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## G = C3×C20.46D4order 480 = 25·3·5

### Direct product of C3 and C20.46D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×C20.46D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C60 — C6×D20 — C3×C20.46D4
 Lower central C5 — C10 — C2×C10 — C3×C20.46D4
 Upper central C1 — C6 — C2×C12 — C3×M4(2)

Generators and relations for C3×C20.46D4
G = < a,b,c,d | a3=b20=d2=1, c4=b10, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=b15c3 >

Subgroups: 416 in 92 conjugacy classes, 38 normal (34 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C8, C2×C4, D4, C23, D5, C10, C10, C12, C2×C6, C2×C6, C15, M4(2), M4(2), C2×D4, C20, D10, C2×C10, C24, C2×C12, C3×D4, C22×C6, C3×D5, C30, C30, C4.D4, C52C8, C40, D20, C2×C20, C22×D5, C3×M4(2), C3×M4(2), C6×D4, C60, C6×D5, C2×C30, C4.Dic5, C5×M4(2), C2×D20, C3×C4.D4, C3×C52C8, C120, C3×D20, C2×C60, D5×C2×C6, C20.46D4, C3×C4.Dic5, C15×M4(2), C6×D20, C3×C20.46D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, D5, C12, C2×C6, C22⋊C4, D10, C2×C12, C3×D4, C3×D5, C4.D4, C4×D5, D20, C5⋊D4, C3×C22⋊C4, C6×D5, D10⋊C4, C3×C4.D4, D5×C12, C3×D20, C3×C5⋊D4, C20.46D4, C3×D10⋊C4, C3×C20.46D4

Smallest permutation representation of C3×C20.46D4
On 120 points
Generators in S120
(1 48 31)(2 49 32)(3 50 33)(4 51 34)(5 52 35)(6 53 36)(7 54 37)(8 55 38)(9 56 39)(10 57 40)(11 58 21)(12 59 22)(13 60 23)(14 41 24)(15 42 25)(16 43 26)(17 44 27)(18 45 28)(19 46 29)(20 47 30)(61 118 95)(62 119 96)(63 120 97)(64 101 98)(65 102 99)(66 103 100)(67 104 81)(68 105 82)(69 106 83)(70 107 84)(71 108 85)(72 109 86)(73 110 87)(74 111 88)(75 112 89)(76 113 90)(77 114 91)(78 115 92)(79 116 93)(80 117 94)
(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 61 16 66 11 71 6 76)(2 80 17 65 12 70 7 75)(3 79 18 64 13 69 8 74)(4 78 19 63 14 68 9 73)(5 77 20 62 15 67 10 72)(21 85 36 90 31 95 26 100)(22 84 37 89 32 94 27 99)(23 83 38 88 33 93 28 98)(24 82 39 87 34 92 29 97)(25 81 40 86 35 91 30 96)(41 105 56 110 51 115 46 120)(42 104 57 109 52 114 47 119)(43 103 58 108 53 113 48 118)(44 102 59 107 54 112 49 117)(45 101 60 106 55 111 50 116)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(56 60)(57 59)(61 71)(62 70)(63 69)(64 68)(65 67)(72 80)(73 79)(74 78)(75 77)(81 99)(82 98)(83 97)(84 96)(85 95)(86 94)(87 93)(88 92)(89 91)(101 105)(102 104)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)

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

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

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

93 conjugacy classes

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

93 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D4 D5 D10 C3×D4 C3×D5 D20 C5⋊D4 C4×D5 C6×D5 C3×D20 C3×C5⋊D4 D5×C12 C4.D4 C3×C4.D4 C20.46D4 C3×C20.46D4 kernel C3×C20.46D4 C3×C4.Dic5 C15×M4(2) C6×D20 C20.46D4 D5×C2×C6 C4.Dic5 C5×M4(2) C2×D20 C22×D5 C60 C3×M4(2) C2×C12 C20 M4(2) C12 C12 C2×C6 C2×C4 C4 C4 C22 C15 C5 C3 C1 # reps 1 1 1 1 2 4 2 2 2 8 2 2 2 4 4 4 4 4 4 8 8 8 1 2 4 8

Matrix representation of C3×C20.46D4 in GL4(𝔽241) generated by

 225 0 0 0 0 225 0 0 0 0 225 0 0 0 0 225
,
 85 41 0 0 200 122 0 0 0 0 85 41 0 0 200 122
,
 0 0 240 0 0 0 52 1 200 156 0 0 119 41 0 0
,
 240 0 0 0 52 1 0 0 0 0 240 0 0 0 52 1
G:=sub<GL(4,GF(241))| [225,0,0,0,0,225,0,0,0,0,225,0,0,0,0,225],[85,200,0,0,41,122,0,0,0,0,85,200,0,0,41,122],[0,0,200,119,0,0,156,41,240,52,0,0,0,1,0,0],[240,52,0,0,0,1,0,0,0,0,240,52,0,0,0,1] >;

C3×C20.46D4 in GAP, Magma, Sage, TeX

C_3\times C_{20}._{46}D_4
% in TeX

G:=Group("C3xC20.46D4");
// GroupNames label

G:=SmallGroup(480,101);
// by ID

G=gap.SmallGroup(480,101);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,365,92,1683,136,1271,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^20=d^2=1,c^4=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=b^15*c^3>;
// generators/relations

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