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## G = C3×C23⋊D10order 480 = 25·3·5

### Direct product of C3 and C23⋊D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×C23⋊D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — D5×C2×C6 — D5×C22×C6 — C3×C23⋊D10
 Lower central C5 — C2×C10 — C3×C23⋊D10
 Upper central C1 — C2×C6 — C6×D4

Generators and relations for C3×C23⋊D10
G = < a,b,c,d,e,f | a3=b2=c2=d2=e10=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 992 in 260 conjugacy classes, 74 normal (34 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, C2×D4, 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, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C3×C22⋊C4, C6×D4, C6×D4, C23×C6, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C2×C30, C2×C30, D10⋊C4, C23.D5, C2×C5⋊D4, D4×C10, C23×D5, C3×C22≀C2, C6×Dic5, C3×C5⋊D4, C2×C60, D4×C15, D5×C2×C6, D5×C2×C6, C22×C30, C23⋊D10, C3×D10⋊C4, C3×C23.D5, C6×C5⋊D4, D4×C30, D5×C22×C6, C3×C23⋊D10
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C22≀C2, C5⋊D4, C22×D5, C6×D4, C6×D5, D4×D5, C2×C5⋊D4, C3×C22≀C2, C3×C5⋊D4, D5×C2×C6, C23⋊D10, C3×D4×D5, C6×C5⋊D4, C3×C23⋊D10

Smallest permutation representation of C3×C23⋊D10
On 120 points
Generators in S120
(1 42 22)(2 43 23)(3 44 24)(4 45 25)(5 41 21)(6 48 28)(7 49 29)(8 50 30)(9 46 26)(10 47 27)(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)(61 107 87)(62 108 88)(63 109 89)(64 110 90)(65 101 81)(66 102 82)(67 103 83)(68 104 84)(69 105 85)(70 106 86)(71 112 91)(72 113 92)(73 114 93)(74 115 94)(75 116 95)(76 117 96)(77 118 97)(78 119 98)(79 120 99)(80 111 100)
(1 61)(2 67)(3 63)(4 69)(5 65)(6 70)(7 66)(8 62)(9 68)(10 64)(11 71)(12 77)(13 73)(14 79)(15 75)(16 74)(17 80)(18 76)(19 72)(20 78)(21 81)(22 87)(23 83)(24 89)(25 85)(26 84)(27 90)(28 86)(29 82)(30 88)(31 91)(32 97)(33 93)(34 99)(35 95)(36 94)(37 100)(38 96)(39 92)(40 98)(41 101)(42 107)(43 103)(44 109)(45 105)(46 104)(47 110)(48 106)(49 102)(50 108)(51 112)(52 118)(53 114)(54 120)(55 116)(56 115)(57 111)(58 117)(59 113)(60 119)
(1 12)(2 13)(3 14)(4 15)(5 11)(6 18)(7 19)(8 20)(9 16)(10 17)(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 77)(62 78)(63 79)(64 80)(65 71)(66 72)(67 73)(68 74)(69 75)(70 76)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 112)(102 113)(103 114)(104 115)(105 116)(106 117)(107 118)(108 119)(109 120)(110 111)
(1 7)(2 8)(3 9)(4 10)(5 6)(11 18)(12 19)(13 20)(14 16)(15 17)(21 28)(22 29)(23 30)(24 26)(25 27)(31 38)(32 39)(33 40)(34 36)(35 37)(41 48)(42 49)(43 50)(44 46)(45 47)(51 58)(52 59)(53 60)(54 56)(55 57)(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 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 6)(2 10)(3 9)(4 8)(5 7)(11 19)(12 18)(13 17)(14 16)(15 20)(21 29)(22 28)(23 27)(24 26)(25 30)(31 39)(32 38)(33 37)(34 36)(35 40)(41 49)(42 48)(43 47)(44 46)(45 50)(51 59)(52 58)(53 57)(54 56)(55 60)(61 71)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 100)(89 99)(90 98)(101 118)(102 117)(103 116)(104 115)(105 114)(106 113)(107 112)(108 111)(109 120)(110 119)

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

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

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

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 6L 6M ··· 6T 10A ··· 10F 10G ··· 10N 12A 12B 12C 12D 12E 12F 15A 15B 15C 15D 20A 20B 20C 20D 30A ··· 30L 30M ··· 30AB 60A ··· 60H 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 12 12 12 12 12 12 15 15 15 15 20 20 20 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 4 10 10 10 10 1 1 4 20 20 2 2 1 ··· 1 2 2 2 2 4 4 10 ··· 10 2 ··· 2 4 ··· 4 4 4 20 20 20 20 2 2 2 2 4 4 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 C5⋊D4 C6×D5 C6×D5 C3×C5⋊D4 D4×D5 C3×D4×D5 kernel C3×C23⋊D10 C3×D10⋊C4 C3×C23.D5 C6×C5⋊D4 D4×C30 D5×C22×C6 C23⋊D10 D10⋊C4 C23.D5 C2×C5⋊D4 D4×C10 C23×D5 C6×D5 C2×C30 C6×D4 C2×C12 C22×C6 D10 C2×C10 C2×D4 C2×C6 C2×C4 C23 C22 C6 C2 # reps 1 2 1 2 1 1 2 4 2 4 2 2 4 2 2 2 4 8 4 4 8 4 8 16 4 8

Matrix representation of C3×C23⋊D10 in GL4(𝔽61) generated by

 47 0 0 0 0 47 0 0 0 0 13 0 0 0 0 13
,
 1 53 0 0 0 60 0 0 0 0 31 1 0 0 16 30
,
 1 0 0 0 0 1 0 0 0 0 60 0 0 0 0 60
,
 60 0 0 0 0 60 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 46 60 0 0 0 0 0 18 0 0 44 43
,
 60 0 0 0 15 1 0 0 0 0 1 0 0 0 60 60
G:=sub<GL(4,GF(61))| [47,0,0,0,0,47,0,0,0,0,13,0,0,0,0,13],[1,0,0,0,53,60,0,0,0,0,31,16,0,0,1,30],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,46,0,0,0,60,0,0,0,0,0,44,0,0,18,43],[60,15,0,0,0,1,0,0,0,0,1,60,0,0,0,60] >;

C3×C23⋊D10 in GAP, Magma, Sage, TeX

C_3\times C_2^3\rtimes D_{10}
% in TeX

G:=Group("C3xC2^3:D10");
// GroupNames label

G:=SmallGroup(480,730);
// by ID

G=gap.SmallGroup(480,730);
# by ID

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

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

׿
×
𝔽