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G = C5×C4○D12order 240 = 24·3·5

Direct product of C5 and C4○D12

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

Aliases: C5×C4○D12, D125C10, C20.60D6, Dic65C10, C30.51C23, C60.68C22, (C2×C20)⋊7S3, (S3×C20)⋊9C2, (C4×S3)⋊4C10, (C2×C12)⋊4C10, (C2×C60)⋊11C2, C3⋊D43C10, (C5×D12)⋊11C2, C1513(C4○D4), C4.16(S3×C10), D6.1(C2×C10), (C2×C10).20D6, C12.16(C2×C10), (C5×Dic6)⋊11C2, C22.2(S3×C10), C6.4(C22×C10), C10.41(C22×S3), (C2×C30).50C22, Dic3.2(C2×C10), (S3×C10).12C22, (C5×Dic3).14C22, C31(C5×C4○D4), (C2×C4)⋊3(C5×S3), C2.5(S3×C2×C10), (C5×C3⋊D4)⋊7C2, (C2×C6).11(C2×C10), SmallGroup(240,168)

Series: Derived Chief Lower central Upper central

C1C6 — C5×C4○D12
C1C3C6C30S3×C10S3×C20 — C5×C4○D12
C3C6 — C5×C4○D12
C1C20C2×C20

Generators and relations for C5×C4○D12
 G = < a,b,c,d | a5=b4=d2=1, c6=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c5 >

Subgroups: 152 in 80 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C5, S3 [×2], C6, C6, C2×C4, C2×C4 [×2], D4 [×3], Q8, C10, C10 [×3], Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C15, C4○D4, C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12, C5×S3 [×2], C30, C30, C2×C20, C2×C20 [×2], C5×D4 [×3], C5×Q8, C4○D12, C5×Dic3 [×2], C60 [×2], S3×C10 [×2], C2×C30, C5×C4○D4, C5×Dic6, S3×C20 [×2], C5×D12, C5×C3⋊D4 [×2], C2×C60, C5×C4○D12
Quotients: C1, C2 [×7], C22 [×7], C5, S3, C23, C10 [×7], D6 [×3], C4○D4, C2×C10 [×7], C22×S3, C5×S3, C22×C10, C4○D12, S3×C10 [×3], C5×C4○D4, S3×C2×C10, C5×C4○D12

Smallest permutation representation of C5×C4○D12
On 120 points
Generators in S120
(1 117 42 79 29)(2 118 43 80 30)(3 119 44 81 31)(4 120 45 82 32)(5 109 46 83 33)(6 110 47 84 34)(7 111 48 73 35)(8 112 37 74 36)(9 113 38 75 25)(10 114 39 76 26)(11 115 40 77 27)(12 116 41 78 28)(13 91 54 107 65)(14 92 55 108 66)(15 93 56 97 67)(16 94 57 98 68)(17 95 58 99 69)(18 96 59 100 70)(19 85 60 101 71)(20 86 49 102 72)(21 87 50 103 61)(22 88 51 104 62)(23 89 52 105 63)(24 90 53 106 64)
(1 85 7 91)(2 86 8 92)(3 87 9 93)(4 88 10 94)(5 89 11 95)(6 90 12 96)(13 29 19 35)(14 30 20 36)(15 31 21 25)(16 32 22 26)(17 33 23 27)(18 34 24 28)(37 108 43 102)(38 97 44 103)(39 98 45 104)(40 99 46 105)(41 100 47 106)(42 101 48 107)(49 112 55 118)(50 113 56 119)(51 114 57 120)(52 115 58 109)(53 116 59 110)(54 117 60 111)(61 75 67 81)(62 76 68 82)(63 77 69 83)(64 78 70 84)(65 79 71 73)(66 80 72 74)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 15)(16 24)(17 23)(18 22)(19 21)(25 35)(26 34)(27 33)(28 32)(29 31)(38 48)(39 47)(40 46)(41 45)(42 44)(50 60)(51 59)(52 58)(53 57)(54 56)(61 71)(62 70)(63 69)(64 68)(65 67)(73 75)(76 84)(77 83)(78 82)(79 81)(85 87)(88 96)(89 95)(90 94)(91 93)(97 107)(98 106)(99 105)(100 104)(101 103)(109 115)(110 114)(111 113)(116 120)(117 119)

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

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

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

C5×C4○D12 is a maximal subgroup of
C60.98D4  C60.99D4  D12.2Dic5  D12.Dic5  D20.34D6  C60.36D4  C20.60D12  C60.38D4  D12.37D10  D12.33D10  D20.39D6  C30.C24  D2024D6  D2025D6  D2029D6  C5×S3×C4○D4
C5×C4○D12 is a maximal quotient of
C20×Dic6  C20×D12  C20×C3⋊D4

90 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B5C5D6A6B6C10A10B10C10D10E10F10G10H10I···10P12A12B12C12D15A15B15C15D20A···20H20I20J20K20L20M···20T30A···30L60A···60P
order122223444445555666101010101010101010···10121212121515151520···202020202020···2030···3060···60
size112662112661111222111122226···6222222221···122226···62···22···2

90 irreducible representations

dim1111111111112222222222
type+++++++++
imageC1C2C2C2C2C2C5C10C10C10C10C10S3D6D6C4○D4C5×S3C4○D12S3×C10S3×C10C5×C4○D4C5×C4○D12
kernelC5×C4○D12C5×Dic6S3×C20C5×D12C5×C3⋊D4C2×C60C4○D12Dic6C4×S3D12C3⋊D4C2×C12C2×C20C20C2×C10C15C2×C4C5C4C22C3C1
# reps11212144848412124484816

Matrix representation of C5×C4○D12 in GL2(𝔽61) generated by

200
020
,
110
011
,
2323
3846
,
060
600
G:=sub<GL(2,GF(61))| [20,0,0,20],[11,0,0,11],[23,38,23,46],[0,60,60,0] >;

C5×C4○D12 in GAP, Magma, Sage, TeX

C_5\times C_4\circ D_{12}
% in TeX

G:=Group("C5xC4oD12");
// GroupNames label

G:=SmallGroup(240,168);
// by ID

G=gap.SmallGroup(240,168);
# by ID

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

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

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