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## G = C5×C23⋊2D6order 480 = 25·3·5

### Direct product of C5 and C23⋊2D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C5×C23⋊2D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — S3×C2×C10 — S3×C22×C10 — C5×C23⋊2D6
 Lower central C3 — C2×C6 — C5×C23⋊2D6
 Upper central C1 — C2×C10 — D4×C10

Generators and relations for C5×C232D6
G = < a,b,c,d,e,f | a5=b2=c2=d2=e6=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: 708 in 260 conjugacy classes, 74 normal (34 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C10, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, C2×D4, C24, C20, C2×C10, C2×C10, C2×C10, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C5×S3, C30, C30, C30, C22≀C2, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, D6⋊C4, C6.D4, C2×C3⋊D4, C6×D4, S3×C23, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C2×C30, C2×C30, C5×C22⋊C4, D4×C10, D4×C10, C23×C10, C232D6, C10×Dic3, C5×C3⋊D4, C2×C60, D4×C15, S3×C2×C10, S3×C2×C10, C22×C30, C5×C22≀C2, C5×D6⋊C4, C5×C6.D4, C10×C3⋊D4, D4×C30, S3×C22×C10, C5×C232D6
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, C3⋊D4, C22×S3, C5×S3, C22≀C2, C5×D4, C22×C10, S3×D4, C2×C3⋊D4, S3×C10, D4×C10, C232D6, C5×C3⋊D4, S3×C2×C10, C5×C22≀C2, C5×S3×D4, C10×C3⋊D4, C5×C232D6

Smallest permutation representation of C5×C232D6
On 120 points
Generators in S120
(1 51 39 25 14)(2 49 37 26 15)(3 50 38 27 13)(4 54 41 29 18)(5 52 42 30 16)(6 53 40 28 17)(7 56 44 32 19)(8 57 45 33 20)(9 55 43 31 21)(10 59 47 35 23)(11 60 48 36 24)(12 58 46 34 22)(61 114 101 89 77)(62 109 102 90 78)(63 110 97 85 73)(64 111 98 86 74)(65 112 99 87 75)(66 113 100 88 76)(67 115 103 91 79)(68 116 104 92 80)(69 117 105 93 81)(70 118 106 94 82)(71 119 107 95 83)(72 120 108 96 84)
(1 61)(2 65)(3 63)(4 66)(5 64)(6 62)(7 67)(8 71)(9 69)(10 70)(11 68)(12 72)(13 73)(14 77)(15 75)(16 74)(17 78)(18 76)(19 79)(20 83)(21 81)(22 84)(23 82)(24 80)(25 89)(26 87)(27 85)(28 90)(29 88)(30 86)(31 93)(32 91)(33 95)(34 96)(35 94)(36 92)(37 99)(38 97)(39 101)(40 102)(41 100)(42 98)(43 105)(44 103)(45 107)(46 108)(47 106)(48 104)(49 112)(50 110)(51 114)(52 111)(53 109)(54 113)(55 117)(56 115)(57 119)(58 120)(59 118)(60 116)
(1 8)(2 9)(3 7)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 24)(17 22)(18 23)(25 33)(26 31)(27 32)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 60)(53 58)(54 59)(61 71)(62 72)(63 67)(64 68)(65 69)(66 70)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)(109 120)(110 115)(111 116)(112 117)(113 118)(114 119)
(1 5)(2 6)(3 4)(7 10)(8 11)(9 12)(13 18)(14 16)(15 17)(19 23)(20 24)(21 22)(25 30)(26 28)(27 29)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)(43 46)(44 47)(45 48)(49 53)(50 54)(51 52)(55 58)(56 59)(57 60)(61 64)(62 65)(63 66)(67 70)(68 71)(69 72)(73 76)(74 77)(75 78)(79 82)(80 83)(81 84)(85 88)(86 89)(87 90)(91 94)(92 95)(93 96)(97 100)(98 101)(99 102)(103 106)(104 107)(105 108)(109 112)(110 113)(111 114)(115 118)(116 119)(117 120)
(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 4)(2 6)(3 5)(7 11)(8 10)(9 12)(13 16)(14 18)(15 17)(19 24)(20 23)(21 22)(25 29)(26 28)(27 30)(31 34)(32 36)(33 35)(37 40)(38 42)(39 41)(43 46)(44 48)(45 47)(49 53)(50 52)(51 54)(55 58)(56 60)(57 59)(61 67)(62 72)(63 71)(64 70)(65 69)(66 68)(73 83)(74 82)(75 81)(76 80)(77 79)(78 84)(85 95)(86 94)(87 93)(88 92)(89 91)(90 96)(97 107)(98 106)(99 105)(100 104)(101 103)(102 108)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 3 4A 4B 4C 5A 5B 5C 5D 6A 6B 6C 6D 6E 6F 6G 10A ··· 10L 10M ··· 10T 10U 10V 10W 10X 10Y ··· 10AN 12A 12B 15A 15B 15C 15D 20A 20B 20C 20D 20E ··· 20L 30A ··· 30L 30M ··· 30AB 60A ··· 60H order 1 2 2 2 2 2 2 2 2 2 2 3 4 4 4 5 5 5 5 6 6 6 6 6 6 6 10 ··· 10 10 ··· 10 10 10 10 10 10 ··· 10 12 12 15 15 15 15 20 20 20 20 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 4 6 6 6 6 2 4 12 12 1 1 1 1 2 2 2 4 4 4 4 1 ··· 1 2 ··· 2 4 4 4 4 6 ··· 6 4 4 2 2 2 2 4 4 4 4 12 ··· 12 2 ··· 2 4 ··· 4 4 ··· 4

120 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 C5 C10 C10 C10 C10 C10 S3 D4 D4 D6 D6 C3⋊D4 C5×S3 C5×D4 C5×D4 S3×C10 S3×C10 C5×C3⋊D4 S3×D4 C5×S3×D4 kernel C5×C23⋊2D6 C5×D6⋊C4 C5×C6.D4 C10×C3⋊D4 D4×C30 S3×C22×C10 C23⋊2D6 D6⋊C4 C6.D4 C2×C3⋊D4 C6×D4 S3×C23 D4×C10 S3×C10 C2×C30 C2×C20 C22×C10 C2×C10 C2×D4 D6 C2×C6 C2×C4 C23 C22 C10 C2 # reps 1 2 1 2 1 1 4 8 4 8 4 4 1 4 2 1 2 4 4 16 8 4 8 16 2 8

Matrix representation of C5×C232D6 in GL4(𝔽61) generated by

 9 0 0 0 0 9 0 0 0 0 9 0 0 0 0 9
,
 9 18 0 0 43 52 0 0 0 0 21 16 0 0 3 40
,
 60 0 0 0 0 60 0 0 0 0 60 0 0 0 0 60
,
 1 0 0 0 0 1 0 0 0 0 60 0 0 0 0 60
,
 0 1 0 0 60 60 0 0 0 0 1 47 0 0 0 60
,
 0 1 0 0 1 0 0 0 0 0 60 0 0 0 0 60
G:=sub<GL(4,GF(61))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[9,43,0,0,18,52,0,0,0,0,21,3,0,0,16,40],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[0,60,0,0,1,60,0,0,0,0,1,0,0,0,47,60],[0,1,0,0,1,0,0,0,0,0,60,0,0,0,0,60] >;

C5×C232D6 in GAP, Magma, Sage, TeX

C_5\times C_2^3\rtimes_2D_6
% in TeX

G:=Group("C5xC2^3:2D6");
// GroupNames label

G:=SmallGroup(480,816);
// by ID

G=gap.SmallGroup(480,816);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,926,891,15686]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^6=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