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## G = C3⋊(C23⋊F5)  order 480 = 25·3·5

### The semidirect product of C3 and C23⋊F5 acting via C23⋊F5/C2×C5⋊D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C3⋊(C23⋊F5)
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×C2×C6 — D10.D6 — C3⋊(C23⋊F5)
 Lower central C15 — C30 — C2×C30 — C3⋊(C23⋊F5)
 Upper central C1 — C2 — C22 — C23

Generators and relations for C3⋊(C23⋊F5)
G = < a,b,c,d,e,f | a2=b2=c2=d3=e5=f4=1, faf-1=ab=ba, ac=ca, ad=da, ae=ea, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, fdf-1=d-1, fef-1=e3 >

Subgroups: 652 in 104 conjugacy classes, 29 normal (23 characteristic)
C1, C2, C2 [×4], C3, C4 [×3], C22, C22 [×5], C5, C6, C6 [×4], C2×C4 [×3], D4 [×2], C23, C23, D5 [×2], C10, C10 [×2], Dic3 [×2], C12, C2×C6, C2×C6 [×5], C15, C22⋊C4 [×2], C2×D4, Dic5, F5 [×2], D10 [×2], D10, C2×C10, C2×C10 [×2], C2×Dic3 [×2], C2×C12, C3×D4 [×2], C22×C6, C22×C6, C3×D5 [×2], C30, C30 [×2], C23⋊C4, C2×Dic5, C5⋊D4 [×2], C2×F5 [×2], C22×D5, C22×C10, C6.D4 [×2], C6×D4, C3×Dic5, C3⋊F5 [×2], C6×D5 [×2], C6×D5, C2×C30, C2×C30 [×2], C22⋊F5 [×2], C2×C5⋊D4, C23.7D6, C6×Dic5, C3×C5⋊D4 [×2], C2×C3⋊F5 [×2], D5×C2×C6, C22×C30, C23⋊F5, D10.D6 [×2], C6×C5⋊D4, C3⋊(C23⋊F5)
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, F5, C2×Dic3, C3⋊D4 [×2], C23⋊C4, C2×F5, C6.D4, C3⋊F5, C22⋊F5, C23.7D6, C2×C3⋊F5, C23⋊F5, D10.D6, C3⋊(C23⋊F5)

Smallest permutation representation of C3⋊(C23⋊F5)
On 120 points
Generators in S120
(1 64)(2 65)(3 61)(4 62)(5 63)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)
(1 34)(2 35)(3 31)(4 32)(5 33)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(61 76)(62 77)(63 78)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)(61 66 71)(62 67 72)(63 68 73)(64 69 74)(65 70 75)(76 81 86)(77 82 87)(78 83 88)(79 84 89)(80 85 90)(91 96 101)(92 97 102)(93 98 103)(94 99 104)(95 100 105)(106 111 116)(107 112 117)(108 113 118)(109 114 119)(110 115 120)
(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)
(1 109 34 64)(2 106 33 62)(3 108 32 65)(4 110 31 63)(5 107 35 61)(6 118 37 75)(7 120 36 73)(8 117 40 71)(9 119 39 74)(10 116 38 72)(11 113 42 70)(12 115 41 68)(13 112 45 66)(14 114 44 69)(15 111 43 67)(16 93 47 80)(17 95 46 78)(18 92 50 76)(19 94 49 79)(20 91 48 77)(21 103 52 90)(22 105 51 88)(23 102 55 86)(24 104 54 89)(25 101 53 87)(26 98 57 85)(27 100 56 83)(28 97 60 81)(29 99 59 84)(30 96 58 82)

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 5 6A 6B 6C 6D 6E 6F 6G 10A ··· 10G 12A 12B 15A 15B 30A ··· 30N order 1 2 2 2 2 2 3 4 4 4 4 4 5 6 6 6 6 6 6 6 10 ··· 10 12 12 15 15 30 ··· 30 size 1 1 2 4 10 10 2 20 60 60 60 60 4 2 2 2 4 4 20 20 4 ··· 4 20 20 4 4 4 ··· 4

45 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 type + + + + + - + - + + + + image C1 C2 C2 C4 C4 S3 D4 Dic3 D6 Dic3 C3⋊D4 F5 C23⋊C4 C2×F5 C3⋊F5 C22⋊F5 C23.7D6 C2×C3⋊F5 C23⋊F5 D10.D6 C3⋊(C23⋊F5) kernel C3⋊(C23⋊F5) D10.D6 C6×C5⋊D4 C6×Dic5 C22×C30 C2×C5⋊D4 C6×D5 C2×Dic5 C22×D5 C22×C10 D10 C22×C6 C15 C2×C6 C23 C6 C5 C22 C3 C2 C1 # reps 1 2 1 2 2 1 2 1 1 1 4 1 1 1 2 2 2 2 4 4 8

Matrix representation of C3⋊(C23⋊F5) in GL6(𝔽61)

 9 43 0 0 0 0 18 52 0 0 0 0 0 0 34 3 48 3 0 0 7 37 51 51 0 0 10 10 24 54 0 0 58 13 58 27
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 7 47 0 14 0 0 0 54 47 14 0 0 14 47 54 0 0 0 14 0 47 7
,
 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
,
 0 60 0 0 0 0 1 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 1 0 0 0 0 60 0 1 0 0 0 60 0 0 1 0 0 60 0 0 0
,
 15 23 0 0 0 0 38 46 0 0 0 0 0 0 10 10 24 54 0 0 34 3 48 3 0 0 58 13 58 27 0 0 7 37 51 51

G:=sub<GL(6,GF(61))| [9,18,0,0,0,0,43,52,0,0,0,0,0,0,34,7,10,58,0,0,3,37,10,13,0,0,48,51,24,58,0,0,3,51,54,27],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,7,0,14,14,0,0,47,54,47,0,0,0,0,47,54,47,0,0,14,14,0,7],[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],[0,1,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,60,60,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[15,38,0,0,0,0,23,46,0,0,0,0,0,0,10,34,58,7,0,0,10,3,13,37,0,0,24,48,58,51,0,0,54,3,27,51] >;

C3⋊(C23⋊F5) in GAP, Magma, Sage, TeX

C_3\rtimes (C_2^3\rtimes F_5)
% in TeX

G:=Group("C3:(C2^3:F5)");
// GroupNames label

G:=SmallGroup(480,316);
// by ID

G=gap.SmallGroup(480,316);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,219,675,2693,14118,4724]);
// Polycyclic

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

׿
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