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G = C3×D10⋊C4order 240 = 24·3·5

Direct product of C3 and D10⋊C4

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

Aliases: C3×D10⋊C4, D101C12, C30.27D4, C6.17D20, (C2×C20)⋊1C6, (C2×C60)⋊1C2, (C6×D5)⋊3C4, (C2×C12)⋊1D5, C2.5(D5×C12), C2.2(C3×D20), C6.19(C4×D5), C10.6(C3×D4), C158(C22⋊C4), C30.44(C2×C4), (C6×Dic5)⋊7C2, (C2×Dic5)⋊1C6, (C2×C6).34D10, C22.6(C6×D5), C10.11(C2×C12), C6.22(C5⋊D4), (C22×D5).2C6, (C2×C30).35C22, (C2×C4)⋊1(C3×D5), C52(C3×C22⋊C4), (D5×C2×C6).3C2, C2.2(C3×C5⋊D4), (C2×C10).6(C2×C6), SmallGroup(240,43)

Series: Derived Chief Lower central Upper central

C1C10 — C3×D10⋊C4
C1C5C10C2×C10C2×C30D5×C2×C6 — C3×D10⋊C4
C5C10 — C3×D10⋊C4
C1C2×C6C2×C12

Generators and relations for C3×D10⋊C4
 G = < a,b,c,d | a3=b10=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b5c >

Subgroups: 212 in 68 conjugacy classes, 34 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C22, C22 [×4], C5, C6 [×3], C6 [×2], C2×C4, C2×C4, C23, D5 [×2], C10 [×3], C12 [×2], C2×C6, C2×C6 [×4], C15, C22⋊C4, Dic5, C20, D10 [×2], D10 [×2], C2×C10, C2×C12, C2×C12, C22×C6, C3×D5 [×2], C30 [×3], C2×Dic5, C2×C20, C22×D5, C3×C22⋊C4, C3×Dic5, C60, C6×D5 [×2], C6×D5 [×2], C2×C30, D10⋊C4, C6×Dic5, C2×C60, D5×C2×C6, C3×D10⋊C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], D5, C12 [×2], C2×C6, C22⋊C4, D10, C2×C12, C3×D4 [×2], C3×D5, C4×D5, D20, C5⋊D4, C3×C22⋊C4, C6×D5, D10⋊C4, D5×C12, C3×D20, C3×C5⋊D4, C3×D10⋊C4

Smallest permutation representation of C3×D10⋊C4
On 120 points
Generators in S120
(1 47 27)(2 48 28)(3 49 29)(4 50 30)(5 41 21)(6 42 22)(7 43 23)(8 44 24)(9 45 25)(10 46 26)(11 51 31)(12 52 32)(13 53 33)(14 54 34)(15 55 35)(16 56 36)(17 57 37)(18 58 38)(19 59 39)(20 60 40)(61 101 81)(62 102 82)(63 103 83)(64 104 84)(65 105 85)(66 106 86)(67 107 87)(68 108 88)(69 109 89)(70 110 90)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 12)(13 20)(14 19)(15 18)(16 17)(21 22)(23 30)(24 29)(25 28)(26 27)(31 32)(33 40)(34 39)(35 38)(36 37)(41 42)(43 50)(44 49)(45 48)(46 47)(51 52)(53 60)(54 59)(55 58)(56 57)(61 67)(62 66)(63 65)(68 70)(71 77)(72 76)(73 75)(78 80)(81 87)(82 86)(83 85)(88 90)(91 97)(92 96)(93 95)(98 100)(101 107)(102 106)(103 105)(108 110)(111 117)(112 116)(113 115)(118 120)
(1 77 17 67)(2 78 18 68)(3 79 19 69)(4 80 20 70)(5 71 11 61)(6 72 12 62)(7 73 13 63)(8 74 14 64)(9 75 15 65)(10 76 16 66)(21 91 31 81)(22 92 32 82)(23 93 33 83)(24 94 34 84)(25 95 35 85)(26 96 36 86)(27 97 37 87)(28 98 38 88)(29 99 39 89)(30 100 40 90)(41 111 51 101)(42 112 52 102)(43 113 53 103)(44 114 54 104)(45 115 55 105)(46 116 56 106)(47 117 57 107)(48 118 58 108)(49 119 59 109)(50 120 60 110)

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

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

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

C3×D10⋊C4 is a maximal subgroup of
Dic3.D20  D30.34D4  D30.D4  (C2×C12).D10  (C2×C60).C22  (C4×Dic3)⋊D5  (C4×Dic15)⋊C2  (D5×Dic3)⋊C4  D10.19(C4×S3)  Dic34D20  Dic1513D4  D10.16D12  D10.17D12  D101Dic6  D102Dic6  Dic15.D4  D104Dic6  C1517(C4×D4)  Dic159D4  D10⋊C4⋊S3  Dic152D4  D6⋊D20  (C2×Dic6)⋊D5  D6.9D20  D302D4  D3012D4  Dic15.10D4  Dic15.31D4  D30.27D4  D304D4  D305D4  C12×D20  C3×D5×C22⋊C4  C12×C5⋊D4

78 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D5A5B6A···6F6G6H6I6J10A···10F12A12B12C12D12E12F12G12H15A15B15C15D20A···20H30A···30L60A···60P
order122222334444556···6666610···1012121212121212121515151520···2030···3060···60
size1111101011221010221···1101010102···222221010101022222···22···22···2

78 irreducible representations

dim1111111111222222222222
type++++++++
imageC1C2C2C2C3C4C6C6C6C12D4D5D10C3×D4C3×D5C4×D5D20C5⋊D4C6×D5D5×C12C3×D20C3×C5⋊D4
kernelC3×D10⋊C4C6×Dic5C2×C60D5×C2×C6D10⋊C4C6×D5C2×Dic5C2×C20C22×D5D10C30C2×C12C2×C6C10C2×C4C6C6C6C22C2C2C2
# reps1111242228222444444888

Matrix representation of C3×D10⋊C4 in GL3(𝔽61) generated by

100
0470
0047
,
100
01844
0180
,
6000
0601
001
,
1100
03116
0130
G:=sub<GL(3,GF(61))| [1,0,0,0,47,0,0,0,47],[1,0,0,0,18,18,0,44,0],[60,0,0,0,60,0,0,1,1],[11,0,0,0,31,1,0,16,30] >;

C3×D10⋊C4 in GAP, Magma, Sage, TeX

C_3\times D_{10}\rtimes C_4
% in TeX

G:=Group("C3xD10:C4");
// GroupNames label

G:=SmallGroup(240,43);
// by ID

G=gap.SmallGroup(240,43);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,313,79,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^10=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^5*c>;
// generators/relations

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