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## G = D5×C2×C12order 240 = 24·3·5

### Direct product of C2×C12 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C2×C12
 Chief series C1 — C5 — C10 — C30 — C6×D5 — D5×C2×C6 — D5×C2×C12
 Lower central C5 — D5×C2×C12
 Upper central C1 — C2×C12

Generators and relations for D5×C2×C12
G = < a,b,c,d | a2=b12=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 260 in 108 conjugacy classes, 70 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C22×C4, Dic5, C20, D10, C2×C10, C2×C12, C2×C12, C22×C6, C3×D5, C30, C30, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×C12, C3×Dic5, C60, C6×D5, C2×C30, C2×C4×D5, D5×C12, C6×Dic5, C2×C60, D5×C2×C6, D5×C2×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, D5, C12, C2×C6, C22×C4, D10, C2×C12, C22×C6, C3×D5, C4×D5, C22×D5, C22×C12, C6×D5, C2×C4×D5, D5×C12, D5×C2×C6, D5×C2×C12

Smallest permutation representation of D5×C2×C12
On 120 points
Generators in S120
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 100)(26 101)(27 102)(28 103)(29 104)(30 105)(31 106)(32 107)(33 108)(34 97)(35 98)(36 99)(37 71)(38 72)(39 61)(40 62)(41 63)(42 64)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(49 87)(50 88)(51 89)(52 90)(53 91)(54 92)(55 93)(56 94)(57 95)(58 96)(59 85)(60 86)(73 118)(74 119)(75 120)(76 109)(77 110)(78 111)(79 112)(80 113)(81 114)(82 115)(83 116)(84 117)
(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)
(1 104 46 119 88)(2 105 47 120 89)(3 106 48 109 90)(4 107 37 110 91)(5 108 38 111 92)(6 97 39 112 93)(7 98 40 113 94)(8 99 41 114 95)(9 100 42 115 96)(10 101 43 116 85)(11 102 44 117 86)(12 103 45 118 87)(13 35 62 80 56)(14 36 63 81 57)(15 25 64 82 58)(16 26 65 83 59)(17 27 66 84 60)(18 28 67 73 49)(19 29 68 74 50)(20 30 69 75 51)(21 31 70 76 52)(22 32 71 77 53)(23 33 72 78 54)(24 34 61 79 55)
(1 88)(2 89)(3 90)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 85)(11 86)(12 87)(13 56)(14 57)(15 58)(16 59)(17 60)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 82)(26 83)(27 84)(28 73)(29 74)(30 75)(31 76)(32 77)(33 78)(34 79)(35 80)(36 81)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)(106 109)(107 110)(108 111)

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

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

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

96 conjugacy classes

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

96 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 D5 D10 D10 C3×D5 C4×D5 C6×D5 C6×D5 D5×C12 kernel D5×C2×C12 D5×C12 C6×Dic5 C2×C60 D5×C2×C6 C2×C4×D5 C6×D5 C4×D5 C2×Dic5 C2×C20 C22×D5 D10 C2×C12 C12 C2×C6 C2×C4 C6 C4 C22 C2 # reps 1 4 1 1 1 2 8 8 2 2 2 16 2 4 2 4 8 8 4 16

Matrix representation of D5×C2×C12 in GL3(𝔽61) generated by

 60 0 0 0 60 0 0 0 60
,
 21 0 0 0 47 0 0 0 47
,
 1 0 0 0 0 1 0 60 17
,
 60 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(61))| [60,0,0,0,60,0,0,0,60],[21,0,0,0,47,0,0,0,47],[1,0,0,0,0,60,0,1,17],[60,0,0,0,0,1,0,1,0] >;

D5×C2×C12 in GAP, Magma, Sage, TeX

D_5\times C_2\times C_{12}
% in TeX

G:=Group("D5xC2xC12");
// GroupNames label

G:=SmallGroup(240,156);
// by ID

G=gap.SmallGroup(240,156);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-5,122,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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