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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C30
 Chief series C1 — C2 — C10 — C30 — C2×C30 — D4×C15 — D4×C30
 Lower central C1 — C2 — D4×C30
 Upper central C1 — C2×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 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 5A 5B 5C 5D 6A ··· 6F 6G ··· 6N 10A ··· 10L 10M ··· 10AB 12A 12B 12C 12D 15A ··· 15H 20A ··· 20H 30A ··· 30X 30Y ··· 30BD 60A ··· 60P order 1 2 2 2 2 2 2 2 3 3 4 4 5 5 5 5 6 ··· 6 6 ··· 6 10 ··· 10 10 ··· 10 12 12 12 12 15 ··· 15 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 2 1 1 2 2 1 1 1 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 2 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

150 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C2 C3 C5 C6 C6 C6 C10 C10 C10 C15 C30 C30 C30 D4 C3×D4 C5×D4 D4×C15 kernel D4×C30 C2×C60 D4×C15 C22×C30 D4×C10 C6×D4 C2×C20 C5×D4 C22×C10 C2×C12 C3×D4 C22×C6 C2×D4 C2×C4 D4 C23 C30 C10 C6 C2 # reps 1 1 4 2 2 4 2 8 4 4 16 8 8 8 32 16 2 4 8 16

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

 60 0 0 0 49 0 0 0 49
,
 1 0 0 0 24 59 0 14 37
,
 60 0 0 0 24 59 0 13 37
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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