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G = C6×C4⋊F5  order 480 = 25·3·5

Direct product of C6 and C4⋊F5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6×C4⋊F5, C4⋊2(C6×F5), (C2×C60)⋊8C4, C60⋊9(C2×C4), (C2×C12)⋊8F5, C12⋊9(C2×F5), C30⋊3(C4⋊C4), (C2×C20)⋊3C12, C20⋊2(C2×C12), (C4×D5)⋊4C12, D5.1(C6×D4), D5.1(C6×Q8), (D5×C12)⋊11C4, (C6×D5).56D4, (C6×D5).14Q8, D10.4(C3×Q8), (C6×Dic5)⋊18C4, Dic5⋊7(C2×C12), (C2×Dic5)⋊8C12, D10.10(C3×D4), (C22×F5).2C6, C6.49(C22×F5), C22.18(C6×F5), D10.15(C2×C12), C30.87(C22×C4), C10.5(C22×C12), (C6×D5).68C23, D10.9(C22×C6), (C6×F5).14C22, (D5×C12).133C22, C5⋊(C6×C4⋊C4), C10⋊(C3×C4⋊C4), D5⋊(C3×C4⋊C4), C15⋊4(C2×C4⋊C4), C2.6(C2×C6×F5), (C2×C4)⋊3(C3×F5), (C2×C6×F5).5C2, (C2×C4×D5).14C6, (C3×D5)⋊5(C4⋊C4), (D5×C2×C12).35C2, (C2×F5).1(C2×C6), (C2×C6).61(C2×F5), (C3×D5).6(C2×Q8), (C2×C30).60(C2×C4), (C3×D5).10(C2×D4), (C6×D5).64(C2×C4), (C4×D5).31(C2×C6), (C2×C10).17(C2×C12), (C3×Dic5)⋊28(C2×C4), (D5×C2×C6).150C22, (C22×D5).39(C2×C6), SmallGroup(480,1051)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — C6×C4⋊F5
Chief series
C1 — C5 — C10 — D10 — C6×D5 — C6×F5 — C2×C6×F5 — C6×C4⋊F5
Lower central
C5 — C10 — C6×C4⋊F5

Subgroups: 584 in 184 conjugacy classes, 92 normal (32 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×6], C22, C22 [×6], C5, C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×13], C23, D5 [×2], D5 [×2], C10, C10 [×2], C12 [×2], C12 [×6], C2×C6, C2×C6 [×6], C15, C4⋊C4 [×4], C22×C4 [×3], Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×4], C2×C10, C2×C12, C2×C12 [×13], C22×C6, C3×D5 [×2], C3×D5 [×2], C30, C30 [×2], C2×C4⋊C4, C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×4], C2×F5 [×4], C22×D5, C3×C4⋊C4 [×4], C22×C12 [×3], C3×Dic5 [×2], C60 [×2], C3×F5 [×4], C6×D5 [×2], C6×D5 [×4], C2×C30, C4⋊F5 [×4], C2×C4×D5, C22×F5 [×2], C6×C4⋊C4, D5×C12 [×4], C6×Dic5, C2×C60, C6×F5 [×4], C6×F5 [×4], D5×C2×C6, C2×C4⋊F5, C3×C4⋊F5 [×4], D5×C2×C12, C2×C6×F5 [×2], C6×C4⋊F5

Quotients:
C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C12 [×4], C2×C6 [×7], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, F5, C2×C12 [×6], C3×D4 [×2], C3×Q8 [×2], C22×C6, C2×C4⋊C4, C2×F5 [×3], C3×C4⋊C4 [×4], C22×C12, C6×D4, C6×Q8, C3×F5, C4⋊F5 [×2], C22×F5, C6×C4⋊C4, C6×F5 [×3], C2×C4⋊F5, C3×C4⋊F5 [×2], C2×C6×F5, C6×C4⋊F5

Generators and relations
 G = < a,b,c,d | a6=b4=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

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

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

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

Matrix representation ►G ⊆ GL8(𝔽61)

480000000
048000000
004700000
000470000
000047000
000004700
000000470
000000047
,
4256000000
4819000000
0042560000
0048190000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00000100
00000010
00000001
000060606060
,
2655000000
135000000
003560000
0060260000
00001000
00000001
00000100
000060606060

G:=sub<GL(8,GF(61))| [48,0,0,0,0,0,0,0,0,48,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,0,47],[42,48,0,0,0,0,0,0,56,19,0,0,0,0,0,0,0,0,42,48,0,0,0,0,0,0,56,19,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,60,0,0,0,0,1,0,0,60,0,0,0,0,0,1,0,60,0,0,0,0,0,0,1,60],[26,1,0,0,0,0,0,0,55,35,0,0,0,0,0,0,0,0,35,60,0,0,0,0,0,0,6,26,0,0,0,0,0,0,0,0,1,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,60,0,0,0,0,0,1,0,60] >;
 

84 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B4C···4L 5 6A···6F6G···6N10A10B10C12A12B12C12D12E···12X15A15B20A20B20C20D30A···30F60A···60H
order1222222233444···456···66···61010101212121212···1215152020202030···3060···60
size11115555112210···1041···15···5444222210···104444444···44···4

84 irreducible representations

dim11111111111111222244444444
type+++++-+++
imageC1C2C2C2C3C4C4C4C6C6C6C12C12C12D4Q8C3×D4C3×Q8F5C2×F5C2×F5C3×F5C4⋊F5C6×F5C6×F5C3×C4⋊F5
kernelC6×C4⋊F5C3×C4⋊F5D5×C2×C12C2×C6×F5C2×C4⋊F5D5×C12C6×Dic5C2×C60C4⋊F5C2×C4×D5C22×F5C4×D5C2×Dic5C2×C20C6×D5C6×D5D10D10C2×C12C12C2×C6C2×C4C6C4C22C2
# reps14122422824844224412124428

In GAP, Magma, Sage, TeX 

C_6\times C_4\rtimes F_5
 
% in TeX
 
G:=Group("C6xC4:F5");
 
// GroupNames label
 
G:=SmallGroup(480,1051);
 
// by ID
 
G=gap.SmallGroup(480,1051);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,1094,268,9414,818]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^4=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
 
// generators/relations
 

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