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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 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×3], C6, C6, C8, C8, C2×C4 [×2], D4 [×5], Q8, C23, D5 [×2], C10, C10 [×2], Dic3, C12, C12, D6 [×5], C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5 [×2], C20, D10 [×4], C2×C10 [×2], C3⋊C8, C24, C4×S3 [×2], D12 [×2], D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3 [×2], C3×D5, D15, C30, C8⋊C22, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4 [×2], C5×D4 [×2], C22×D5, C8⋊S3, D24, D4⋊S3, Q82S3, C3×SD16, S3×D4, Q83S3, C3×Dic5, Dic15, C60, S3×D5 [×2], C6×D5, S3×C10 [×2], 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 [×2], 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 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], 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 42)(26 41)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(43 48)(44 47)(45 46)(49 54)(50 53)(51 52)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)(73 86)(74 85)(75 84)(76 83)(77 82)(78 81)(79 80)(87 96)(88 95)(89 94)(90 93)(91 92)(97 108)(98 107)(99 106)(100 105)(101 104)(102 103)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)
(1 52 115 46 92)(2 53 116 47 93)(3 54 117 48 94)(4 55 118 25 95)(5 56 119 26 96)(6 57 120 27 73)(7 58 97 28 74)(8 59 98 29 75)(9 60 99 30 76)(10 61 100 31 77)(11 62 101 32 78)(12 63 102 33 79)(13 64 103 34 80)(14 65 104 35 81)(15 66 105 36 82)(16 67 106 37 83)(17 68 107 38 84)(18 69 108 39 85)(19 70 109 40 86)(20 71 110 41 87)(21 72 111 42 88)(22 49 112 43 89)(23 50 113 44 90)(24 51 114 45 91)
(1 92)(2 87)(3 82)(4 77)(5 96)(6 91)(7 86)(8 81)(9 76)(10 95)(11 90)(12 85)(13 80)(14 75)(15 94)(16 89)(17 84)(18 79)(19 74)(20 93)(21 88)(22 83)(23 78)(24 73)(25 61)(26 56)(27 51)(28 70)(29 65)(30 60)(31 55)(32 50)(33 69)(34 64)(35 59)(36 54)(37 49)(38 68)(39 63)(40 58)(41 53)(42 72)(43 67)(44 62)(45 57)(46 52)(47 71)(48 66)(97 109)(98 104)(100 118)(101 113)(102 108)(105 117)(106 112)(110 116)(114 120)

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

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

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

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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