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## G = C3×C22⋊D20order 480 = 25·3·5

### Direct product of C3 and C22⋊D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×C22⋊D20
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — D5×C2×C6 — D5×C22×C6 — C3×C22⋊D20
 Lower central C5 — C2×C10 — C3×C22⋊D20
 Upper central C1 — C2×C6 — C3×C22⋊C4

Generators and relations for C3×C22⋊D20
G = < a,b,c,d,e | a3=b2=c2=d20=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1088 in 260 conjugacy classes, 74 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22⋊C4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C30, C22≀C2, D20, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×D5, C22×D5, C22×C10, C3×C22⋊C4, C3×C22⋊C4, C6×D4, C23×C6, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C2×C30, C2×C30, D10⋊C4, C5×C22⋊C4, C2×D20, C2×C5⋊D4, C23×D5, C3×C22≀C2, C3×D20, C6×Dic5, C3×C5⋊D4, C2×C60, D5×C2×C6, D5×C2×C6, D5×C2×C6, C22×C30, C22⋊D20, C3×D10⋊C4, C15×C22⋊C4, C6×D20, C6×C5⋊D4, D5×C22×C6, C3×C22⋊D20
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C22≀C2, D20, C22×D5, C6×D4, C6×D5, C2×D20, D4×D5, C3×C22≀C2, C3×D20, D5×C2×C6, C22⋊D20, C6×D20, C3×D4×D5, C3×C22⋊D20

Smallest permutation representation of C3×C22⋊D20
On 120 points
Generators in S120
(1 117 67)(2 118 68)(3 119 69)(4 120 70)(5 101 71)(6 102 72)(7 103 73)(8 104 74)(9 105 75)(10 106 76)(11 107 77)(12 108 78)(13 109 79)(14 110 80)(15 111 61)(16 112 62)(17 113 63)(18 114 64)(19 115 65)(20 116 66)(21 91 49)(22 92 50)(23 93 51)(24 94 52)(25 95 53)(26 96 54)(27 97 55)(28 98 56)(29 99 57)(30 100 58)(31 81 59)(32 82 60)(33 83 41)(34 84 42)(35 85 43)(36 86 44)(37 87 45)(38 88 46)(39 89 47)(40 90 48)
(1 11)(2 45)(3 13)(4 47)(5 15)(6 49)(7 17)(8 51)(9 19)(10 53)(12 55)(14 57)(16 59)(18 41)(20 43)(21 102)(22 32)(23 104)(24 34)(25 106)(26 36)(27 108)(28 38)(29 110)(30 40)(31 112)(33 114)(35 116)(37 118)(39 120)(42 52)(44 54)(46 56)(48 58)(50 60)(61 71)(62 81)(63 73)(64 83)(65 75)(66 85)(67 77)(68 87)(69 79)(70 89)(72 91)(74 93)(76 95)(78 97)(80 99)(82 92)(84 94)(86 96)(88 98)(90 100)(101 111)(103 113)(105 115)(107 117)(109 119)
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 41)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 112)(22 113)(23 114)(24 115)(25 116)(26 117)(27 118)(28 119)(29 120)(30 101)(31 102)(32 103)(33 104)(34 105)(35 106)(36 107)(37 108)(38 109)(39 110)(40 111)(61 90)(62 91)(63 92)(64 93)(65 94)(66 95)(67 96)(68 97)(69 98)(70 99)(71 100)(72 81)(73 82)(74 83)(75 84)(76 85)(77 86)(78 87)(79 88)(80 89)
(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 53)(2 52)(3 51)(4 50)(5 49)(6 48)(7 47)(8 46)(9 45)(10 44)(11 43)(12 42)(13 41)(14 60)(15 59)(16 58)(17 57)(18 56)(19 55)(20 54)(21 101)(22 120)(23 119)(24 118)(25 117)(26 116)(27 115)(28 114)(29 113)(30 112)(31 111)(32 110)(33 109)(34 108)(35 107)(36 106)(37 105)(38 104)(39 103)(40 102)(61 81)(62 100)(63 99)(64 98)(65 97)(66 96)(67 95)(68 94)(69 93)(70 92)(71 91)(72 90)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)

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

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

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

102 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 3A 3B 4A 4B 4C 5A 5B 6A ··· 6F 6G 6H 6I 6J 6K ··· 6R 6S 6T 10A ··· 10F 10G 10H 10I 10J 12A 12B 12C 12D 12E 12F 15A 15B 15C 15D 20A ··· 20H 30A ··· 30L 30M ··· 30T 60A ··· 60P order 1 2 2 2 2 2 2 2 2 2 2 3 3 4 4 4 5 5 6 ··· 6 6 6 6 6 6 ··· 6 6 6 10 ··· 10 10 10 10 10 12 12 12 12 12 12 15 15 15 15 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 10 10 10 10 20 1 1 4 4 20 2 2 1 ··· 1 2 2 2 2 10 ··· 10 20 20 2 ··· 2 4 4 4 4 4 4 4 4 20 20 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

102 irreducible representations

 dim 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 C6 C6 C6 C6 C6 D4 D4 D5 D10 D10 C3×D4 C3×D4 C3×D5 D20 C6×D5 C6×D5 C3×D20 D4×D5 C3×D4×D5 kernel C3×C22⋊D20 C3×D10⋊C4 C15×C22⋊C4 C6×D20 C6×C5⋊D4 D5×C22×C6 C22⋊D20 D10⋊C4 C5×C22⋊C4 C2×D20 C2×C5⋊D4 C23×D5 C6×D5 C2×C30 C3×C22⋊C4 C2×C12 C22×C6 D10 C2×C10 C22⋊C4 C2×C6 C2×C4 C23 C22 C6 C2 # reps 1 2 1 2 1 1 2 4 2 4 2 2 4 2 2 4 2 8 4 4 8 8 4 16 4 8

Matrix representation of C3×C22⋊D20 in GL4(𝔽61) generated by

 13 0 0 0 0 13 0 0 0 0 13 0 0 0 0 13
,
 60 0 0 0 0 60 0 0 0 0 60 0 0 0 55 1
,
 1 0 0 0 0 1 0 0 0 0 60 0 0 0 0 60
,
 2 29 0 0 32 7 0 0 0 0 1 20 0 0 6 60
,
 2 29 0 0 2 59 0 0 0 0 60 41 0 0 0 1
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[60,0,0,0,0,60,0,0,0,0,60,55,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[2,32,0,0,29,7,0,0,0,0,1,6,0,0,20,60],[2,2,0,0,29,59,0,0,0,0,60,0,0,0,41,1] >;

C3×C22⋊D20 in GAP, Magma, Sage, TeX

C_3\times C_2^2\rtimes D_{20}
% in TeX

G:=Group("C3xC2^2:D20");
// GroupNames label

G:=SmallGroup(480,675);
// by ID

G=gap.SmallGroup(480,675);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,590,555,142,18822]);
// Polycyclic

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

׿
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𝔽