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## G = C3×D4.D5order 240 = 24·3·5

### Direct product of C3 and D4.D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D4.D5
 Chief series C1 — C5 — C10 — C20 — C60 — C3×Dic10 — C3×D4.D5
 Lower central C5 — C10 — C20 — C3×D4.D5
 Upper central C1 — C6 — C12 — C3×D4

Generators and relations for C3×D4.D5
G = < a,b,c,d,e | a3=b4=c2=d5=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Smallest permutation representation of C3×D4.D5
On 120 points
Generators in S120
(1 41 21)(2 42 22)(3 43 23)(4 44 24)(5 45 25)(6 46 26)(7 47 27)(8 48 28)(9 49 29)(10 50 30)(11 51 31)(12 52 32)(13 53 33)(14 54 34)(15 55 35)(16 56 36)(17 57 37)(18 58 38)(19 59 39)(20 60 40)(61 101 81)(62 102 82)(63 103 83)(64 104 84)(65 105 85)(66 106 86)(67 107 87)(68 108 88)(69 109 89)(70 110 90)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)(81 91 86 96)(82 92 87 97)(83 93 88 98)(84 94 89 99)(85 95 90 100)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 66)(62 67)(63 68)(64 69)(65 70)(81 86)(82 87)(83 88)(84 89)(85 90)(101 106)(102 107)(103 108)(104 109)(105 110)
(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 68 6 63)(2 67 7 62)(3 66 8 61)(4 70 9 65)(5 69 10 64)(11 78 16 73)(12 77 17 72)(13 76 18 71)(14 80 19 75)(15 79 20 74)(21 88 26 83)(22 87 27 82)(23 86 28 81)(24 90 29 85)(25 89 30 84)(31 98 36 93)(32 97 37 92)(33 96 38 91)(34 100 39 95)(35 99 40 94)(41 108 46 103)(42 107 47 102)(43 106 48 101)(44 110 49 105)(45 109 50 104)(51 118 56 113)(52 117 57 112)(53 116 58 111)(54 120 59 115)(55 119 60 114)

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

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

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

C3×D4.D5 is a maximal subgroup of
C60.10C23  Dic10⋊D6  D30.9D4  C60.19C23  D12.9D10  D30.11D4  D125D10  C3×D5×SD16

51 conjugacy classes

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

51 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + - image C1 C2 C2 C2 C3 C6 C6 C6 D4 D5 SD16 D10 C3×D4 C3×D5 C5⋊D4 C3×SD16 C6×D5 C3×C5⋊D4 D4.D5 C3×D4.D5 kernel C3×D4.D5 C3×C5⋊2C8 C3×Dic10 D4×C15 D4.D5 C5⋊2C8 Dic10 C5×D4 C30 C3×D4 C15 C12 C10 D4 C6 C5 C4 C2 C3 C1 # reps 1 1 1 1 2 2 2 2 1 2 2 2 2 4 4 4 4 8 2 4

Matrix representation of C3×D4.D5 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 225 0 0 0 0 225
,
 0 1 0 0 240 0 0 0 0 0 240 0 0 0 0 240
,
 0 1 0 0 1 0 0 0 0 0 240 85 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 91 177 0 0 0 98
,
 222 19 0 0 19 19 0 0 0 0 23 156 0 0 182 218
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,225,0,0,0,0,225],[0,240,0,0,1,0,0,0,0,0,240,0,0,0,0,240],[0,1,0,0,1,0,0,0,0,0,240,0,0,0,85,1],[1,0,0,0,0,1,0,0,0,0,91,0,0,0,177,98],[222,19,0,0,19,19,0,0,0,0,23,182,0,0,156,218] >;

C3×D4.D5 in GAP, Magma, Sage, TeX

C_3\times D_4.D_5
% in TeX

G:=Group("C3xD4.D5");
// GroupNames label

G:=SmallGroup(240,45);
// by ID

G=gap.SmallGroup(240,45);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,144,169,867,441,69,6917]);
// Polycyclic

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

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