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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, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, C24, C24, Dic6, C4×S3, D12, D12, C3⋊D4, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C8⋊C22, C52C8, C40, C4×D5, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C24⋊C2, C24⋊C2, D24, C3×M4(2), C2×D12, C4○D12, C5×Dic3, C3×Dic5, C60, S3×D5, C6×D5, S3×C10, D30, 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, 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, C22, S3, D4, C23, D5, D6, C2×D4, D10, D12, 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)
(1 119 73 64 33 7 113 79 58 39)(2 106 74 51 34 18 114 90 59 26)(3 117 75 62 35 5 115 77 60 37)(4 104 76 49 36 16 116 88 61 48)(6 102 78 71 38 14 118 86 63 46)(8 100 80 69 40 12 120 84 65 44)(9 111 81 56 41 23 97 95 66 31)(10 98 82 67 42)(11 109 83 54 43 21 99 93 68 29)(13 107 85 52 45 19 101 91 70 27)(15 105 87 50 47 17 103 89 72 25)(20 112 92 57 28 24 108 96 53 32)(22 110 94 55 30)
(1 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)(49 104)(50 103)(51 102)(52 101)(53 100)(54 99)(55 98)(56 97)(57 120)(58 119)(59 118)(60 117)(61 116)(62 115)(63 114)(64 113)(65 112)(66 111)(67 110)(68 109)(69 108)(70 107)(71 106)(72 105)(73 79)(74 78)(75 77)(80 96)(81 95)(82 94)(83 93)(84 92)(85 91)(86 90)(87 89)

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

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

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

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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