052189
G:=sub<GL(3,GF(241))| [225,0,0,0,1,0,0,0,1],[1,0,0,0,216,101,0,140,25],[1,0,0,0,0,1,0,240,189],[240,0,0,0,52,52,0,240,189] >;

C3×C8⋊D5 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes D_5
% in TeX

G:=Group("C3xC8:D5");
// GroupNames label

G:=SmallGroup(240,34);
// by ID

G=gap.SmallGroup(240,34);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,313,79,69,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations

Export

Subgroup lattice of C3×C8⋊D5 in TeX

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