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## G = C3×D5⋊M4(2)  order 480 = 25·3·5

### Direct product of C3 and D5⋊M4(2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×D5⋊M4(2)
 Chief series C1 — C5 — C10 — Dic5 — C3×Dic5 — C3×C5⋊C8 — C3×D5⋊C8 — C3×D5⋊M4(2)
 Lower central C5 — C10 — C3×D5⋊M4(2)
 Upper central C1 — C12 — C2×C12

Generators and relations for C3×D5⋊M4(2)
G = < a,b,c,d,e | a3=b5=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=b3, be=eb, dcd-1=b2c, ce=ec, ede=d5 >

Subgroups: 392 in 136 conjugacy classes, 72 normal (48 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C24, C2×C12, C2×C12, C22×C6, C3×D5, C3×D5, C30, C30, C2×M4(2), C5⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C24, C3×M4(2), C22×C12, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, D5⋊C8, C4.F5, C22.F5, C2×C4×D5, C6×M4(2), C3×C5⋊C8, D5×C12, C6×Dic5, C2×C60, D5×C2×C6, D5⋊M4(2), C3×D5⋊C8, C3×C4.F5, C3×C22.F5, D5×C2×C12, C3×D5⋊M4(2)
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, M4(2), C22×C4, F5, C2×C12, C22×C6, C2×M4(2), C2×F5, C3×M4(2), C22×C12, C3×F5, C22×F5, C6×M4(2), C6×F5, D5⋊M4(2), C2×C6×F5, C3×D5⋊M4(2)

Smallest permutation representation of C3×D5⋊M4(2)
On 120 points
Generators in S120
(1 104 60)(2 97 61)(3 98 62)(4 99 63)(5 100 64)(6 101 57)(7 102 58)(8 103 59)(9 82 111)(10 83 112)(11 84 105)(12 85 106)(13 86 107)(14 87 108)(15 88 109)(16 81 110)(17 46 67)(18 47 68)(19 48 69)(20 41 70)(21 42 71)(22 43 72)(23 44 65)(24 45 66)(25 54 75)(26 55 76)(27 56 77)(28 49 78)(29 50 79)(30 51 80)(31 52 73)(32 53 74)(33 92 113)(34 93 114)(35 94 115)(36 95 116)(37 96 117)(38 89 118)(39 90 119)(40 91 120)
(1 105 33 42 54)(2 43 106 55 34)(3 56 44 35 107)(4 36 49 108 45)(5 109 37 46 50)(6 47 110 51 38)(7 52 48 39 111)(8 40 53 112 41)(9 102 73 69 90)(10 70 103 91 74)(11 92 71 75 104)(12 76 93 97 72)(13 98 77 65 94)(14 66 99 95 78)(15 96 67 79 100)(16 80 89 101 68)(17 29 64 88 117)(18 81 30 118 57)(19 119 82 58 31)(20 59 120 32 83)(21 25 60 84 113)(22 85 26 114 61)(23 115 86 62 27)(24 63 116 28 87)
(1 54)(2 34)(3 107)(4 45)(5 50)(6 38)(7 111)(8 41)(9 102)(10 91)(11 71)(13 98)(14 95)(15 67)(17 88)(18 30)(20 59)(21 84)(22 26)(24 63)(25 60)(27 115)(29 64)(31 119)(35 56)(36 108)(39 52)(40 112)(42 105)(43 55)(46 109)(47 51)(57 118)(58 82)(61 114)(62 86)(66 99)(68 80)(70 103)(72 76)(73 90)(75 104)(77 94)(79 100)(83 120)(87 116)(89 101)(93 97)
(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 5)(3 7)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(42 46)(44 48)(50 54)(52 56)(58 62)(60 64)(65 69)(67 71)(73 77)(75 79)(82 86)(84 88)(90 94)(92 96)(98 102)(100 104)(105 109)(107 111)(113 117)(115 119)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 5 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 8A ··· 8H 10A 10B 10C 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L 15A 15B 20A 20B 20C 20D 24A ··· 24P 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 3 3 4 4 4 4 4 4 5 6 6 6 6 6 6 6 6 6 6 8 ··· 8 10 10 10 12 12 12 12 12 12 12 12 12 12 12 12 15 15 20 20 20 20 24 ··· 24 30 ··· 30 60 ··· 60 size 1 1 2 5 5 10 1 1 1 1 2 5 5 10 4 1 1 2 2 5 5 5 5 10 10 10 ··· 10 4 4 4 1 1 1 1 2 2 5 5 5 5 10 10 4 4 4 4 4 4 10 ··· 10 4 ··· 4 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + image C1 C2 C2 C2 C2 C3 C4 C4 C4 C6 C6 C6 C6 C12 C12 C12 M4(2) C3×M4(2) F5 C2×F5 C2×F5 C3×F5 C6×F5 C6×F5 D5⋊M4(2) C3×D5⋊M4(2) kernel C3×D5⋊M4(2) C3×D5⋊C8 C3×C4.F5 C3×C22.F5 D5×C2×C12 D5⋊M4(2) D5×C12 C2×C60 D5×C2×C6 D5⋊C8 C4.F5 C22.F5 C2×C4×D5 C4×D5 C2×C20 C22×D5 C3×D5 D5 C2×C12 C12 C2×C6 C2×C4 C4 C22 C3 C1 # reps 1 2 2 2 1 2 4 2 2 4 4 4 2 8 4 4 4 8 1 2 1 2 4 2 4 8

Matrix representation of C3×D5⋊M4(2) in GL6(𝔽241)

 225 0 0 0 0 0 0 225 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 240 240 240 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 240 240 240 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0
,
 240 2 0 0 0 0 152 1 0 0 0 0 0 0 43 0 117 117 0 0 117 117 0 43 0 0 124 167 124 0 0 0 198 74 74 198
,
 240 0 0 0 0 0 240 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(241))| [225,0,0,0,0,0,0,225,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,1,0,0,0,0,240,0,1,0,0,0,240,0,0,1,0,0,240,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,240,0,0,1,0,0,240,0,1,0,0,0,240,1,0,0],[240,152,0,0,0,0,2,1,0,0,0,0,0,0,43,117,124,198,0,0,0,117,167,74,0,0,117,0,124,74,0,0,117,43,0,198],[240,240,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C3×D5⋊M4(2) in GAP, Magma, Sage, TeX

C_3\times D_5\rtimes M_4(2)
% in TeX

G:=Group("C3xD5:M4(2)");
// GroupNames label

G:=SmallGroup(480,1049);
// by ID

G=gap.SmallGroup(480,1049);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,344,1094,102,9414,818]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^5=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d^-1=b^3,b*e=e*b,d*c*d^-1=b^2*c,c*e=e*c,e*d*e=d^5>;
// generators/relations

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