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## G = C3×D5×C22⋊C4order 480 = 25·3·5

### Direct product of C3, D5 and C22⋊C4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×D5×C22⋊C4
G = < a,b,c,d,e,f | a3=b5=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, ef=fe >

Subgroups: 944 in 264 conjugacy classes, 98 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C23, C23, D5, D5, C10, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22⋊C4, C22×C4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C22×C6, C22×C6, C3×D5, C3×D5, C30, C30, C30, C2×C22⋊C4, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×D5, C22×D5, C22×C10, C3×C22⋊C4, C3×C22⋊C4, C22×C12, C23×C6, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C2×C30, C2×C30, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C23×D5, C6×C22⋊C4, D5×C12, C6×Dic5, C2×C60, D5×C2×C6, D5×C2×C6, D5×C2×C6, C22×C30, D5×C22⋊C4, C3×D10⋊C4, C3×C23.D5, C15×C22⋊C4, D5×C2×C12, D5×C22×C6, C3×D5×C22⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, D5, C12, C2×C6, C22⋊C4, C22×C4, C2×D4, D10, C2×C12, C3×D4, C22×C6, C3×D5, C2×C22⋊C4, C4×D5, C22×D5, C3×C22⋊C4, C22×C12, C6×D4, C6×D5, C2×C4×D5, D4×D5, C6×C22⋊C4, D5×C12, D5×C2×C6, D5×C22⋊C4, D5×C2×C12, C3×D4×D5, C3×D5×C22⋊C4

Smallest permutation representation of C3×D5×C22⋊C4
On 120 points
Generators in S120
(1 44 24)(2 45 25)(3 41 21)(4 42 22)(5 43 23)(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 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 18)(2 17)(3 16)(4 20)(5 19)(6 11)(7 15)(8 14)(9 13)(10 12)(21 36)(22 40)(23 39)(24 38)(25 37)(26 31)(27 35)(28 34)(29 33)(30 32)(41 56)(42 60)(43 59)(44 58)(45 57)(46 51)(47 55)(48 54)(49 53)(50 52)(61 76)(62 80)(63 79)(64 78)(65 77)(66 71)(67 75)(68 74)(69 73)(70 72)(81 96)(82 100)(83 99)(84 98)(85 97)(86 91)(87 95)(88 94)(89 93)(90 92)(101 116)(102 120)(103 119)(104 118)(105 117)(106 111)(107 115)(108 114)(109 113)(110 112)
(1 14)(2 15)(3 11)(4 12)(5 13)(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 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)(81 96)(82 97)(83 98)(84 99)(85 100)(86 91)(87 92)(88 93)(89 94)(90 95)(101 116)(102 117)(103 118)(104 119)(105 120)(106 111)(107 112)(108 113)(109 114)(110 115)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)(81 86)(82 87)(83 88)(84 89)(85 90)(91 96)(92 97)(93 98)(94 99)(95 100)(101 106)(102 107)(103 108)(104 109)(105 110)(111 116)(112 117)(113 118)(114 119)(115 120)
(1 74 14 64)(2 75 15 65)(3 71 11 61)(4 72 12 62)(5 73 13 63)(6 76 16 66)(7 77 17 67)(8 78 18 68)(9 79 19 69)(10 80 20 70)(21 91 31 81)(22 92 32 82)(23 93 33 83)(24 94 34 84)(25 95 35 85)(26 96 36 86)(27 97 37 87)(28 98 38 88)(29 99 39 89)(30 100 40 90)(41 111 51 101)(42 112 52 102)(43 113 53 103)(44 114 54 104)(45 115 55 105)(46 116 56 106)(47 117 57 107)(48 118 58 108)(49 119 59 109)(50 120 60 110)

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

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

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

120 conjugacy classes

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

120 irreducible representations

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

Matrix representation of C3×D5×C22⋊C4 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 13 0 0 0 0 13
,
 0 1 0 0 60 43 0 0 0 0 1 0 0 0 0 1
,
 0 60 0 0 60 0 0 0 0 0 60 0 0 0 0 60
,
 60 0 0 0 0 60 0 0 0 0 1 0 0 0 29 60
,
 1 0 0 0 0 1 0 0 0 0 60 0 0 0 0 60
,
 11 0 0 0 0 11 0 0 0 0 29 59 0 0 54 32
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[0,60,0,0,1,43,0,0,0,0,1,0,0,0,0,1],[0,60,0,0,60,0,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,29,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[11,0,0,0,0,11,0,0,0,0,29,54,0,0,59,32] >;

C3×D5×C22⋊C4 in GAP, Magma, Sage, TeX

C_3\times D_5\times C_2^2\rtimes C_4
% in TeX

G:=Group("C3xD5xC2^2:C4");
// GroupNames label

G:=SmallGroup(480,673);
// by ID

G=gap.SmallGroup(480,673);
# by ID

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

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

׿
×
𝔽