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G = C5×Q83S3order 240 = 24·3·5

Direct product of C5 and Q83S3

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

Aliases: C5×Q83S3, D124C10, C20.42D6, C60.49C22, C30.55C23, Q83(C5×S3), (C5×Q8)⋊7S3, (C4×S3)⋊3C10, (S3×C20)⋊8C2, C4.7(S3×C10), (Q8×C15)⋊9C2, (C3×Q8)⋊3C10, (C5×D12)⋊10C2, C1519(C4○D4), C12.7(C2×C10), D6.3(C2×C10), C6.8(C22×C10), C10.45(C22×S3), Dic3.5(C2×C10), (S3×C10).14C22, (C5×Dic3).17C22, C33(C5×C4○D4), C2.9(S3×C2×C10), (C5×Q8)(C5×Dic3), SmallGroup(240,172)

Series: Derived Chief Lower central Upper central

C1C6 — C5×Q83S3
C1C3C6C30S3×C10S3×C20 — C5×Q83S3
C3C6 — C5×Q83S3
C1C10C5×Q8

Generators and relations for C5×Q83S3
 G = < a,b,c,d,e | a5=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 160 in 80 conjugacy classes, 46 normal (16 characteristic)
C1, C2, C2 [×3], C3, C4 [×3], C4, C22 [×3], C5, S3 [×3], C6, C2×C4 [×3], D4 [×3], Q8, C10, C10 [×3], Dic3, C12 [×3], D6 [×3], C15, C4○D4, C20 [×3], C20, C2×C10 [×3], C4×S3 [×3], D12 [×3], C3×Q8, C5×S3 [×3], C30, C2×C20 [×3], C5×D4 [×3], C5×Q8, Q83S3, C5×Dic3, C60 [×3], S3×C10 [×3], C5×C4○D4, S3×C20 [×3], C5×D12 [×3], Q8×C15, C5×Q83S3
Quotients: C1, C2 [×7], C22 [×7], C5, S3, C23, C10 [×7], D6 [×3], C4○D4, C2×C10 [×7], C22×S3, C5×S3, C22×C10, Q83S3, S3×C10 [×3], C5×C4○D4, S3×C2×C10, C5×Q83S3

Smallest permutation representation of C5×Q83S3
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 56 26 71)(2 57 27 72)(3 58 28 73)(4 59 29 74)(5 60 30 75)(6 109 21 95)(7 110 22 91)(8 106 23 92)(9 107 24 93)(10 108 25 94)(11 84 114 99)(12 85 115 100)(13 81 111 96)(14 82 112 97)(15 83 113 98)(16 89 119 104)(17 90 120 105)(18 86 116 101)(19 87 117 102)(20 88 118 103)(31 80 50 61)(32 76 46 62)(33 77 47 63)(34 78 48 64)(35 79 49 65)(36 51 43 66)(37 52 44 67)(38 53 45 68)(39 54 41 69)(40 55 42 70)
(1 86 26 101)(2 87 27 102)(3 88 28 103)(4 89 29 104)(5 90 30 105)(6 65 21 79)(7 61 22 80)(8 62 23 76)(9 63 24 77)(10 64 25 78)(11 69 114 54)(12 70 115 55)(13 66 111 51)(14 67 112 52)(15 68 113 53)(16 74 119 59)(17 75 120 60)(18 71 116 56)(19 72 117 57)(20 73 118 58)(31 110 50 91)(32 106 46 92)(33 107 47 93)(34 108 48 94)(35 109 49 95)(36 81 43 96)(37 82 44 97)(38 83 45 98)(39 84 41 99)(40 85 42 100)
(1 33 39)(2 34 40)(3 35 36)(4 31 37)(5 32 38)(6 13 20)(7 14 16)(8 15 17)(9 11 18)(10 12 19)(21 111 118)(22 112 119)(23 113 120)(24 114 116)(25 115 117)(26 47 41)(27 48 42)(28 49 43)(29 50 44)(30 46 45)(51 58 79)(52 59 80)(53 60 76)(54 56 77)(55 57 78)(61 67 74)(62 68 75)(63 69 71)(64 70 72)(65 66 73)(81 88 109)(82 89 110)(83 90 106)(84 86 107)(85 87 108)(91 97 104)(92 98 105)(93 99 101)(94 100 102)(95 96 103)
(6 111)(7 112)(8 113)(9 114)(10 115)(11 24)(12 25)(13 21)(14 22)(15 23)(16 119)(17 120)(18 116)(19 117)(20 118)(31 37)(32 38)(33 39)(34 40)(35 36)(41 47)(42 48)(43 49)(44 50)(45 46)(51 65)(52 61)(53 62)(54 63)(55 64)(56 71)(57 72)(58 73)(59 74)(60 75)(66 79)(67 80)(68 76)(69 77)(70 78)(81 109)(82 110)(83 106)(84 107)(85 108)(91 97)(92 98)(93 99)(94 100)(95 96)

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

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

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

C5×Q83S3 is a maximal subgroup of
D12⋊D10  Dic10.26D6  D20.27D6  Dic10.27D6  D12.29D10  D2016D6  D2017D6  C5×S3×C4○D4
C5×Q83S3 is a maximal quotient of
C5×Q8×Dic3

75 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B5C5D 6 10A10B10C10D10E···10P12A12B12C15A15B15C15D20A···20L20M···20T30A30B30C30D60A···60L
order12222344444555561010101010···101212121515151520···2020···203030303060···60
size116662222331111211116···644422222···23···322224···4

75 irreducible representations

dim1111111122222244
type+++++++
imageC1C2C2C2C5C10C10C10S3D6C4○D4C5×S3S3×C10C5×C4○D4Q83S3C5×Q83S3
kernelC5×Q83S3S3×C20C5×D12Q8×C15Q83S3C4×S3D12C3×Q8C5×Q8C20C15Q8C4C3C5C1
# reps1331412124132412814

Matrix representation of C5×Q83S3 in GL4(𝔽61) generated by

58000
05800
00340
00034
,
60000
06000
0001
00600
,
60000
06000
00500
00011
,
60100
60000
0010
0001
,
0100
1000
0010
00060
G:=sub<GL(4,GF(61))| [58,0,0,0,0,58,0,0,0,0,34,0,0,0,0,34],[60,0,0,0,0,60,0,0,0,0,0,60,0,0,1,0],[60,0,0,0,0,60,0,0,0,0,50,0,0,0,0,11],[60,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,60] >;

C5×Q83S3 in GAP, Magma, Sage, TeX

C_5\times Q_8\rtimes_3S_3
% in TeX

G:=Group("C5xQ8:3S3");
// GroupNames label

G:=SmallGroup(240,172);
// by ID

G=gap.SmallGroup(240,172);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-3,247,794,404,194,5765]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=d^3=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=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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