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G = C24⋊D10order 480 = 25·3·5

1st semidirect product of C24 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C407D6, C241D10, D12014C2, C1207C22, Dic68D10, D10.12D12, D12.19D10, D6019C22, C60.118C23, Dic5.14D12, C81(S3×D5), C52C81D6, C6.3(D4×D5), (D5×D12)⋊9C2, C8⋊D51S3, C24⋊C21D5, C51(C8⋊D6), (C6×D5).1D4, (C4×D5).1D6, C30.3(C2×D4), C2.8(D5×D12), C5⋊D2410C2, C31(D40⋊C2), C151(C8⋊C22), C10.3(C2×D12), C12.28D108C2, (C3×Dic5).1D4, Dic6⋊D510C2, C20.66(C22×S3), (C5×Dic6)⋊13C22, (D5×C12).24C22, (C5×D12).21C22, C12.141(C22×D5), C4.66(C2×S3×D5), (C5×C24⋊C2)⋊1C2, (C3×C8⋊D5)⋊1C2, (C3×C52C8)⋊15C22, SmallGroup(480,325)

Series: Derived Chief Lower central Upper central

C1C60 — C24⋊D10
C1C5C15C30C60D5×C12D5×D12 — C24⋊D10
C15C30C60 — C24⋊D10
C1C2C4C8

Generators and relations for C24⋊D10
 G = < a,b,c | a24=b10=c2=1, bab-1=a11, cac=a-1, cbc=b-1 >

Subgroups: 1052 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×3], C6, C6, C8, C8, C2×C4 [×2], D4 [×5], Q8, C23, D5 [×3], C10, C10, Dic3, C12, C12, D6 [×5], C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20, C20, D10, D10 [×4], C2×C10, C24, C24, Dic6, C4×S3, D12, D12 [×3], C3⋊D4, C2×C12, C22×S3, C5×S3, C3×D5, D15 [×2], C30, C8⋊C22, C52C8, C40, C4×D5, C4×D5, D20 [×3], C5⋊D4, C5×D4, C5×Q8, C22×D5, C24⋊C2, C24⋊C2, D24 [×2], C3×M4(2), C2×D12, C4○D12, C5×Dic3, C3×Dic5, C60, S3×D5 [×2], C6×D5, S3×C10, D30 [×2], C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q82D5, C8⋊D6, C3×C52C8, C120, D30.C2, C3⋊D20, C5⋊D12, D5×C12, C5×Dic6, C5×D12, D60 [×2], C2×S3×D5, D40⋊C2, C5⋊D24, Dic6⋊D5, C3×C8⋊D5, C5×C24⋊C2, D120, C12.28D10, D5×D12, C24⋊D10
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], D12 [×2], C22×S3, C8⋊C22, C22×D5, C2×D12, S3×D5, D4×D5, C8⋊D6, C2×S3×D5, D40⋊C2, D5×D12, C24⋊D10

Smallest permutation representation of C24⋊D10
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 39 84 105 51 7 33 90 99 57)(2 26 85 116 52 18 34 77 100 68)(3 37 86 103 53 5 35 88 101 55)(4 48 87 114 54 16 36 75 102 66)(6 46 89 112 56 14 38 73 104 64)(8 44 91 110 58 12 40 95 106 62)(9 31 92 97 59 23 41 82 107 49)(10 42 93 108 60)(11 29 94 119 61 21 43 80 109 71)(13 27 96 117 63 19 45 78 111 69)(15 25 74 115 65 17 47 76 113 67)(20 32 79 98 70 24 28 83 118 50)(22 30 81 120 72)
(1 57)(2 56)(3 55)(4 54)(5 53)(6 52)(7 51)(8 50)(9 49)(10 72)(11 71)(12 70)(13 69)(14 68)(15 67)(16 66)(17 65)(18 64)(19 63)(20 62)(21 61)(22 60)(23 59)(24 58)(25 113)(26 112)(27 111)(28 110)(29 109)(30 108)(31 107)(32 106)(33 105)(34 104)(35 103)(36 102)(37 101)(38 100)(39 99)(40 98)(41 97)(42 120)(43 119)(44 118)(45 117)(46 116)(47 115)(48 114)(73 77)(74 76)(78 96)(79 95)(80 94)(81 93)(82 92)(83 91)(84 90)(85 89)(86 88)

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

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

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

51 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B8A8B10A10B10C10D12A12B12C15A15B20A20B20C20D24A24B24C24D30A30B40A40B40C40D60A60B60C60D120A···120H
order1222223444556688101010101212121515202020202424242430304040404060606060120···120
size11101260602210122222042022242422204444242444202044444444444···4

51 irreducible representations

dim1111111122222222222244444444
type++++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10D10D12D12C8⋊C22S3×D5D4×D5C8⋊D6C2×S3×D5D40⋊C2D5×D12C24⋊D10
kernelC24⋊D10C5⋊D24Dic6⋊D5C3×C8⋊D5C5×C24⋊C2D120C12.28D10D5×D12C8⋊D5C3×Dic5C6×D5C24⋊C2C52C8C40C4×D5C24Dic6D12Dic5D10C15C8C6C5C4C3C2C1
# reps1111111111121112222212222448

Matrix representation of C24⋊D10 in GL6(𝔽241)

24000000
02400000
0014310022
00141432390
001707198141
0017099100198
,
01890000
511900000
009919800
009914200
00141432400
0014310011
,
511890000
501900000
009919800
009914200
000010
0000240240

G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,143,141,170,170,0,0,100,43,71,99,0,0,2,239,98,100,0,0,2,0,141,198],[0,51,0,0,0,0,189,190,0,0,0,0,0,0,99,99,141,143,0,0,198,142,43,100,0,0,0,0,240,1,0,0,0,0,0,1],[51,50,0,0,0,0,189,190,0,0,0,0,0,0,99,99,0,0,0,0,198,142,0,0,0,0,0,0,1,240,0,0,0,0,0,240] >;

C24⋊D10 in GAP, Magma, Sage, TeX

C_{24}\rtimes D_{10}
% in TeX

G:=Group("C24:D10");
// GroupNames label

G:=SmallGroup(480,325);
// by ID

G=gap.SmallGroup(480,325);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,135,58,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^24=b^10=c^2=1,b*a*b^-1=a^11,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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