Copied to
clipboard

G = C30.4C42order 480 = 25·3·5

4th non-split extension by C30 of C42 acting via C42/C2=C2×C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C30.4C42, C3⋊C84F5, C5⋊C81Dic3, C153C86C4, C6.9(C4×F5), D5⋊C8.2S3, C32(C8⋊F5), C51(C24⋊C4), C152(C8⋊C4), C4.25(S3×F5), C20.25(C4×S3), C60.25(C2×C4), (C4×D5).71D6, D10.8(C4×S3), C12.32(C2×F5), C2.5(Dic3×F5), C10.4(C4×Dic3), D5.1(C8⋊S3), (C3×D5).2M4(2), Dic5.8(C2×Dic3), (D5×C12).63C22, (C5×C3⋊C8)⋊6C4, (C3×C5⋊C8)⋊2C4, (D5×C3⋊C8).9C2, (C4×C3⋊F5).3C2, (C2×C3⋊F5).2C4, (C3×D5⋊C8).3C2, (C6×D5).12(C2×C4), (C3×Dic5).16(C2×C4), SmallGroup(480,226)

Series: Derived Chief Lower central Upper central

C1C30 — C30.4C42
C1C5C15C30C3×Dic5D5×C12C3×D5⋊C8 — C30.4C42
C15C30 — C30.4C42
C1C4

Generators and relations for C30.4C42
 G = < a,b,c | a30=1, b4=c4=a15, bab-1=a13, cac-1=a11, cbc-1=a15b >

Subgroups: 356 in 80 conjugacy classes, 36 normal (26 characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×3], C22, C5, C6, C6 [×2], C8 [×4], C2×C4 [×3], D5 [×2], C10, Dic3 [×2], C12, C12, C2×C6, C15, C42, C2×C8 [×2], Dic5, C20, F5 [×2], D10, C3⋊C8, C3⋊C8, C24 [×2], C2×Dic3 [×2], C2×C12, C3×D5 [×2], C30, C8⋊C4, C52C8, C40, C5⋊C8 [×2], C4×D5, C2×F5 [×2], C2×C3⋊C8, C4×Dic3, C2×C24, C3×Dic5, C60, C3⋊F5 [×2], C6×D5, C8×D5, D5⋊C8, C4×F5, C24⋊C4, C5×C3⋊C8, C153C8, C3×C5⋊C8 [×2], D5×C12, C2×C3⋊F5 [×2], C8⋊F5, D5×C3⋊C8, C3×D5⋊C8, C4×C3⋊F5, C30.4C42
Quotients: C1, C2 [×3], C4 [×6], C22, S3, C2×C4 [×3], Dic3 [×2], D6, C42, M4(2) [×2], F5, C4×S3 [×2], C2×Dic3, C8⋊C4, C2×F5, C8⋊S3 [×2], C4×Dic3, C4×F5, C24⋊C4, S3×F5, C8⋊F5, Dic3×F5, C30.4C42

Smallest permutation representation of C30.4C42
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 76 46 105 16 61 31 120)(2 83 35 118 17 68 50 103)(3 90 54 101 18 75 39 116)(4 67 43 114 19 82 58 99)(5 74 32 97 20 89 47 112)(6 81 51 110 21 66 36 95)(7 88 40 93 22 73 55 108)(8 65 59 106 23 80 44 91)(9 72 48 119 24 87 33 104)(10 79 37 102 25 64 52 117)(11 86 56 115 26 71 41 100)(12 63 45 98 27 78 60 113)(13 70 34 111 28 85 49 96)(14 77 53 94 29 62 38 109)(15 84 42 107 30 69 57 92)
(1 105 46 61 16 120 31 76)(2 116 47 72 17 101 32 87)(3 97 48 83 18 112 33 68)(4 108 49 64 19 93 34 79)(5 119 50 75 20 104 35 90)(6 100 51 86 21 115 36 71)(7 111 52 67 22 96 37 82)(8 92 53 78 23 107 38 63)(9 103 54 89 24 118 39 74)(10 114 55 70 25 99 40 85)(11 95 56 81 26 110 41 66)(12 106 57 62 27 91 42 77)(13 117 58 73 28 102 43 88)(14 98 59 84 29 113 44 69)(15 109 60 65 30 94 45 80)

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

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

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

48 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H 5 6A6B6C8A8B8C8D8E8F8G8H 10 12A12B12C12D 15 20A20B24A···24H 30 40A40B40C40D60A60B
order1222344444444566688888888101212121215202024···2430404040406060
size1155211553030303042101066101010103030422101084410···1081212121288

48 irreducible representations

dim1111111122222224444888
type+++++-++++-
imageC1C2C2C2C4C4C4C4S3Dic3D6M4(2)C4×S3C4×S3C8⋊S3F5C2×F5C4×F5C8⋊F5S3×F5Dic3×F5C30.4C42
kernelC30.4C42D5×C3⋊C8C3×D5⋊C8C4×C3⋊F5C5×C3⋊C8C153C8C3×C5⋊C8C2×C3⋊F5D5⋊C8C5⋊C8C4×D5C3×D5C20D10D5C3⋊C8C12C6C3C4C2C1
# reps1111224412142281124112

Matrix representation of C30.4C42 in GL6(𝔽241)

2402400000
100000
00024000
00002400
00000240
001111
,
100000
010000
002130185185
001851850213
005628560
002821321328
,
6400000
1771770000
0013603131
002101052100
000210105210
0031310136

G:=sub<GL(6,GF(241))| [240,1,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,1,0,0,240,0,0,1,0,0,0,240,0,1,0,0,0,0,240,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,213,185,56,28,0,0,0,185,28,213,0,0,185,0,56,213,0,0,185,213,0,28],[64,177,0,0,0,0,0,177,0,0,0,0,0,0,136,210,0,31,0,0,0,105,210,31,0,0,31,210,105,0,0,0,31,0,210,136] >;

C30.4C42 in GAP, Magma, Sage, TeX

C_{30}._4C_4^2
% in TeX

G:=Group("C30.4C4^2");
// GroupNames label

G:=SmallGroup(480,226);
// by ID

G=gap.SmallGroup(480,226);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,64,100,80,1356,9414,4724]);
// Polycyclic

G:=Group<a,b,c|a^30=1,b^4=c^4=a^15,b*a*b^-1=a^13,c*a*c^-1=a^11,c*b*c^-1=a^15*b>;
// generators/relations

׿
×
𝔽