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G = D4×C30order 240 = 24·3·5

Direct product of C30 and D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C30, C232C30, C6014C22, C30.58C23, C4⋊(C2×C30), (C2×C4)⋊2C30, (C2×C20)⋊6C6, C204(C2×C6), (C2×C60)⋊14C2, (C2×C12)⋊6C10, C124(C2×C10), C222(C2×C30), (C2×C30)⋊8C22, (C22×C10)⋊3C6, (C22×C30)⋊1C2, (C22×C6)⋊1C10, C2.1(C22×C30), C6.11(C22×C10), C10.11(C22×C6), (C2×C6)⋊2(C2×C10), (C2×C10)⋊4(C2×C6), SmallGroup(240,186)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C30
C1C2C10C30C2×C30D4×C15 — D4×C30
C1C2 — D4×C30
C1C2×C30 — D4×C30

Generators and relations for D4×C30
 G = < a,b,c | a30=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 140 in 108 conjugacy classes, 76 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×4], C22 [×4], C5, C6, C6 [×2], C6 [×4], C2×C4, D4 [×4], C23 [×2], C10, C10 [×2], C10 [×4], C12 [×2], C2×C6, C2×C6 [×4], C2×C6 [×4], C15, C2×D4, C20 [×2], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C12, C3×D4 [×4], C22×C6 [×2], C30, C30 [×2], C30 [×4], C2×C20, C5×D4 [×4], C22×C10 [×2], C6×D4, C60 [×2], C2×C30, C2×C30 [×4], C2×C30 [×4], D4×C10, C2×C60, D4×C15 [×4], C22×C30 [×2], D4×C30
Quotients: C1, C2 [×7], C3, C22 [×7], C5, C6 [×7], D4 [×2], C23, C10 [×7], C2×C6 [×7], C15, C2×D4, C2×C10 [×7], C3×D4 [×2], C22×C6, C30 [×7], C5×D4 [×2], C22×C10, C6×D4, C2×C30 [×7], D4×C10, D4×C15 [×2], C22×C30, D4×C30

Smallest permutation representation of D4×C30
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 37 65 106)(2 38 66 107)(3 39 67 108)(4 40 68 109)(5 41 69 110)(6 42 70 111)(7 43 71 112)(8 44 72 113)(9 45 73 114)(10 46 74 115)(11 47 75 116)(12 48 76 117)(13 49 77 118)(14 50 78 119)(15 51 79 120)(16 52 80 91)(17 53 81 92)(18 54 82 93)(19 55 83 94)(20 56 84 95)(21 57 85 96)(22 58 86 97)(23 59 87 98)(24 60 88 99)(25 31 89 100)(26 32 90 101)(27 33 61 102)(28 34 62 103)(29 35 63 104)(30 36 64 105)
(1 106)(2 107)(3 108)(4 109)(5 110)(6 111)(7 112)(8 113)(9 114)(10 115)(11 116)(12 117)(13 118)(14 119)(15 120)(16 91)(17 92)(18 93)(19 94)(20 95)(21 96)(22 97)(23 98)(24 99)(25 100)(26 101)(27 102)(28 103)(29 104)(30 105)(31 89)(32 90)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)(41 69)(42 70)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(50 78)(51 79)(52 80)(53 81)(54 82)(55 83)(56 84)(57 85)(58 86)(59 87)(60 88)

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

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

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

D4×C30 is a maximal subgroup of
D4⋊Dic15  C60.8D4  C23.7D30  D4.D30  C23.22D30  C60.17D4  D3017D4  C602D4  Dic1512D4  C603D4  D46D30

150 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B5A5B5C5D6A···6F6G···6N10A···10L10M···10AB12A12B12C12D15A···15H20A···20H30A···30X30Y···30BD60A···60P
order12222222334455556···66···610···1010···101212121215···1520···2030···3030···3060···60
size11112222112211111···12···21···12···222221···12···21···12···22···2

150 irreducible representations

dim11111111111111112222
type+++++
imageC1C2C2C2C3C5C6C6C6C10C10C10C15C30C30C30D4C3×D4C5×D4D4×C15
kernelD4×C30C2×C60D4×C15C22×C30D4×C10C6×D4C2×C20C5×D4C22×C10C2×C12C3×D4C22×C6C2×D4C2×C4D4C23C30C10C6C2
# reps114224284416888321624816

Matrix representation of D4×C30 in GL3(𝔽61) generated by

6000
0490
0049
,
100
02459
01437
,
6000
02459
01337
G:=sub<GL(3,GF(61))| [60,0,0,0,49,0,0,0,49],[1,0,0,0,24,14,0,59,37],[60,0,0,0,24,13,0,59,37] >;

D4×C30 in GAP, Magma, Sage, TeX

D_4\times C_{30}
% in TeX

G:=Group("D4xC30");
// GroupNames label

G:=SmallGroup(240,186);
// by ID

G=gap.SmallGroup(240,186);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-5,-2,1465]);
// Polycyclic

G:=Group<a,b,c|a^30=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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