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## G = C3×D20⋊C4order 480 = 25·3·5

### Direct product of C3 and D20⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D20⋊C4
 Chief series C1 — C5 — C10 — C20 — C4×D5 — D5×C12 — C3×C4⋊F5 — C3×D20⋊C4
 Lower central C5 — C10 — C20 — C3×D20⋊C4
 Upper central C1 — C6 — C12 — C3×D4

Generators and relations for C3×D20⋊C4
G = < a,b,c,d | a3=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b17c >

Subgroups: 472 in 100 conjugacy classes, 36 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D4, D4, C23, D5, D5, C10, C10, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20, F5, D10, D10, C2×C10, C24, C2×C12, C3×D4, C3×D4, C22×C6, C3×D5, C3×D5, C30, C30, D4⋊C4, C5⋊C8, C4×D5, D20, C5⋊D4, C5×D4, C2×F5, C22×D5, C3×C4⋊C4, C2×C24, C6×D4, C3×Dic5, C60, C3×F5, C6×D5, C6×D5, C2×C30, D5⋊C8, C4⋊F5, D4×D5, C3×D4⋊C4, C3×C5⋊C8, D5×C12, C3×D20, C3×C5⋊D4, D4×C15, C6×F5, D5×C2×C6, D20⋊C4, C3×D5⋊C8, C3×C4⋊F5, C3×D4×D5, C3×D20⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, D8, SD16, F5, C2×C12, C3×D4, D4⋊C4, C2×F5, C3×C22⋊C4, C3×D8, C3×SD16, C3×F5, C22⋊F5, C3×D4⋊C4, C6×F5, D20⋊C4, C3×C22⋊F5, C3×D20⋊C4

Smallest permutation representation of C3×D20⋊C4
On 120 points
Generators in S120
(1 41 36)(2 42 37)(3 43 38)(4 44 39)(5 45 40)(6 46 21)(7 47 22)(8 48 23)(9 49 24)(10 50 25)(11 51 26)(12 52 27)(13 53 28)(14 54 29)(15 55 30)(16 56 31)(17 57 32)(18 58 33)(19 59 34)(20 60 35)(61 101 86)(62 102 87)(63 103 88)(64 104 89)(65 105 90)(66 106 91)(67 107 92)(68 108 93)(69 109 94)(70 110 95)(71 111 96)(72 112 97)(73 113 98)(74 114 99)(75 115 100)(76 116 81)(77 117 82)(78 118 83)(79 119 84)(80 120 85)
(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 80)(2 79)(3 78)(4 77)(5 76)(6 75)(7 74)(8 73)(9 72)(10 71)(11 70)(12 69)(13 68)(14 67)(15 66)(16 65)(17 64)(18 63)(19 62)(20 61)(21 100)(22 99)(23 98)(24 97)(25 96)(26 95)(27 94)(28 93)(29 92)(30 91)(31 90)(32 89)(33 88)(34 87)(35 86)(36 85)(37 84)(38 83)(39 82)(40 81)(41 120)(42 119)(43 118)(44 117)(45 116)(46 115)(47 114)(48 113)(49 112)(50 111)(51 110)(52 109)(53 108)(54 107)(55 106)(56 105)(57 104)(58 103)(59 102)(60 101)
(2 8 10 4)(3 15 19 7)(5 9 17 13)(6 16)(12 18 20 14)(21 31)(22 38 30 34)(23 25 39 37)(24 32 28 40)(27 33 35 29)(42 48 50 44)(43 55 59 47)(45 49 57 53)(46 56)(52 58 60 54)(61 66)(62 73 70 69)(63 80 79 72)(64 67 68 75)(65 74 77 78)(71 76)(81 96)(82 83 90 99)(84 97 88 85)(86 91)(87 98 95 94)(89 92 93 100)(101 106)(102 113 110 109)(103 120 119 112)(104 107 108 115)(105 114 117 118)(111 116)

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

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

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

57 conjugacy classes

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

57 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 8 8 type + + + + + + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 D4 D4 D8 SD16 C3×D4 C3×D4 C3×D8 C3×SD16 F5 C2×F5 C3×F5 C22⋊F5 C6×F5 C3×C22⋊F5 D20⋊C4 C3×D20⋊C4 kernel C3×D20⋊C4 C3×D5⋊C8 C3×C4⋊F5 C3×D4×D5 D20⋊C4 C3×D20 D4×C15 D5⋊C8 C4⋊F5 D4×D5 D20 C5×D4 C3×Dic5 C6×D5 C3×D5 C3×D5 Dic5 D10 D5 D5 C3×D4 C12 D4 C6 C4 C2 C3 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 1 1 2 2 2 2 4 4 1 1 2 2 2 4 1 2

Matrix representation of C3×D20⋊C4 in GL8(𝔽241)

 15 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 0 0 0 0 240 0 0 0 0 1 0 0 240 0 0 0 0 0 1 0 240 0 0 0 0 0 0 1 240
,
 20 231 0 0 0 0 0 0 64 221 0 0 0 0 0 0 0 0 230 11 0 0 0 0 0 0 11 11 0 0 0 0 0 0 0 0 0 0 1 240 0 0 0 0 0 1 0 240 0 0 0 0 1 0 0 240 0 0 0 0 0 0 0 240
,
 177 0 0 0 0 0 0 0 226 64 0 0 0 0 0 0 0 0 177 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0

G:=sub<GL(8,GF(241))| [15,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,240,240,240,240],[20,64,0,0,0,0,0,0,231,221,0,0,0,0,0,0,0,0,230,11,0,0,0,0,0,0,11,11,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,240,240,240,240],[177,226,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;

C3×D20⋊C4 in GAP, Magma, Sage, TeX

C_3\times D_{20}\rtimes C_4
% in TeX

G:=Group("C3xD20:C4");
// GroupNames label

G:=SmallGroup(480,287);
// by ID

G=gap.SmallGroup(480,287);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,2524,1271,102,9414,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^20=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^17*c>;
// generators/relations

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