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G = C12.5D20  order 480 = 25·3·5

5th non-split extension by C12 of D20 acting via D20/C10=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.5D20, C60.28D4, (C6×D20).1C2, (C2×D20).6S3, (C2×C20).40D6, C4.Dic3⋊2D5, C60.7C4⋊5C2, (C2×C12).41D10, C12.5(C5⋊D4), C15⋊2(C4.D4), C5⋊2(C12.D4), C4.19(C15⋊D4), C20.77(C3⋊D4), C4.19(C3⋊D20), C3⋊3(C20.46D4), (C2×C60).29C22, C22.3(D5×Dic3), C30.41(C22⋊C4), C6.24(D10⋊C4), (C22×D5).1Dic3, C2.3(D10⋊Dic3), C10.13(C6.D4), (D5×C2×C6).1C4, (C2×C4).1(S3×D5), (C2×C6).45(C4×D5), (C2×C30).80(C2×C4), (C5×C4.Dic3)⋊1C2, (C2×C10).21(C2×Dic3), SmallGroup(480,34)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C30 — C12.5D20
Chief series
C1 — C5 — C15 — C30 — C60 — C2×C60 — C6×D20 — C12.5D20
Lower central
C15 — C30 — C2×C30 — C12.5D20
Upper central
C1 — C2 — C2×C4

Generators and relations for C12.5D20
 G = < a,b,c | a12=1, b20=a6, c2=a9, bab-1=a-1, cac-1=a5, cbc-1=a3b19 >

Subgroups: 476 in 92 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C22, C22 [×4], C5, C6, C6 [×3], C8 [×2], C2×C4, D4 [×2], C23 [×2], D5 [×2], C10, C10, C12 [×2], C2×C6, C2×C6 [×4], C15, M4(2) [×2], C2×D4, C20 [×2], D10 [×4], C2×C10, C3⋊C8 [×2], C2×C12, C3×D4 [×2], C22×C6 [×2], C3×D5 [×2], C30, C30, C4.D4, C5⋊2C8, C40, D20 [×2], C2×C20, C22×D5 [×2], C4.Dic3, C4.Dic3, C6×D4, C60 [×2], C6×D5 [×4], C2×C30, C4.Dic5, C5×M4(2), C2×D20, C12.D4, C5×C3⋊C8, C15⋊3C8, C3×D20 [×2], C2×C60, D5×C2×C6 [×2], C20.46D4, C5×C4.Dic3, C60.7C4, C6×D20, C12.5D20
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, Dic3 [×2], D6, C22⋊C4, D10, C2×Dic3, C3⋊D4 [×2], C4.D4, C4×D5, D20, C5⋊D4, C6.D4, S3×D5, D10⋊C4, C12.D4, D5×Dic3, C15⋊D4, C3⋊D20, C20.46D4, D10⋊Dic3, C12.5D20

Smallest permutation representation of C12.5D20
►On 120 points
Generators in S120
(1 41 114 31 71 104 21 61 94 11 51 84)(2 85 52 12 95 62 22 105 72 32 115 42)(3 43 116 33 73 106 23 63 96 13 53 86)(4 87 54 14 97 64 24 107 74 34 117 44)(5 45 118 35 75 108 25 65 98 15 55 88)(6 89 56 16 99 66 26 109 76 36 119 46)(7 47 120 37 77 110 27 67 100 17 57 90)(8 91 58 18 101 68 28 111 78 38 81 48)(9 49 82 39 79 112 29 69 102 19 59 92)(10 93 60 20 103 70 30 113 80 40 83 50)

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

57 conjugacy classes

class 1 2A2B2C2D 3 4A4B5A5B6A6B6C6D6E6F6G8A8B8C8D10A10B10C10D12A12B15A15B20A20B20C20D20E20F30A···30F40A···40H60A···60H
order122223445566666668888101010101212151520202020202030···3040···4060···60
size1122020222222222020202012126060224444442222444···412···124···4

57 irreducible representations

dim11111222222222244444444
type++++++++-++++-+-+
imageC1C2C2C2C4S3D4D5D6Dic3D10C3⋊D4D20C5⋊D4C4×D5C4.D4S3×D5C12.D4C15⋊D4C3⋊D20D5×Dic3C20.46D4C12.5D20
kernelC12.5D20C5×C4.Dic3C60.7C4C6×D20D5×C2×C6C2×D20C60C4.Dic3C2×C20C22×D5C2×C12C20C12C12C2×C6C15C2×C4C5C4C4C22C3C1
# reps11114122122444412222248

Matrix representation of C12.5D20 ►in GL4(𝔽241) generated by

1784500
1966300
0022248
0019319
,
0001
0024051
23819700
4416300
,
0001
0010
23819700
44300
G:=sub<GL(4,GF(241))| [178,196,0,0,45,63,0,0,0,0,222,193,0,0,48,19],[0,0,238,44,0,0,197,163,0,240,0,0,1,51,0,0],[0,0,238,44,0,0,197,3,0,1,0,0,1,0,0,0] >;
 

C12.5D20 in GAP, Magma, Sage, TeX 

C_{12}._5D_{20}
 
% in TeX
 
G:=Group("C12.5D20");
 
// GroupNames label
 
G:=SmallGroup(480,34);
 
// by ID
 
G=gap.SmallGroup(480,34);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,422,100,346,1356,18822]);
 
// Polycyclic
 
G:=Group<a,b,c|a^12=1,b^20=a^6,c^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^3*b^19>;
 
// generators/relations
 

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