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G = C30.3C42order 480 = 25·3·5

3rd non-split extension by C30 of C42 acting via C42/C2=C2×C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C30.3C42, C3⋊C83F5, C153C85C4, C15⋊C82C4, C6.8(C4×F5), C33(C8⋊F5), C151(C8⋊C4), (C2×F5).Dic3, (C4×F5).2S3, (C6×F5).2C4, C4.24(S3×F5), C20.24(C4×S3), C60.24(C2×C4), (C4×D5).70D6, (C12×F5).3C2, C12.31(C2×F5), C2.4(Dic3×F5), C10.3(C4×Dic3), D5.(C4.Dic3), C60.C4.3C2, C51(C42.S3), D10.6(C2×Dic3), Dic5.10(C4×S3), (C3×D5).1M4(2), (D5×C12).62C22, (C5×C3⋊C8)⋊5C4, (D5×C3⋊C8).8C2, (C6×D5).11(C2×C4), (C3×Dic5).15(C2×C4), SmallGroup(480,225)

Series: Derived Chief Lower central Upper central

C1C30 — C30.3C42
C1C5C15C30C6×D5D5×C12C12×F5 — C30.3C42
C15C30 — C30.3C42
C1C4

Generators and relations for C30.3C42
 G = < a,b,c | a30=b4=1, c4=a15, bab-1=a13, cac-1=a11, cbc-1=a15b >

Subgroups: 308 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, C12, C12 [×3], C2×C6, C15, C42, C2×C8 [×2], Dic5, C20, F5 [×2], D10, C3⋊C8, C3⋊C8 [×3], C2×C12 [×3], C3×D5 [×2], C30, C8⋊C4, C52C8, C40, C5⋊C8 [×2], C4×D5, C2×F5 [×2], C2×C3⋊C8 [×2], C4×C12, C3×Dic5, C60, C3×F5 [×2], C6×D5, C8×D5, D5⋊C8, C4×F5, C42.S3, C5×C3⋊C8, C153C8, C15⋊C8 [×2], D5×C12, C6×F5 [×2], C8⋊F5, D5×C3⋊C8, C12×F5, C60.C4, C30.3C42
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, C4.Dic3 [×2], C4×Dic3, C4×F5, C42.S3, S3×F5, C8⋊F5, Dic3×F5, C30.3C42

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

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

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

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

48 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H 5 6A6B6C8A8B8C···8H 10 12A12B12C···12L 15 20A20B 30 40A40B40C40D60A60B
order12223444444445666888···810121212···1215202030404040406060
size115521155101010104210106630···3042210···1084481212121288

48 irreducible representations

dim1111111122222224444888
type++++++-+++-
imageC1C2C2C2C4C4C4C4S3D6Dic3M4(2)C4×S3C4×S3C4.Dic3F5C2×F5C4×F5C8⋊F5S3×F5Dic3×F5C30.3C42
kernelC30.3C42D5×C3⋊C8C12×F5C60.C4C5×C3⋊C8C153C8C15⋊C8C6×F5C4×F5C4×D5C2×F5C3×D5Dic5C20D5C3⋊C8C12C6C3C4C2C1
# reps1111224411242281124112

Matrix representation of C30.3C42 in GL6(𝔽241)

02400000
12400000
00024000
00002400
00000240
001111
,
100000
010000
000010
001000
000001
000100
,
24000000
24010000
002130185185
005628560
000562856
001851850213

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

C30.3C42 in GAP, Magma, Sage, TeX

C_{30}._3C_4^2
% in TeX

G:=Group("C30.3C4^2");
// GroupNames label

G:=SmallGroup(480,225);
// by ID

G=gap.SmallGroup(480,225);
# by ID

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

G:=Group<a,b,c|a^30=b^4=1,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

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