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## G = C5×U2(𝔽3)  order 480 = 25·3·5

### Direct product of C5 and U2(𝔽3)

Aliases: C5×U2(𝔽3), C20.11S4, SL2(𝔽3)⋊2C20, C4.5(C5×S4), Q8.(C5×Dic3), C4.A4.2C10, C10.9(A4⋊C4), (C5×Q8).3Dic3, (C5×SL2(𝔽3))⋊8C4, C2.3(C5×A4⋊C4), C4○D4.1(C5×S3), (C5×C4.A4).5C2, (C5×C4○D4).3S3, SmallGroup(480,257)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C5×U2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C4.A4 — C5×C4.A4 — C5×U2(𝔽3)
 Lower central SL2(𝔽3) — C5×U2(𝔽3)
 Upper central C1 — C20

Generators and relations for C5×U2(𝔽3)
G = < a,b,c,d,e,f | a5=b4=e3=1, c2=d2=b2, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=b2c, ece-1=b2cd, fcf-1=cd, ede-1=c, fdf-1=b2d, fef-1=e-1 >

Smallest permutation representation of C5×U2(𝔽3)
On 120 points
Generators in S120
(1 98 74 50 26)(2 99 75 51 27)(3 100 76 52 28)(4 101 77 53 29)(5 102 78 54 30)(6 103 79 55 31)(7 104 80 56 32)(8 97 73 49 25)(9 105 81 57 33)(10 106 82 58 34)(11 107 83 59 35)(12 108 84 60 36)(13 109 85 61 37)(14 110 86 62 38)(15 111 87 63 39)(16 112 88 64 40)(17 113 89 65 41)(18 114 90 66 42)(19 115 91 67 43)(20 116 92 68 44)(21 117 93 69 45)(22 118 94 70 46)(23 119 95 71 47)(24 120 96 72 48)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)
(1 2 5 6)(3 4 7 8)(9 15 13 11)(10 18 14 22)(12 20 16 24)(17 19 21 23)(25 28 29 32)(26 27 30 31)(33 39 37 35)(34 42 38 46)(36 44 40 48)(41 43 45 47)(49 52 53 56)(50 51 54 55)(57 63 61 59)(58 66 62 70)(60 68 64 72)(65 67 69 71)(73 76 77 80)(74 75 78 79)(81 87 85 83)(82 90 86 94)(84 92 88 96)(89 91 93 95)(97 100 101 104)(98 99 102 103)(105 111 109 107)(106 114 110 118)(108 116 112 120)(113 115 117 119)
(1 7 5 3)(2 4 6 8)(9 19 13 23)(10 24 14 20)(11 21 15 17)(12 18 16 22)(25 27 29 31)(26 32 30 28)(33 43 37 47)(34 48 38 44)(35 45 39 41)(36 42 40 46)(49 51 53 55)(50 56 54 52)(57 67 61 71)(58 72 62 68)(59 69 63 65)(60 66 64 70)(73 75 77 79)(74 80 78 76)(81 91 85 95)(82 96 86 92)(83 93 87 89)(84 90 88 94)(97 99 101 103)(98 104 102 100)(105 115 109 119)(106 120 110 116)(107 117 111 113)(108 114 112 118)
(1 18 15)(2 16 19)(3 20 9)(4 10 21)(5 22 11)(6 12 23)(7 24 13)(8 14 17)(25 38 41)(26 42 39)(27 40 43)(28 44 33)(29 34 45)(30 46 35)(31 36 47)(32 48 37)(49 62 65)(50 66 63)(51 64 67)(52 68 57)(53 58 69)(54 70 59)(55 60 71)(56 72 61)(73 86 89)(74 90 87)(75 88 91)(76 92 81)(77 82 93)(78 94 83)(79 84 95)(80 96 85)(97 110 113)(98 114 111)(99 112 115)(100 116 105)(101 106 117)(102 118 107)(103 108 119)(104 120 109)
(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)

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

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

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

80 conjugacy classes

 class 1 2A 2B 3 4A 4B 4C ··· 4G 5A 5B 5C 5D 6 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 12A 12B 15A 15B 15C 15D 20A ··· 20H 20I ··· 20AB 30A 30B 30C 30D 40A ··· 40H 60A ··· 60H order 1 2 2 3 4 4 4 ··· 4 5 5 5 5 6 8 8 10 10 10 10 10 10 10 10 12 12 15 15 15 15 20 ··· 20 20 ··· 20 30 30 30 30 40 ··· 40 60 ··· 60 size 1 1 6 8 1 1 6 ··· 6 1 1 1 1 8 12 12 1 1 1 1 6 6 6 6 8 8 8 8 8 8 1 ··· 1 6 ··· 6 8 8 8 8 12 ··· 12 8 ··· 8

80 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 4 4 type + + + - + image C1 C2 C4 C5 C10 C20 S3 Dic3 C5×S3 C5×Dic3 U2(𝔽3) C5×U2(𝔽3) S4 A4⋊C4 C5×S4 C5×A4⋊C4 U2(𝔽3) C5×U2(𝔽3) kernel C5×U2(𝔽3) C5×C4.A4 C5×SL2(𝔽3) U2(𝔽3) C4.A4 SL2(𝔽3) C5×C4○D4 C5×Q8 C4○D4 Q8 C5 C1 C20 C10 C4 C2 C5 C1 # reps 1 1 2 4 4 8 1 1 4 4 4 16 2 2 8 8 2 8

Matrix representation of C5×U2(𝔽3) in GL2(𝔽41) generated by

 16 0 0 16
,
 32 0 0 32
,
 1 4 20 40
,
 32 5 0 9
,
 8 16 39 32
,
 9 20 16 32
G:=sub<GL(2,GF(41))| [16,0,0,16],[32,0,0,32],[1,20,4,40],[32,0,5,9],[8,39,16,32],[9,16,20,32] >;

C5×U2(𝔽3) in GAP, Magma, Sage, TeX

C_5\times {\rm U}_2({\mathbb F}_3)
% in TeX

G:=Group("C5xU(2,3)");
// GroupNames label

G:=SmallGroup(480,257);
// by ID

G=gap.SmallGroup(480,257);
# by ID

G:=PCGroup([7,-2,-5,-2,-3,-2,2,-2,70,520,1123,4204,655,172,2525,404,285,124]);
// Polycyclic

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

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