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G = C30.38D4order 240 = 24·3·5

7th non-split extension by C30 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C30.38D4, C23.2D15, C22.7D30, C222Dic15, (C2×C30)⋊6C4, (C2×C6)⋊2Dic5, C30.53(C2×C4), (C2×C10)⋊7Dic3, (C2×C6).25D10, (C2×C10).25D6, C32(C23.D5), (C22×C6).2D5, C1510(C22⋊C4), (C2×Dic15)⋊2C2, C6.20(C5⋊D4), C53(C6.D4), C2.3(C157D4), (C22×C10).4S3, (C22×C30).2C2, C6.10(C2×Dic5), C2.5(C2×Dic15), C10.20(C3⋊D4), (C2×C30).26C22, C10.17(C2×Dic3), SmallGroup(240,80)

Series: Derived Chief Lower central Upper central

C1C30 — C30.38D4
C1C5C15C30C2×C30C2×Dic15 — C30.38D4
C15C30 — C30.38D4
C1C22C23

Generators and relations for C30.38D4
 G = < a,b,c | a30=b4=1, c2=a15, bab-1=cac-1=a-1, cbc-1=a15b-1 >

Subgroups: 232 in 68 conjugacy classes, 35 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C22, C22 [×2], C22 [×2], C5, C6, C6 [×2], C6 [×2], C2×C4 [×2], C23, C10, C10 [×2], C10 [×2], Dic3 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22⋊C4, Dic5 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×Dic3 [×2], C22×C6, C30, C30 [×2], C30 [×2], C2×Dic5 [×2], C22×C10, C6.D4, Dic15 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C23.D5, C2×Dic15 [×2], C22×C30, C30.38D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, Dic3 [×2], D6, C22⋊C4, Dic5 [×2], D10, C2×Dic3, C3⋊D4 [×2], D15, C2×Dic5, C5⋊D4 [×2], C6.D4, Dic15 [×2], D30, C23.D5, C2×Dic15, C157D4 [×2], C30.38D4

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

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

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

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

C30.38D4 is a maximal subgroup of
(C2×C6).D20  C158(C23⋊C4)  C23.6D30  C23.7D30  C23.D5⋊S3  (C6×Dic5)⋊7C4  C23.26(S3×D5)  C23.13(S3×D5)  C23.14(S3×D5)  C6.(D4×D5)  (C2×C30).D4  C30.(C2×D4)  D5×C6.D4  C23.17(S3×D5)  (C6×D5)⋊D4  Dic5×C3⋊D4  Dic3×C5⋊D4  S3×C23.D5  (S3×C10).D4  (C2×C30)⋊D4  (S3×C10)⋊D4  C15⋊C22≀C2  (C2×C30)⋊Q8  (C2×C10)⋊8Dic6  C23.15D30  C222Dic30  C23.8D30  C22⋊C4×D15  D30.28D4  C23.11D30  C60.205D4  C23.26D30  C4×C157D4  C23.28D30  D4×Dic15  C23.22D30  C60.17D4  D3017D4  C602D4  Dic1512D4  C245D15
C30.38D4 is a maximal quotient of
C60.212D4  C30.29C42  D4⋊Dic15  C60.8D4  C23.7D30  Q82Dic15  C60.10D4  Q83Dic15

66 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D5A5B6A···6G10A···10N15A15B15C15D30A···30AB
order12222234444556···610···101515151530···30
size111122230303030222···22···222222···2

66 irreducible representations

dim11112222222222222
type++++++-+-++-+
imageC1C2C2C4S3D4D5Dic3D6Dic5D10C3⋊D4D15C5⋊D4Dic15D30C157D4
kernelC30.38D4C2×Dic15C22×C30C2×C30C22×C10C30C22×C6C2×C10C2×C10C2×C6C2×C6C10C23C6C22C22C2
# reps121412221424488416

Matrix representation of C30.38D4 in GL3(𝔽61) generated by

6000
0490
005
,
1100
001
010
,
5000
001
0600
G:=sub<GL(3,GF(61))| [60,0,0,0,49,0,0,0,5],[11,0,0,0,0,1,0,1,0],[50,0,0,0,0,60,0,1,0] >;

C30.38D4 in GAP, Magma, Sage, TeX

C_{30}._{38}D_4
% in TeX

G:=Group("C30.38D4");
// GroupNames label

G:=SmallGroup(240,80);
// by ID

G=gap.SmallGroup(240,80);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,964,6917]);
// Polycyclic

G:=Group<a,b,c|a^30=b^4=1,c^2=a^15,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^15*b^-1>;
// generators/relations

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