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G = C8⋊D30order 480 = 25·3·5

1st semidirect product of C8 and D30 acting via D30/C15=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C81D30, C403D6, C243D10, D1202C2, C4.14D60, C60.12D4, C1201C22, C12.24D20, C20.24D12, C22.5D60, D6021C22, M4(2)⋊1D15, C60.248C23, Dic3019C22, C53(C8⋊D6), (C2×C30).5D4, C24⋊D51C2, (C2×D60)⋊10C2, C33(C8⋊D10), (C2×C4).14D30, C6.41(C2×D20), C2.15(C2×D60), (C2×C6).10D20, C1525(C8⋊C22), (C2×C10).10D12, C30.269(C2×D4), (C2×C20).142D6, C10.42(C2×D12), (C5×M4(2))⋊1S3, (C3×M4(2))⋊1D5, (C2×C12).141D10, (C15×M4(2))⋊1C2, (C2×C60).68C22, D6011C212C2, C4.29(C22×D15), C20.219(C22×S3), C12.221(C22×D5), SmallGroup(480,873)

Series: Derived Chief Lower central Upper central

C1C60 — C8⋊D30
C1C5C15C30C60D60C2×D60 — C8⋊D30
C15C30C60 — C8⋊D30
C1C2C2×C4M4(2)

Generators and relations for C8⋊D30
 G = < a,b,c | a8=b30=c2=1, bab-1=a5, cac=a3, cbc=b-1 >

Subgroups: 1172 in 136 conjugacy classes, 47 normal (33 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], C5, S3 [×3], C6, C6, C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, D5 [×3], C10, C10, Dic3, C12 [×2], D6 [×5], C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20 [×2], D10 [×5], C2×C10, C24 [×2], Dic6, C4×S3, D12 [×4], C3⋊D4, C2×C12, C22×S3, D15 [×3], C30, C30, C8⋊C22, C40 [×2], Dic10, C4×D5, D20 [×4], C5⋊D4, C2×C20, C22×D5, C24⋊C2 [×2], D24 [×2], C3×M4(2), C2×D12, C4○D12, Dic15, C60 [×2], D30 [×5], C2×C30, C40⋊C2 [×2], D40 [×2], C5×M4(2), C2×D20, C4○D20, C8⋊D6, C120 [×2], Dic30, C4×D15, D60, D60 [×2], D60, C157D4, C2×C60, C22×D15, C8⋊D10, C24⋊D5 [×2], D120 [×2], C15×M4(2), C2×D60, D6011C2, C8⋊D30
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], D12 [×2], C22×S3, D15, C8⋊C22, D20 [×2], C22×D5, C2×D12, D30 [×3], C2×D20, C8⋊D6, D60 [×2], C22×D15, C8⋊D10, C2×D60, C8⋊D30

Smallest permutation representation of C8⋊D30
On 120 points
Generators in S120
(1 111 31 61 21 96 51 76)(2 97 32 77 22 112 52 62)(3 113 33 63 23 98 53 78)(4 99 34 79 24 114 54 64)(5 115 35 65 25 100 55 80)(6 101 36 81 26 116 56 66)(7 117 37 67 27 102 57 82)(8 103 38 83 28 118 58 68)(9 119 39 69 29 104 59 84)(10 105 40 85 30 120 60 70)(11 91 41 71 16 106 46 86)(12 107 42 87 17 92 47 72)(13 93 43 73 18 108 48 88)(14 109 44 89 19 94 49 74)(15 95 45 75 20 110 50 90)
(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 5)(2 4)(6 15)(7 14)(8 13)(9 12)(10 11)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 60)(42 59)(43 58)(44 57)(45 56)(61 115)(62 114)(63 113)(64 112)(65 111)(66 110)(67 109)(68 108)(69 107)(70 106)(71 105)(72 104)(73 103)(74 102)(75 101)(76 100)(77 99)(78 98)(79 97)(80 96)(81 95)(82 94)(83 93)(84 92)(85 91)(86 120)(87 119)(88 118)(89 117)(90 116)

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

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

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

81 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B8A8B10A10B10C10D12A12B12C15A15B15C15D20A20B20C20D20E20F24A24B24C24D30A30B30C30D30E30F30G30H40A···40H60A···60H60I60J60K60L120A···120P
order1222223444556688101010101212121515151520202020202024242424303030303030303040···4060···6060606060120···120
size11260606022260222444224422422222222444444222244444···42···244444···4

81 irreducible representations

dim111111222222222222222224444
type+++++++++++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D5D6D6D10D10D12D12D15D20D20D30D30D60D60C8⋊C22C8⋊D6C8⋊D10C8⋊D30
kernelC8⋊D30C24⋊D5D120C15×M4(2)C2×D60D6011C2C5×M4(2)C60C2×C30C3×M4(2)C40C2×C20C24C2×C12C20C2×C10M4(2)C12C2×C6C8C2×C4C4C22C15C5C3C1
# reps122111111221422244484881248

Matrix representation of C8⋊D30 in GL6(𝔽241)

24000000
02400000
001092157
000184146
00161912400
001502230240
,
2081870000
177340000
005124000
001000
0041171901
002241852400
,
100000
302400000
0013114700
008011000
001088078197
004513378163

G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,161,150,0,0,0,1,91,223,0,0,92,84,240,0,0,0,157,146,0,240],[208,177,0,0,0,0,187,34,0,0,0,0,0,0,51,1,41,224,0,0,240,0,17,185,0,0,0,0,190,240,0,0,0,0,1,0],[1,30,0,0,0,0,0,240,0,0,0,0,0,0,131,80,108,45,0,0,147,110,80,133,0,0,0,0,78,78,0,0,0,0,197,163] >;

C8⋊D30 in GAP, Magma, Sage, TeX

C_8\rtimes D_{30}
% in TeX

G:=Group("C8:D30");
// GroupNames label

G:=SmallGroup(480,873);
// by ID

G=gap.SmallGroup(480,873);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,58,675,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c|a^8=b^30=c^2=1,b*a*b^-1=a^5,c*a*c=a^3,c*b*c=b^-1>;
// generators/relations

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