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G = C5⋊(C12.D4)  order 480 = 25·3·5

The semidirect product of C5 and C12.D4 acting via C12.D4/C22×C6=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5⋊(C12.D4), C32(C23.F5), C23.2(C3⋊F5), (C22×C6).2F5, C155(C4.D4), (C22×C30).5C4, C158M4(2)⋊5C2, (C2×Dic5).74D6, (C3×Dic5).39D4, C6.26(C22⋊F5), C30.26(C22⋊C4), Dic5.8(C3⋊D4), (C22×D5).2Dic3, (C22×C10).7Dic3, C10.11(C6.D4), (C6×Dic5).169C22, C2.11(D10.D6), (D5×C2×C6).3C4, C22.5(C2×C3⋊F5), (C2×C5⋊D4).8S3, (C2×C6).42(C2×F5), (C2×C30).36(C2×C4), (C6×C5⋊D4).15C2, (C2×C10).12(C2×Dic3), SmallGroup(480,318)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C5⋊(C12.D4)
C1C5C15C30C3×Dic5C6×Dic5C158M4(2) — C5⋊(C12.D4)
C15C30C2×C30 — C5⋊(C12.D4)
C1C2C22C23

Generators and relations for C5⋊(C12.D4)
 G = < a,b,c,d | a5=b12=1, c4=b6, d2=b9, bab-1=a-1, cac-1=dad-1=a2, cbc-1=b-1, dbd-1=b5, dcd-1=b3c3 >

Subgroups: 460 in 92 conjugacy classes, 29 normal (23 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C22, C22 [×4], C5, C6, C6 [×3], C8 [×2], C2×C4, D4 [×2], C23, C23, D5, C10, C10 [×2], C12 [×2], C2×C6, C2×C6 [×4], C15, M4(2) [×2], C2×D4, Dic5 [×2], D10 [×2], C2×C10, C2×C10 [×2], C3⋊C8 [×2], C2×C12, C3×D4 [×2], C22×C6, C22×C6, C3×D5, C30, C30 [×2], C4.D4, C5⋊C8 [×2], C2×Dic5, C5⋊D4 [×2], C22×D5, C22×C10, C4.Dic3 [×2], C6×D4, C3×Dic5 [×2], C6×D5 [×2], C2×C30, C2×C30 [×2], C22.F5 [×2], C2×C5⋊D4, C12.D4, C15⋊C8 [×2], C6×Dic5, C3×C5⋊D4 [×2], D5×C2×C6, C22×C30, C23.F5, C158M4(2) [×2], C6×C5⋊D4, C5⋊(C12.D4)
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, F5, C2×Dic3, C3⋊D4 [×2], C4.D4, C2×F5, C6.D4, C3⋊F5, C22⋊F5, C12.D4, C2×C3⋊F5, C23.F5, D10.D6, C5⋊(C12.D4)

Smallest permutation representation of C5⋊(C12.D4)
On 120 points
Generators in S120
(1 92 102 83 64)(2 65 84 103 93)(3 94 104 73 66)(4 67 74 105 95)(5 96 106 75 68)(6 69 76 107 85)(7 86 108 77 70)(8 71 78 97 87)(9 88 98 79 72)(10 61 80 99 89)(11 90 100 81 62)(12 63 82 101 91)(13 115 49 28 46)(14 47 29 50 116)(15 117 51 30 48)(16 37 31 52 118)(17 119 53 32 38)(18 39 33 54 120)(19 109 55 34 40)(20 41 35 56 110)(21 111 57 36 42)(22 43 25 58 112)(23 113 59 26 44)(24 45 27 60 114)
(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 38 10 41 7 44 4 47)(2 37 11 40 8 43 5 46)(3 48 12 39 9 42 6 45)(13 103 16 100 19 97 22 106)(14 102 17 99 20 108 23 105)(15 101 18 98 21 107 24 104)(25 75 28 84 31 81 34 78)(26 74 29 83 32 80 35 77)(27 73 30 82 33 79 36 76)(49 93 52 90 55 87 58 96)(50 92 53 89 56 86 59 95)(51 91 54 88 57 85 60 94)(61 110 70 113 67 116 64 119)(62 109 71 112 68 115 65 118)(63 120 72 111 69 114 66 117)
(1 44 10 41 7 38 4 47)(2 37 11 46 8 43 5 40)(3 42 12 39 9 48 6 45)(13 97 22 106 19 103 16 100)(14 102 23 99 20 108 17 105)(15 107 24 104 21 101 18 98)(25 75 34 84 31 81 28 78)(26 80 35 77 32 74 29 83)(27 73 36 82 33 79 30 76)(49 87 58 96 55 93 52 90)(50 92 59 89 56 86 53 95)(51 85 60 94 57 91 54 88)(61 110 70 119 67 116 64 113)(62 115 71 112 68 109 65 118)(63 120 72 117 69 114 66 111)

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

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

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

45 conjugacy classes

class 1 2A2B2C2D 3 4A4B 5 6A6B6C6D6E6F6G8A8B8C8D10A···10G12A12B15A15B30A···30N
order1222234456666666888810···101212151530···30
size112420210104222442020606060604···42020444···4

45 irreducible representations

dim111112222224444444444
type++++++--++++
imageC1C2C2C4C4S3D4D6Dic3Dic3C3⋊D4F5C4.D4C2×F5C3⋊F5C22⋊F5C12.D4C2×C3⋊F5C23.F5D10.D6C5⋊(C12.D4)
kernelC5⋊(C12.D4)C158M4(2)C6×C5⋊D4D5×C2×C6C22×C30C2×C5⋊D4C3×Dic5C2×Dic5C22×D5C22×C10Dic5C22×C6C15C2×C6C23C6C5C22C3C2C1
# reps121221211141112222448

Matrix representation of C5⋊(C12.D4) in GL4(𝔽241) generated by

91000
09800
002050
00087
,
022500
16000
00015
002260
,
0010
000240
0100
1000
,
0010
0001
0100
240000
G:=sub<GL(4,GF(241))| [91,0,0,0,0,98,0,0,0,0,205,0,0,0,0,87],[0,16,0,0,225,0,0,0,0,0,0,226,0,0,15,0],[0,0,0,1,0,0,1,0,1,0,0,0,0,240,0,0],[0,0,0,240,0,0,1,0,1,0,0,0,0,1,0,0] >;

C5⋊(C12.D4) in GAP, Magma, Sage, TeX

C_5\rtimes (C_{12}.D_4)
% in TeX

G:=Group("C5:(C12.D4)");
// GroupNames label

G:=SmallGroup(480,318);
// by ID

G=gap.SmallGroup(480,318);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,219,100,675,2693,14118,4724]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^12=1,c^4=b^6,d^2=b^9,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a^2,c*b*c^-1=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^3*c^3>;
// generators/relations

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