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G = D246D5order 480 = 25·3·5

6th semidirect product of D24 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D246D5, C4011D6, C248D10, D122D10, D30.1D4, D20.2D6, Dic102D6, C12019C22, Dic15.1D4, C60.141C23, C85(S3×D5), (C5×D24)⋊8C2, C40⋊C23S3, C52(Q83D6), C6.30(D4×D5), C40⋊S37C2, C20⋊D69C2, C15⋊D810C2, C32(D8⋊D5), C155(C8⋊C22), C10.30(S3×D4), C30.11(C2×D4), D12⋊D58C2, C153C819C22, C20.D610C2, C2.8(C20⋊D6), (C5×D12)⋊17C22, C20.70(C22×S3), C12.70(C22×D5), (C4×D15).30C22, (C3×D20).26C22, (C3×Dic10)⋊15C22, C4.114(C2×S3×D5), (C3×C40⋊C2)⋊7C2, SmallGroup(480,333)

Series: Derived Chief Lower central Upper central

C1C60 — D246D5
C1C5C15C30C60C3×D20C20⋊D6 — D246D5
C15C30C60 — D246D5
C1C2C4C8

Generators and relations for D246D5
 G = < a,b,c,d | a24=b2=c5=d2=1, bab=a-1, ac=ca, dad=a19, bc=cb, dbd=a18b, dcd=c-1 >

Subgroups: 940 in 136 conjugacy classes, 38 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, D10, C2×C10, C3⋊C8, C24, C4×S3, D12, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3, C3×D5, D15, C30, C8⋊C22, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C5×D4, C22×D5, C8⋊S3, D24, D4⋊S3, Q82S3, C3×SD16, S3×D4, Q83S3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D42D5, Q83D6, C153C8, C120, S3×Dic5, C15⋊D4, C5⋊D12, C3×Dic10, C3×D20, C5×D12, C4×D15, C2×S3×D5, D8⋊D5, C15⋊D8, C20.D6, C3×C40⋊C2, C5×D24, C40⋊S3, D12⋊D5, C20⋊D6, D246D5
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C8⋊C22, C22×D5, S3×D4, S3×D5, D4×D5, Q83D6, C2×S3×D5, D8⋊D5, C20⋊D6, D246D5

Smallest permutation representation of D246D5
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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(25 26)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 40)(36 39)(37 38)(49 68)(50 67)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 60)(58 59)(69 72)(70 71)(73 96)(74 95)(75 94)(76 93)(77 92)(78 91)(79 90)(80 89)(81 88)(82 87)(83 86)(84 85)(97 104)(98 103)(99 102)(100 101)(105 120)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)
(1 85 101 59 38)(2 86 102 60 39)(3 87 103 61 40)(4 88 104 62 41)(5 89 105 63 42)(6 90 106 64 43)(7 91 107 65 44)(8 92 108 66 45)(9 93 109 67 46)(10 94 110 68 47)(11 95 111 69 48)(12 96 112 70 25)(13 73 113 71 26)(14 74 114 72 27)(15 75 115 49 28)(16 76 116 50 29)(17 77 117 51 30)(18 78 118 52 31)(19 79 119 53 32)(20 80 120 54 33)(21 81 97 55 34)(22 82 98 56 35)(23 83 99 57 36)(24 84 100 58 37)
(1 38)(2 33)(3 28)(4 47)(5 42)(6 37)(7 32)(8 27)(9 46)(10 41)(11 36)(12 31)(13 26)(14 45)(15 40)(16 35)(17 30)(18 25)(19 44)(20 39)(21 34)(22 29)(23 48)(24 43)(49 87)(50 82)(51 77)(52 96)(53 91)(54 86)(55 81)(56 76)(57 95)(58 90)(59 85)(60 80)(61 75)(62 94)(63 89)(64 84)(65 79)(66 74)(67 93)(68 88)(69 83)(70 78)(71 73)(72 92)(98 116)(99 111)(100 106)(102 120)(103 115)(104 110)(107 119)(108 114)(112 118)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B8A8B10A10B10C10D10E10F12A12B15A15B20A20B24A24B30A30B40A40B40C40D60A60B60C60D120A···120H
order1222223444556688101010101010121215152020242430304040404060606060120···120
size111212203022203022240460222424242444044444444444444444···4

48 irreducible representations

dim11111111222222222444444444
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10C8⋊C22S3×D4S3×D5D4×D5Q83D6C2×S3×D5D8⋊D5C20⋊D6D246D5
kernelD246D5C15⋊D8C20.D6C3×C40⋊C2C5×D24C40⋊S3D12⋊D5C20⋊D6C40⋊C2Dic15D30D24C40Dic10D20C24D12C15C10C8C6C5C4C3C2C1
# reps11111111111211124112222448

Matrix representation of D246D5 in GL4(𝔽241) generated by

23203184
023257229
2385722957
184121843
,
23203184
023257229
1218490
5723809
,
0100
2405100
0001
0024051
,
0100
1000
0001
0010
G:=sub<GL(4,GF(241))| [232,0,238,184,0,232,57,12,3,57,229,184,184,229,57,3],[232,0,12,57,0,232,184,238,3,57,9,0,184,229,0,9],[0,240,0,0,1,51,0,0,0,0,0,240,0,0,1,51],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

D246D5 in GAP, Magma, Sage, TeX

D_{24}\rtimes_6D_5
% in TeX

G:=Group("D24:6D5");
// GroupNames label

G:=SmallGroup(480,333);
// by ID

G=gap.SmallGroup(480,333);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,303,58,675,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^24=b^2=c^5=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^19,b*c=c*b,d*b*d=a^18*b,d*c*d=c^-1>;
// generators/relations

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