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G = D5.D24order 480 = 25·3·5

The non-split extension by D5 of D24 acting via D24/C24=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C241F5, C1201C4, D5.1D24, C401Dic3, D10.9D12, D5.1Dic12, Dic5.9Dic6, C81(C3⋊F5), C5⋊(C241C4), C31(D5.D8), (C3×D5).4D8, (C8×D5).3S3, C152(C2.D8), C6.2(C4⋊F5), C30.9(C4⋊C4), C60.48(C2×C4), (C4×D5).87D6, (C6×D5).52D4, (D5×C24).5C2, C52C86Dic3, C12.48(C2×F5), (C3×D5).4Q16, C60⋊C4.8C2, C20.9(C2×Dic3), C2.5(C60⋊C4), C10.2(C4⋊Dic3), (C3×Dic5).10Q8, (D5×C12).112C22, C4.9(C2×C3⋊F5), (C3×C52C8)⋊10C4, SmallGroup(480,299)

Series: Derived Chief Lower central Upper central

C1C60 — D5.D24
C1C5C15C30C6×D5D5×C12C60⋊C4 — D5.D24
C15C30C60 — D5.D24
C1C2C4C8

Generators and relations for D5.D24
 G = < a,b,c,d | a5=b2=c24=1, d2=a-1b, bab=a-1, ac=ca, dad-1=a2, bc=cb, dbd-1=ab, dcd-1=c-1 >

Subgroups: 460 in 72 conjugacy classes, 33 normal (31 characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×3], C22, C5, C6, C6 [×2], C8, C8, C2×C4 [×3], D5 [×2], C10, Dic3 [×2], C12, C12, C2×C6, C15, C4⋊C4 [×2], C2×C8, Dic5, C20, F5 [×2], D10, C24, C24, C2×Dic3 [×2], C2×C12, C3×D5 [×2], C30, C2.D8, C52C8, C40, C4×D5, C2×F5 [×2], C4⋊Dic3 [×2], C2×C24, C3×Dic5, C60, C3⋊F5 [×2], C6×D5, C8×D5, C4⋊F5 [×2], C241C4, C3×C52C8, C120, D5×C12, C2×C3⋊F5 [×2], D5.D8, D5×C24, C60⋊C4 [×2], D5.D24
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4, Q8, Dic3 [×2], D6, C4⋊C4, D8, Q16, F5, Dic6, D12, C2×Dic3, C2.D8, C2×F5, D24, Dic12, C4⋊Dic3, C3⋊F5, C4⋊F5, C241C4, C2×C3⋊F5, D5.D8, C60⋊C4, D5.D24

Smallest permutation representation of D5.D24
On 120 points
Generators in S120
(1 84 107 27 57)(2 85 108 28 58)(3 86 109 29 59)(4 87 110 30 60)(5 88 111 31 61)(6 89 112 32 62)(7 90 113 33 63)(8 91 114 34 64)(9 92 115 35 65)(10 93 116 36 66)(11 94 117 37 67)(12 95 118 38 68)(13 96 119 39 69)(14 73 120 40 70)(15 74 97 41 71)(16 75 98 42 72)(17 76 99 43 49)(18 77 100 44 50)(19 78 101 45 51)(20 79 102 46 52)(21 80 103 47 53)(22 81 104 48 54)(23 82 105 25 55)(24 83 106 26 56)
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 82)(26 83)(27 84)(28 85)(29 86)(30 87)(31 88)(32 89)(33 90)(34 91)(35 92)(36 93)(37 94)(38 95)(39 96)(40 73)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 81)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 24)(17 23)(18 22)(19 21)(25 49 105 76)(26 72 106 75)(27 71 107 74)(28 70 108 73)(29 69 109 96)(30 68 110 95)(31 67 111 94)(32 66 112 93)(33 65 113 92)(34 64 114 91)(35 63 115 90)(36 62 116 89)(37 61 117 88)(38 60 118 87)(39 59 119 86)(40 58 120 85)(41 57 97 84)(42 56 98 83)(43 55 99 82)(44 54 100 81)(45 53 101 80)(46 52 102 79)(47 51 103 78)(48 50 104 77)

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

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

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

54 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F 5 6A6B6C8A8B8C8D 10 12A12B12C12D15A15B20A20B24A24B24C24D24E24F24G24H30A30B40A40B40C40D60A60B60C60D120A···120H
order1222344444456668888101212121215152020242424242424242430304040404060606060120···120
size11552210606060604210102210104221010444422221010101044444444444···4

54 irreducible representations

dim1111122222222222244444444
type++++-+--++--++-++
imageC1C2C2C4C4S3Q8D4Dic3Dic3D6D8Q16Dic6D12D24Dic12F5C2×F5C3⋊F5C4⋊F5C2×C3⋊F5D5.D8C60⋊C4D5.D24
kernelD5.D24D5×C24C60⋊C4C3×C52C8C120C8×D5C3×Dic5C6×D5C52C8C40C4×D5C3×D5C3×D5Dic5D10D5D5C24C12C8C6C4C3C2C1
# reps1122211111122224411222448

Matrix representation of D5.D24 in GL8(𝔽241)

10000000
01000000
00100000
00010000
00000100
00000010
00000001
0000240240240240
,
2400000000
0240000000
0024000000
0002400000
00000100
00001000
0000240240240240
00000001
,
145175000000
10474000000
0024010000
0024000000
00002240207207
00003417340
00000341734
00002072070224
,
144143000000
3797000000
0017700000
00177640000
0000240000
0000002400
00001111
0000024000

G:=sub<GL(8,GF(241))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,1,0,0,240,0,0,0,0,0,1,0,240,0,0,0,0,0,0,1,240],[240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,1,240,0,0,0,0,0,1,0,240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,240,1],[145,104,0,0,0,0,0,0,175,74,0,0,0,0,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,224,34,0,207,0,0,0,0,0,17,34,207,0,0,0,0,207,34,17,0,0,0,0,0,207,0,34,224],[144,37,0,0,0,0,0,0,143,97,0,0,0,0,0,0,0,0,177,177,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,240,0,1,0,0,0,0,0,0,0,1,240,0,0,0,0,0,240,1,0,0,0,0,0,0,0,1,0] >;

D5.D24 in GAP, Magma, Sage, TeX

D_5.D_{24}
% in TeX

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

G:=SmallGroup(480,299);
// by ID

G=gap.SmallGroup(480,299);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,176,675,80,2693,14118,4724]);
// Polycyclic

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

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