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## G = C5⋊2U2(𝔽3)  order 480 = 25·3·5

### The semidirect product of C5 and U2(𝔽3) acting via U2(𝔽3)/C4.A4=C2

Aliases: C20.5S4, C52U2(𝔽3), Q8.Dic15, SL2(𝔽3)⋊2Dic5, C4.5(C5⋊S4), C4.A4.2D5, C4○D4.1D15, C10.6(A4⋊C4), (C5×Q8).2Dic3, C2.3(A4⋊Dic5), (C5×SL2(𝔽3))⋊5C4, (C5×C4.A4).2C2, (C5×C4○D4).1S3, SmallGroup(480,261)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C5×SL2(𝔽3) — C5⋊2U2(𝔽3)
 Chief series C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — C5×C4.A4 — C5⋊2U2(𝔽3)
 Lower central C5×SL2(𝔽3) — C5⋊2U2(𝔽3)
 Upper central C1 — C4

Generators and relations for C52U2(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, faf-1=a-1, 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 C52U2(𝔽3)
On 120 points
Generators in S120
(1 38 10 30 18)(2 19 31 11 39)(3 40 12 32 20)(4 21 25 13 33)(5 34 14 26 22)(6 23 27 15 35)(7 36 16 28 24)(8 17 29 9 37)(41 85 73 93 65)(42 66 94 74 86)(43 87 75 95 67)(44 68 96 76 88)(45 81 77 89 69)(46 70 90 78 82)(47 83 79 91 71)(48 72 92 80 84)(49 117 101 57 109)(50 110 58 102 118)(51 119 103 59 111)(52 112 60 104 120)(53 113 97 61 105)(54 106 62 98 114)(55 115 99 63 107)(56 108 64 100 116)
(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 4 5 8)(2 3 6 7)(9 30 13 26)(10 25 14 29)(11 32 15 28)(12 27 16 31)(17 38 21 34)(18 33 22 37)(19 40 23 36)(20 35 24 39)(41 100 45 104)(42 44 46 48)(43 102 47 98)(49 55 53 51)(50 79 54 75)(52 73 56 77)(57 63 61 59)(58 71 62 67)(60 65 64 69)(66 68 70 72)(74 76 78 80)(81 120 85 116)(82 84 86 88)(83 114 87 118)(89 112 93 108)(90 92 94 96)(91 106 95 110)(97 103 101 99)(105 111 109 107)(113 119 117 115)
(1 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)(17 23 21 19)(18 20 22 24)(25 31 29 27)(26 28 30 32)(33 39 37 35)(34 36 38 40)(41 102 45 98)(42 99 46 103)(43 104 47 100)(44 101 48 97)(49 80 53 76)(50 77 54 73)(51 74 55 78)(52 79 56 75)(57 72 61 68)(58 69 62 65)(59 66 63 70)(60 71 64 67)(81 114 85 118)(82 119 86 115)(83 116 87 120)(84 113 88 117)(89 106 93 110)(90 111 94 107)(91 108 95 112)(92 105 96 109)
(1 102 42)(2 43 103)(3 104 44)(4 45 97)(5 98 46)(6 47 99)(7 100 48)(8 41 101)(9 93 49)(10 50 94)(11 95 51)(12 52 96)(13 89 53)(14 54 90)(15 91 55)(16 56 92)(17 85 57)(18 58 86)(19 87 59)(20 60 88)(21 81 61)(22 62 82)(23 83 63)(24 64 84)(25 77 105)(26 106 78)(27 79 107)(28 108 80)(29 73 109)(30 110 74)(31 75 111)(32 112 76)(33 69 113)(34 114 70)(35 71 115)(36 116 72)(37 65 117)(38 118 66)(39 67 119)(40 120 68)
(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)| (1,38,10,30,18)(2,19,31,11,39)(3,40,12,32,20)(4,21,25,13,33)(5,34,14,26,22)(6,23,27,15,35)(7,36,16,28,24)(8,17,29,9,37)(41,85,73,93,65)(42,66,94,74,86)(43,87,75,95,67)(44,68,96,76,88)(45,81,77,89,69)(46,70,90,78,82)(47,83,79,91,71)(48,72,92,80,84)(49,117,101,57,109)(50,110,58,102,118)(51,119,103,59,111)(52,112,60,104,120)(53,113,97,61,105)(54,106,62,98,114)(55,115,99,63,107)(56,108,64,100,116), (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), (1,4,5,8)(2,3,6,7)(9,30,13,26)(10,25,14,29)(11,32,15,28)(12,27,16,31)(17,38,21,34)(18,33,22,37)(19,40,23,36)(20,35,24,39)(41,100,45,104)(42,44,46,48)(43,102,47,98)(49,55,53,51)(50,79,54,75)(52,73,56,77)(57,63,61,59)(58,71,62,67)(60,65,64,69)(66,68,70,72)(74,76,78,80)(81,120,85,116)(82,84,86,88)(83,114,87,118)(89,112,93,108)(90,92,94,96)(91,106,95,110)(97,103,101,99)(105,111,109,107)(113,119,117,115), (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16)(17,23,21,19)(18,20,22,24)(25,31,29,27)(26,28,30,32)(33,39,37,35)(34,36,38,40)(41,102,45,98)(42,99,46,103)(43,104,47,100)(44,101,48,97)(49,80,53,76)(50,77,54,73)(51,74,55,78)(52,79,56,75)(57,72,61,68)(58,69,62,65)(59,66,63,70)(60,71,64,67)(81,114,85,118)(82,119,86,115)(83,116,87,120)(84,113,88,117)(89,106,93,110)(90,111,94,107)(91,108,95,112)(92,105,96,109), (1,102,42)(2,43,103)(3,104,44)(4,45,97)(5,98,46)(6,47,99)(7,100,48)(8,41,101)(9,93,49)(10,50,94)(11,95,51)(12,52,96)(13,89,53)(14,54,90)(15,91,55)(16,56,92)(17,85,57)(18,58,86)(19,87,59)(20,60,88)(21,81,61)(22,62,82)(23,83,63)(24,64,84)(25,77,105)(26,106,78)(27,79,107)(28,108,80)(29,73,109)(30,110,74)(31,75,111)(32,112,76)(33,69,113)(34,114,70)(35,71,115)(36,116,72)(37,65,117)(38,118,66)(39,67,119)(40,120,68), (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( (1,38,10,30,18)(2,19,31,11,39)(3,40,12,32,20)(4,21,25,13,33)(5,34,14,26,22)(6,23,27,15,35)(7,36,16,28,24)(8,17,29,9,37)(41,85,73,93,65)(42,66,94,74,86)(43,87,75,95,67)(44,68,96,76,88)(45,81,77,89,69)(46,70,90,78,82)(47,83,79,91,71)(48,72,92,80,84)(49,117,101,57,109)(50,110,58,102,118)(51,119,103,59,111)(52,112,60,104,120)(53,113,97,61,105)(54,106,62,98,114)(55,115,99,63,107)(56,108,64,100,116), (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), (1,4,5,8)(2,3,6,7)(9,30,13,26)(10,25,14,29)(11,32,15,28)(12,27,16,31)(17,38,21,34)(18,33,22,37)(19,40,23,36)(20,35,24,39)(41,100,45,104)(42,44,46,48)(43,102,47,98)(49,55,53,51)(50,79,54,75)(52,73,56,77)(57,63,61,59)(58,71,62,67)(60,65,64,69)(66,68,70,72)(74,76,78,80)(81,120,85,116)(82,84,86,88)(83,114,87,118)(89,112,93,108)(90,92,94,96)(91,106,95,110)(97,103,101,99)(105,111,109,107)(113,119,117,115), (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16)(17,23,21,19)(18,20,22,24)(25,31,29,27)(26,28,30,32)(33,39,37,35)(34,36,38,40)(41,102,45,98)(42,99,46,103)(43,104,47,100)(44,101,48,97)(49,80,53,76)(50,77,54,73)(51,74,55,78)(52,79,56,75)(57,72,61,68)(58,69,62,65)(59,66,63,70)(60,71,64,67)(81,114,85,118)(82,119,86,115)(83,116,87,120)(84,113,88,117)(89,106,93,110)(90,111,94,107)(91,108,95,112)(92,105,96,109), (1,102,42)(2,43,103)(3,104,44)(4,45,97)(5,98,46)(6,47,99)(7,100,48)(8,41,101)(9,93,49)(10,50,94)(11,95,51)(12,52,96)(13,89,53)(14,54,90)(15,91,55)(16,56,92)(17,85,57)(18,58,86)(19,87,59)(20,60,88)(21,81,61)(22,62,82)(23,83,63)(24,64,84)(25,77,105)(26,106,78)(27,79,107)(28,108,80)(29,73,109)(30,110,74)(31,75,111)(32,112,76)(33,69,113)(34,114,70)(35,71,115)(36,116,72)(37,65,117)(38,118,66)(39,67,119)(40,120,68), (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([[(1,38,10,30,18),(2,19,31,11,39),(3,40,12,32,20),(4,21,25,13,33),(5,34,14,26,22),(6,23,27,15,35),(7,36,16,28,24),(8,17,29,9,37),(41,85,73,93,65),(42,66,94,74,86),(43,87,75,95,67),(44,68,96,76,88),(45,81,77,89,69),(46,70,90,78,82),(47,83,79,91,71),(48,72,92,80,84),(49,117,101,57,109),(50,110,58,102,118),(51,119,103,59,111),(52,112,60,104,120),(53,113,97,61,105),(54,106,62,98,114),(55,115,99,63,107),(56,108,64,100,116)], [(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)], [(1,4,5,8),(2,3,6,7),(9,30,13,26),(10,25,14,29),(11,32,15,28),(12,27,16,31),(17,38,21,34),(18,33,22,37),(19,40,23,36),(20,35,24,39),(41,100,45,104),(42,44,46,48),(43,102,47,98),(49,55,53,51),(50,79,54,75),(52,73,56,77),(57,63,61,59),(58,71,62,67),(60,65,64,69),(66,68,70,72),(74,76,78,80),(81,120,85,116),(82,84,86,88),(83,114,87,118),(89,112,93,108),(90,92,94,96),(91,106,95,110),(97,103,101,99),(105,111,109,107),(113,119,117,115)], [(1,3,5,7),(2,8,6,4),(9,15,13,11),(10,12,14,16),(17,23,21,19),(18,20,22,24),(25,31,29,27),(26,28,30,32),(33,39,37,35),(34,36,38,40),(41,102,45,98),(42,99,46,103),(43,104,47,100),(44,101,48,97),(49,80,53,76),(50,77,54,73),(51,74,55,78),(52,79,56,75),(57,72,61,68),(58,69,62,65),(59,66,63,70),(60,71,64,67),(81,114,85,118),(82,119,86,115),(83,116,87,120),(84,113,88,117),(89,106,93,110),(90,111,94,107),(91,108,95,112),(92,105,96,109)], [(1,102,42),(2,43,103),(3,104,44),(4,45,97),(5,98,46),(6,47,99),(7,100,48),(8,41,101),(9,93,49),(10,50,94),(11,95,51),(12,52,96),(13,89,53),(14,54,90),(15,91,55),(16,56,92),(17,85,57),(18,58,86),(19,87,59),(20,60,88),(21,81,61),(22,62,82),(23,83,63),(24,64,84),(25,77,105),(26,106,78),(27,79,107),(28,108,80),(29,73,109),(30,110,74),(31,75,111),(32,112,76),(33,69,113),(34,114,70),(35,71,115),(36,116,72),(37,65,117),(38,118,66),(39,67,119),(40,120,68)], [(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)]])

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 3 3 4 4 6 6 type + + + + - - + - + + - image C1 C2 C4 S3 D5 Dic3 Dic5 D15 Dic15 U2(𝔽3) S4 A4⋊C4 U2(𝔽3) C5⋊2U2(𝔽3) C5⋊S4 A4⋊Dic5 kernel C5⋊2U2(𝔽3) C5×C4.A4 C5×SL2(𝔽3) C5×C4○D4 C4.A4 C5×Q8 SL2(𝔽3) C4○D4 Q8 C5 C20 C10 C5 C1 C4 C2 # reps 1 1 2 1 2 1 2 4 4 4 2 2 2 12 2 2

Matrix representation of C52U2(𝔽3) in GL4(𝔽241) generated by

 0 1 0 0 240 51 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 64 0 0 0 0 64
,
 1 0 0 0 0 1 0 0 0 0 0 177 0 0 177 0
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 240 0
,
 1 0 0 0 0 1 0 0 0 0 88 152 0 0 153 152
,
 1 0 0 0 51 240 0 0 0 0 88 88 0 0 88 153
G:=sub<GL(4,GF(241))| [0,240,0,0,1,51,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,0,177,0,0,177,0],[1,0,0,0,0,1,0,0,0,0,0,240,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,88,153,0,0,152,152],[1,51,0,0,0,240,0,0,0,0,88,88,0,0,88,153] >;

C52U2(𝔽3) in GAP, Magma, Sage, TeX

C_5\rtimes_2{\rm U}_2({\mathbb F}_3)
% in TeX

G:=Group("C5:2U(2,3)");
// GroupNames label

G:=SmallGroup(480,261);
// by ID

G=gap.SmallGroup(480,261);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,14,1688,170,1347,4204,3168,172,2525,1909,285,124]);
// 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,f*a*f^-1=a^-1,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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