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## G = C15⋊C22≀C2order 480 = 25·3·5

### 3rd semidirect product of C15 and C22≀C2 acting via C22≀C2/C23=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C15⋊C22≀C2
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×C15⋊D4 — C15⋊C22≀C2
 Lower central C15 — C2×C30 — C15⋊C22≀C2
 Upper central C1 — C22 — C23

Generators and relations for C15⋊C22≀C2
G = < a,b,c,d,e,f | a15=b2=c2=d2=e2=f2=1, bab=a11, ac=ca, ad=da, ae=ea, faf=a4, bc=cb, fbf=bd=db, be=eb, cd=dc, fcf=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 1180 in 260 conjugacy classes, 60 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×7], C3, C4 [×3], C22, C22 [×2], C22 [×21], C5, S3 [×4], C6, C6 [×2], C6 [×3], C2×C4 [×3], D4 [×6], C23, C23 [×9], D5, C10, C10 [×2], C10 [×6], Dic3 [×2], C12, D6 [×4], D6 [×12], C2×C6, C2×C6 [×2], C2×C6 [×5], C15, C22⋊C4 [×3], C2×D4 [×3], C24, Dic5 [×3], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×18], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3 [×2], C22×S3 [×6], C22×C6, C22×C6, C5×S3 [×4], C3×D5, C30, C30 [×2], C30 [×2], C22≀C2, C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×6], C22×D5, C22×C10, C22×C10 [×8], D6⋊C4 [×2], C6.D4, C2×C3⋊D4 [×2], C6×D4, S3×C23, C3×Dic5, Dic15 [×2], C6×D5 [×3], S3×C10 [×4], S3×C10 [×12], C2×C30, C2×C30 [×2], C2×C30 [×2], C23.D5 [×3], C2×C5⋊D4, C2×C5⋊D4 [×2], C23×C10, C232D6, C15⋊D4 [×4], C6×Dic5, C3×C5⋊D4 [×2], C2×Dic15 [×2], D5×C2×C6, S3×C2×C10 [×2], S3×C2×C10 [×6], C22×C30, C242D5, D6⋊Dic5 [×2], C30.38D4, C2×C15⋊D4 [×2], C6×C5⋊D4, S3×C22×C10, C15⋊C22≀C2
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D5, D6 [×3], C2×D4 [×3], D10 [×3], C3⋊D4 [×2], C22×S3, C22≀C2, C5⋊D4 [×6], C22×D5, S3×D4 [×2], C2×C3⋊D4, S3×D5, C2×C5⋊D4 [×3], C232D6, C15⋊D4 [×2], C2×S3×D5, C242D5, C2×C15⋊D4, S3×C5⋊D4 [×2], C15⋊C22≀C2

Smallest permutation representation of C15⋊C22≀C2
On 120 points
Generators in S120
(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 16)(2 27)(3 23)(4 19)(5 30)(6 26)(7 22)(8 18)(9 29)(10 25)(11 21)(12 17)(13 28)(14 24)(15 20)(31 57)(32 53)(33 49)(34 60)(35 56)(36 52)(37 48)(38 59)(39 55)(40 51)(41 47)(42 58)(43 54)(44 50)(45 46)(61 115)(62 111)(63 107)(64 118)(65 114)(66 110)(67 106)(68 117)(69 113)(70 109)(71 120)(72 116)(73 112)(74 108)(75 119)(76 97)(77 93)(78 104)(79 100)(80 96)(81 92)(82 103)(83 99)(84 95)(85 91)(86 102)(87 98)(88 94)(89 105)(90 101)
(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 76)(68 77)(69 78)(70 79)(71 80)(72 81)(73 82)(74 83)(75 84)(91 115)(92 116)(93 117)(94 118)(95 119)(96 120)(97 106)(98 107)(99 108)(100 109)(101 110)(102 111)(103 112)(104 113)(105 114)
(1 43)(2 44)(3 45)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 46)(24 47)(25 48)(26 49)(27 50)(28 51)(29 52)(30 53)(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 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(31 57)(32 58)(33 59)(34 60)(35 46)(36 47)(37 48)(38 49)(39 50)(40 51)(41 52)(42 53)(43 54)(44 55)(45 56)(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 76)(68 77)(69 78)(70 79)(71 80)(72 81)(73 82)(74 83)(75 84)(91 115)(92 116)(93 117)(94 118)(95 119)(96 120)(97 106)(98 107)(99 108)(100 109)(101 110)(102 111)(103 112)(104 113)(105 114)
(1 61)(2 65)(3 69)(4 73)(5 62)(6 66)(7 70)(8 74)(9 63)(10 67)(11 71)(12 75)(13 64)(14 68)(15 72)(16 85)(17 89)(18 78)(19 82)(20 86)(21 90)(22 79)(23 83)(24 87)(25 76)(26 80)(27 84)(28 88)(29 77)(30 81)(31 103)(32 92)(33 96)(34 100)(35 104)(36 93)(37 97)(38 101)(39 105)(40 94)(41 98)(42 102)(43 91)(44 95)(45 99)(46 113)(47 117)(48 106)(49 110)(50 114)(51 118)(52 107)(53 111)(54 115)(55 119)(56 108)(57 112)(58 116)(59 120)(60 109)

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

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

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

72 conjugacy classes

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

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 D10 D10 C3⋊D4 C5⋊D4 C5⋊D4 S3×D4 S3×D5 C15⋊D4 C2×S3×D5 S3×C5⋊D4 kernel C15⋊C22≀C2 D6⋊Dic5 C30.38D4 C2×C15⋊D4 C6×C5⋊D4 S3×C22×C10 C2×C5⋊D4 S3×C10 C2×C30 S3×C23 C2×Dic5 C22×D5 C22×C10 C22×S3 C22×C6 C2×C10 D6 C2×C6 C10 C23 C22 C22 C2 # reps 1 2 1 2 1 1 1 4 2 2 1 1 1 4 2 4 16 8 2 2 4 2 8

Matrix representation of C15⋊C22≀C2 in GL4(𝔽61) generated by

 1 52 0 0 41 59 0 0 0 0 34 0 0 0 48 9
,
 60 0 0 0 20 1 0 0 0 0 60 0 0 0 0 60
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 38 60
,
 60 0 0 0 0 60 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 60 0 0 0 0 60
,
 34 40 0 0 55 27 0 0 0 0 22 47 0 0 4 39
G:=sub<GL(4,GF(61))| [1,41,0,0,52,59,0,0,0,0,34,48,0,0,0,9],[60,20,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,1,38,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[34,55,0,0,40,27,0,0,0,0,22,4,0,0,47,39] >;

C15⋊C22≀C2 in GAP, Magma, Sage, TeX

C_{15}\rtimes C_2^2\wr C_2
% in TeX

G:=Group("C15:C2^2wrC2");
// GroupNames label

G:=SmallGroup(480,644);
// by ID

G=gap.SmallGroup(480,644);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,422,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^15=b^2=c^2=d^2=e^2=f^2=1,b*a*b=a^11,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f=a^4,b*c=c*b,f*b*f=b*d=d*b,b*e=e*b,c*d=d*c,f*c*f=c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations

׿
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