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G = C5×Q83D6order 480 = 25·3·5

Direct product of C5 and Q83D6

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5×Q83D6, C4021D6, D246C10, C12028C22, C60.221C23, C83(S3×C10), C243(C2×C10), D4⋊S33C10, (S3×D4)⋊3C10, (C5×Q8)⋊18D6, Q83(S3×C10), D6.7(C5×D4), C8⋊S31C10, (C5×D24)⋊14C2, D122(C2×C10), (C5×SD16)⋊5S3, SD161(C5×S3), (C5×D4).27D6, D4.3(S3×C10), C6.31(D4×C10), C1533(C8⋊C22), Q82S32C10, Q83S31C10, (C15×SD16)⋊7C2, (C3×SD16)⋊1C10, (S3×C10).43D4, C10.185(S3×D4), C30.367(C2×D4), Dic3.9(C5×D4), (C5×D12)⋊19C22, C12.5(C22×C10), (C5×Dic3).46D4, (Q8×C15)⋊17C22, (S3×C20).37C22, C20.194(C22×S3), (D4×C15).32C22, C3⋊C82(C2×C10), (C5×S3×D4)⋊10C2, C33(C5×C8⋊C22), C4.5(S3×C2×C10), C2.19(C5×S3×D4), (C5×C8⋊S3)⋊9C2, (C5×D4⋊S3)⋊11C2, (C5×C3⋊C8)⋊24C22, (C3×Q8)⋊2(C2×C10), (C4×S3).2(C2×C10), (C5×Q83S3)⋊8C2, (C3×D4).3(C2×C10), (C5×Q82S3)⋊10C2, SmallGroup(480,793)

Series: Derived Chief Lower central Upper central

C1C12 — C5×Q83D6
C1C3C6C12C60S3×C20C5×S3×D4 — C5×Q83D6
C3C6C12 — C5×Q83D6
C1C10C20C5×SD16

Generators and relations for C5×Q83D6
 G = < a,b,c,d,e | a5=b4=d6=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=d-1 >

Subgroups: 404 in 136 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×3], C6, C6, C8, C8, C2×C4 [×2], D4, D4 [×4], Q8, C23, C10, C10 [×4], Dic3, C12, C12, D6, D6 [×4], C2×C6, C15, M4(2), D8 [×2], SD16, SD16, C2×D4, C4○D4, C20, C20 [×2], C2×C10 [×6], C3⋊C8, C24, C4×S3, C4×S3, D12 [×2], D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3 [×3], C30, C30, C8⋊C22, C40, C40, C2×C20 [×2], C5×D4, C5×D4 [×4], C5×Q8, C22×C10, C8⋊S3, D24, D4⋊S3, Q82S3, C3×SD16, S3×D4, Q83S3, C5×Dic3, C60, C60, S3×C10, S3×C10 [×4], C2×C30, C5×M4(2), C5×D8 [×2], C5×SD16, C5×SD16, D4×C10, C5×C4○D4, Q83D6, C5×C3⋊C8, C120, S3×C20, S3×C20, C5×D12 [×2], C5×D12, C5×C3⋊D4, D4×C15, Q8×C15, S3×C2×C10, C5×C8⋊C22, C5×C8⋊S3, C5×D24, C5×D4⋊S3, C5×Q82S3, C15×SD16, C5×S3×D4, C5×Q83S3, C5×Q83D6
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C2×C10 [×7], C22×S3, C5×S3, C8⋊C22, C5×D4 [×2], C22×C10, S3×D4, S3×C10 [×3], D4×C10, Q83D6, S3×C2×C10, C5×C8⋊C22, C5×S3×D4, C5×Q83D6

Smallest permutation representation of C5×Q83D6
On 120 points
Generators in S120
(1 19 10 6 23)(2 20 11 4 24)(3 21 12 5 22)(7 13 27 17 28)(8 14 25 18 29)(9 15 26 16 30)(31 66 87 104 69)(32 61 88 105 70)(33 62 89 106 71)(34 63 90 107 72)(35 64 85 108 67)(36 65 86 103 68)(37 50 118 94 98)(38 51 119 95 99)(39 52 120 96 100)(40 53 115 91 101)(41 54 116 92 102)(42 49 117 93 97)(43 83 111 73 55)(44 84 112 74 56)(45 79 113 75 57)(46 80 114 76 58)(47 81 109 77 59)(48 82 110 78 60)
(1 114 30 111)(2 112 28 109)(3 110 29 113)(4 44 27 47)(5 48 25 45)(6 46 26 43)(7 77 20 74)(8 75 21 78)(9 73 19 76)(10 58 15 55)(11 56 13 59)(12 60 14 57)(16 83 23 80)(17 81 24 84)(18 79 22 82)(31 34 101 98)(32 99 102 35)(33 36 97 100)(37 66 63 40)(38 41 64 61)(39 62 65 42)(49 52 89 86)(50 87 90 53)(51 54 85 88)(67 70 95 92)(68 93 96 71)(69 72 91 94)(103 117 120 106)(104 107 115 118)(105 119 116 108)
(1 106 30 117)(2 104 28 115)(3 108 29 119)(4 66 27 40)(5 64 25 38)(6 62 26 42)(7 91 20 69)(8 95 21 67)(9 93 19 71)(10 33 15 97)(11 31 13 101)(12 35 14 99)(16 49 23 89)(17 53 24 87)(18 51 22 85)(32 60 102 57)(34 56 98 59)(36 58 100 55)(37 47 63 44)(39 43 65 46)(41 45 61 48)(50 81 90 84)(52 83 86 80)(54 79 88 82)(68 76 96 73)(70 78 92 75)(72 74 94 77)(103 114 120 111)(105 110 116 113)(107 112 118 109)
(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 28)(2 30)(3 29)(4 26)(5 25)(6 27)(7 19)(8 21)(9 20)(10 13)(11 15)(12 14)(16 24)(17 23)(18 22)(31 36)(32 35)(33 34)(37 42)(38 41)(39 40)(43 47)(44 46)(49 50)(51 54)(52 53)(55 59)(56 58)(61 64)(62 63)(65 66)(67 70)(68 69)(71 72)(73 77)(74 76)(80 84)(81 83)(85 88)(86 87)(89 90)(91 96)(92 95)(93 94)(97 98)(99 102)(100 101)(103 104)(105 108)(106 107)(109 111)(112 114)(115 120)(116 119)(117 118)

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

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

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

90 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B5C5D6A6B8A8B10A10B10C10D10E10F10G10H10I10J10K10L10M···10T12A12B15A15B15C15D20A20B20C20D20E20F20G20H20I20J20K20L24A24B30A30B30C30D30E30F30G30H40A40B40C40D40E40F40G40H60A60B60C60D60E60F60G60H120A···120H
order12222234445555668810101010101010101010101010···101212151515152020202020202020202020202424303030303030303040404040404040406060606060606060120···120
size11461212224611112841211114444666612···124822222222444466664422228888444412121212444488884···4

90 irreducible representations

dim1111111111111111222222222222444444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C5C10C10C10C10C10C10C10S3D4D4D6D6D6C5×S3C5×D4C5×D4S3×C10S3×C10S3×C10C8⋊C22S3×D4Q83D6C5×C8⋊C22C5×S3×D4C5×Q83D6
kernelC5×Q83D6C5×C8⋊S3C5×D24C5×D4⋊S3C5×Q82S3C15×SD16C5×S3×D4C5×Q83S3Q83D6C8⋊S3D24D4⋊S3Q82S3C3×SD16S3×D4Q83S3C5×SD16C5×Dic3S3×C10C40C5×D4C5×Q8SD16Dic3D6C8D4Q8C15C10C5C3C2C1
# reps1111111144444444111111444444112448

Matrix representation of C5×Q83D6 in GL4(𝔽241) generated by

205000
020500
002050
000205
,
24002400
02400240
2010
0201
,
001169
00232125
2321800
223900
,
240100
240000
22391240
2010
,
0100
1000
02390240
23902400
G:=sub<GL(4,GF(241))| [205,0,0,0,0,205,0,0,0,0,205,0,0,0,0,205],[240,0,2,0,0,240,0,2,240,0,1,0,0,240,0,1],[0,0,232,223,0,0,18,9,116,232,0,0,9,125,0,0],[240,240,2,2,1,0,239,0,0,0,1,1,0,0,240,0],[0,1,0,239,1,0,239,0,0,0,0,240,0,0,240,0] >;

C5×Q83D6 in GAP, Magma, Sage, TeX

C_5\times Q_8\rtimes_3D_6
% in TeX

G:=Group("C5xQ8:3D6");
// GroupNames label

G:=SmallGroup(480,793);
// by ID

G=gap.SmallGroup(480,793);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1766,471,436,2111,1068,102,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=d^6=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b^-1*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽