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G = C3×D5×M4(2)  order 480 = 25·3·5

Direct product of C3, D5 and M4(2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×D5×M4(2)
 Chief series C1 — C5 — C10 — C20 — C60 — D5×C12 — D5×C2×C12 — C3×D5×M4(2)
 Lower central C5 — C10 — C3×D5×M4(2)
 Upper central C1 — C12 — C3×M4(2)

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, bd=db, be=eb, cd=dc, ce=ec, ede=d5 >

Subgroups: 368 in 136 conjugacy classes, 78 normal (54 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, M4(2), M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C24, C24, C2×C12, C2×C12, C22×C6, C3×D5, C3×D5, C30, C30, C2×M4(2), C52C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C24, C3×M4(2), C3×M4(2), C22×C12, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), C2×C4×D5, C6×M4(2), C3×C52C8, C120, D5×C12, C6×Dic5, C2×C60, D5×C2×C6, D5×M4(2), D5×C24, C3×C8⋊D5, C3×C4.Dic5, C15×M4(2), D5×C2×C12, C3×D5×M4(2)
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, D5, C12, C2×C6, M4(2), C22×C4, D10, C2×C12, C22×C6, C3×D5, C2×M4(2), C4×D5, C22×D5, C3×M4(2), C22×C12, C6×D5, C2×C4×D5, C6×M4(2), D5×C12, D5×C2×C6, D5×M4(2), D5×C2×C12, C3×D5×M4(2)

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

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 10D 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 24A ··· 24H 24I ··· 24P 30A 30B 30C 30D 30E 30F 30G 30H 40A ··· 40H 60A ··· 60H 60I 60J 60K 60L 120A ··· 120P order 1 2 2 2 2 2 3 3 4 4 4 4 4 4 5 5 6 6 6 6 6 6 6 6 6 6 8 8 8 8 8 8 8 8 10 10 10 10 12 12 12 12 12 12 12 12 12 12 12 12 15 15 15 15 20 20 20 20 20 20 24 ··· 24 24 ··· 24 30 30 30 30 30 30 30 30 40 ··· 40 60 ··· 60 60 60 60 60 120 ··· 120 size 1 1 2 5 5 10 1 1 1 1 2 5 5 10 2 2 1 1 2 2 5 5 5 5 10 10 2 2 2 2 10 10 10 10 2 2 4 4 1 1 1 1 2 2 5 5 5 5 10 10 2 2 2 2 2 2 2 2 4 4 2 ··· 2 10 ··· 10 2 2 2 2 4 4 4 4 4 ··· 4 2 ··· 2 4 4 4 4 4 ··· 4

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C4 C4 C4 C6 C6 C6 C6 C6 C12 C12 C12 D5 M4(2) D10 D10 C3×D5 C4×D5 C4×D5 C3×M4(2) C6×D5 C6×D5 D5×C12 D5×C12 D5×M4(2) C3×D5×M4(2) kernel C3×D5×M4(2) D5×C24 C3×C8⋊D5 C3×C4.Dic5 C15×M4(2) D5×C2×C12 D5×M4(2) D5×C12 C6×Dic5 D5×C2×C6 C8×D5 C8⋊D5 C4.Dic5 C5×M4(2) C2×C4×D5 C4×D5 C2×Dic5 C22×D5 C3×M4(2) C3×D5 C24 C2×C12 M4(2) C12 C2×C6 D5 C8 C2×C4 C4 C22 C3 C1 # reps 1 2 2 1 1 1 2 4 2 2 4 4 2 2 2 8 4 4 2 4 4 2 4 4 4 8 8 4 8 8 4 8

Matrix representation of C3×D5×M4(2) in GL4(𝔽241) generated by

 15 0 0 0 0 15 0 0 0 0 225 0 0 0 0 225
,
 0 1 0 0 240 51 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 240 0 0 0 0 240 0 0 0 0 0 1 0 0 64 0
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 240
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,225,0,0,0,0,225],[0,240,0,0,1,51,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,0,240,0,0,0,0,0,64,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,240] >;

C3×D5×M4(2) in GAP, Magma, Sage, TeX

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

G:=Group("C3xD5xM4(2)");
// GroupNames label

G:=SmallGroup(480,699);
// by ID

G=gap.SmallGroup(480,699);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,555,142,102,18822]);
// 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,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^5>;
// generators/relations

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