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## G = C5×C23.7D6order 480 = 25·3·5

### Direct product of C5 and C23.7D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C5×C23.7D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×C30 — C5×C6.D4 — C5×C23.7D6
 Lower central C3 — C6 — C2×C6 — C5×C23.7D6
 Upper central C1 — C10 — C22×C10 — D4×C10

Generators and relations for C5×C23.7D6
G = < a,b,c,d,e,f | a5=b2=c2=d2=e6=1, f2=cb=bc, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=fbf-1=bd=db, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=cde-1 >

Subgroups: 260 in 104 conjugacy classes, 42 normal (30 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, C23, C10, C10, Dic3, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, C20, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×C12, C3×D4, C22×C6, C30, C30, C23⋊C4, C2×C20, C2×C20, C5×D4, C22×C10, C6.D4, C6×D4, C5×Dic3, C60, C2×C30, C2×C30, C2×C30, C5×C22⋊C4, D4×C10, C23.7D6, C10×Dic3, C2×C60, D4×C15, C22×C30, C5×C23⋊C4, C5×C6.D4, D4×C30, C5×C23.7D6
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, C10, Dic3, D6, C22⋊C4, C20, C2×C10, C2×Dic3, C3⋊D4, C5×S3, C23⋊C4, C2×C20, C5×D4, C6.D4, C5×Dic3, S3×C10, C5×C22⋊C4, C23.7D6, C10×Dic3, C5×C3⋊D4, C5×C23⋊C4, C5×C6.D4, C5×C23.7D6

Smallest permutation representation of C5×C23.7D6
On 120 points
Generators in S120
(1 67 89 83 77)(2 68 90 84 78)(3 69 85 79 73)(4 70 86 80 74)(5 71 87 81 75)(6 72 88 82 76)(7 34 28 22 16)(8 35 29 23 17)(9 36 30 24 18)(10 31 25 19 13)(11 32 26 20 14)(12 33 27 21 15)(37 65 55 49 43)(38 66 56 50 44)(39 61 57 51 45)(40 62 58 52 46)(41 63 59 53 47)(42 64 60 54 48)(91 115 109 103 97)(92 116 110 104 98)(93 117 111 105 99)(94 118 112 106 100)(95 119 113 107 101)(96 120 114 108 102)
(1 16)(2 41)(3 18)(4 37)(5 14)(6 39)(7 67)(8 120)(9 69)(10 116)(11 71)(12 118)(13 92)(15 94)(17 96)(19 98)(20 75)(21 100)(22 77)(23 102)(24 73)(25 104)(26 81)(27 106)(28 83)(29 108)(30 79)(31 110)(32 87)(33 112)(34 89)(35 114)(36 85)(38 93)(40 95)(42 91)(43 74)(44 99)(45 76)(46 101)(47 78)(48 97)(49 80)(50 105)(51 82)(52 107)(53 84)(54 103)(55 86)(56 111)(57 88)(58 113)(59 90)(60 109)(61 72)(62 119)(63 68)(64 115)(65 70)(66 117)
(1 16)(2 17)(3 18)(4 13)(5 14)(6 15)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(19 74)(20 75)(21 76)(22 77)(23 78)(24 73)(25 80)(26 81)(27 82)(28 83)(29 84)(30 79)(31 86)(32 87)(33 88)(34 89)(35 90)(36 85)(37 92)(38 93)(39 94)(40 95)(41 96)(42 91)(43 98)(44 99)(45 100)(46 101)(47 102)(48 97)(49 104)(50 105)(51 106)(52 107)(53 108)(54 103)(55 110)(56 111)(57 112)(58 113)(59 114)(60 109)(61 118)(62 119)(63 120)(64 115)(65 116)(66 117)
(1 95)(2 96)(3 91)(4 92)(5 93)(6 94)(7 62)(8 63)(9 64)(10 65)(11 66)(12 61)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(67 119)(68 120)(69 115)(70 116)(71 117)(72 118)(73 97)(74 98)(75 99)(76 100)(77 101)(78 102)(79 103)(80 104)(81 105)(82 106)(83 107)(84 108)(85 109)(86 110)(87 111)(88 112)(89 113)(90 114)
(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 3)(2 17 96 41)(4 15 92 39)(6 13 94 37)(7 64)(8 120 63 68)(9 62)(10 118 65 72)(11 66)(12 116 61 70)(14 38)(16 42)(18 40)(19 100 43 76)(20 44)(21 98 45 74)(22 48)(23 102 47 78)(24 46)(25 106 49 82)(26 50)(27 104 51 80)(28 54)(29 108 53 84)(30 52)(31 112 55 88)(32 56)(33 110 57 86)(34 60)(35 114 59 90)(36 58)(67 69)(73 77)(79 83)(85 89)(91 95)(97 101)(103 107)(109 113)(115 119)

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

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

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

105 conjugacy classes

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

105 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + - - + + image C1 C2 C2 C4 C4 C5 C10 C10 C20 C20 S3 D4 Dic3 Dic3 D6 C3⋊D4 C5×S3 C5×D4 C5×Dic3 C5×Dic3 S3×C10 C5×C3⋊D4 C23⋊C4 C23.7D6 C5×C23⋊C4 C5×C23.7D6 kernel C5×C23.7D6 C5×C6.D4 D4×C30 C2×C60 C22×C30 C23.7D6 C6.D4 C6×D4 C2×C12 C22×C6 D4×C10 C2×C30 C2×C20 C22×C10 C22×C10 C2×C10 C2×D4 C2×C6 C2×C4 C23 C23 C22 C15 C5 C3 C1 # reps 1 2 1 2 2 4 8 4 8 8 1 2 1 1 1 4 4 8 4 4 4 16 1 2 4 8

Matrix representation of C5×C23.7D6 in GL6(𝔽61)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 34 0 0 0 0 0 0 34 0 0 0 0 0 0 34 0 0 0 0 0 0 34
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 21 21 0 0 3 60 1 1 0 0 0 0 0 60 0 0 0 0 60 0
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 3 60 1 1 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 18 9 0 0 0 0 52 9 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 1 0 0 3 60 1 1 0 0 0 1 0 0
,
 50 50 0 0 0 0 0 11 0 0 0 0 0 0 1 40 0 21 0 0 0 60 0 0 0 0 0 0 0 60 0 0 0 0 1 0

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,0,0,0,0,0,0,34,0,0,0,0,0,0,34,0,0,0,0,0,0,34],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,0,60,0,0,0,0,21,1,0,60,0,0,21,1,60,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,3,0,0,0,0,0,60,0,0,0,0,0,1,0,1,0,0,0,1,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[18,52,0,0,0,0,9,9,0,0,0,0,0,0,60,0,3,0,0,0,0,0,60,1,0,0,0,0,1,0,0,0,0,1,1,0],[50,0,0,0,0,0,50,11,0,0,0,0,0,0,1,0,0,0,0,0,40,60,0,0,0,0,0,0,0,1,0,0,21,0,60,0] >;

C5×C23.7D6 in GAP, Magma, Sage, TeX

C_5\times C_2^3._7D_6
% in TeX

G:=Group("C5xC2^3.7D6");
// GroupNames label

G:=SmallGroup(480,153);
// by ID

G=gap.SmallGroup(480,153);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,589,1410,4204,15686]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^6=1,f^2=c*b=b*c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,e*b*e^-1=f*b*f^-1=b*d=d*b,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^-1>;
// generators/relations

׿
×
𝔽