Copied to
clipboard

## G = C15⋊9(C23⋊C4)  order 480 = 25·3·5

### 3rd semidirect product of C15 and C23⋊C4 acting via C23⋊C4/C23=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C15⋊9(C23⋊C4)
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C22×C30 — C3×C23.D5 — C15⋊9(C23⋊C4)
 Lower central C15 — C30 — C2×C30 — C15⋊9(C23⋊C4)
 Upper central C1 — C2 — C23

Generators and relations for C159(C23⋊C4)
G = < a,b,c,d,e | a15=b2=c2=d2=e4=1, bab=a-1, ac=ca, ad=da, eae-1=a11, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >

Subgroups: 684 in 104 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2 [×4], C3, C4 [×3], C22 [×3], C22 [×3], C5, S3, C6, C6 [×3], C2×C4 [×3], D4 [×2], C23, C23, D5, C10, C10 [×3], Dic3 [×2], C12, D6 [×2], C2×C6 [×3], C2×C6, C15, C22⋊C4 [×2], C2×D4, Dic5 [×2], C20, D10 [×2], C2×C10 [×3], C2×C10, C2×Dic3 [×2], C3⋊D4 [×2], C2×C12, C22×S3, C22×C6, D15, C30, C30 [×3], C23⋊C4, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20, C22×D5, C22×C10, C6.D4, C3×C22⋊C4, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15, D30 [×2], C2×C30 [×3], C2×C30, C23.D5, C5×C22⋊C4, C2×C5⋊D4, C23.6D6, C6×Dic5, C10×Dic3, C2×Dic15, C157D4 [×2], C22×D15, C22×C30, C23.1D10, C3×C23.D5, C5×C6.D4, C2×C157D4, C159(C23⋊C4)
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, C23⋊C4, C4×D5, D20, C5⋊D4, D6⋊C4, S3×D5, D10⋊C4, C23.6D6, D30.C2, C3⋊D20, C5⋊D12, C23.1D10, D304C4, C159(C23⋊C4)

Smallest permutation representation of C159(C23⋊C4)
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 63)(2 62)(3 61)(4 75)(5 74)(6 73)(7 72)(8 71)(9 70)(10 69)(11 68)(12 67)(13 66)(14 65)(15 64)(16 82)(17 81)(18 80)(19 79)(20 78)(21 77)(22 76)(23 90)(24 89)(25 88)(26 87)(27 86)(28 85)(29 84)(30 83)(31 113)(32 112)(33 111)(34 110)(35 109)(36 108)(37 107)(38 106)(39 120)(40 119)(41 118)(42 117)(43 116)(44 115)(45 114)(46 95)(47 94)(48 93)(49 92)(50 91)(51 105)(52 104)(53 103)(54 102)(55 101)(56 100)(57 99)(58 98)(59 97)(60 96)
(1 58)(2 59)(3 60)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 35)(17 36)(18 37)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)(25 44)(26 45)(27 31)(28 32)(29 33)(30 34)(61 96)(62 97)(63 98)(64 99)(65 100)(66 101)(67 102)(68 103)(69 104)(70 105)(71 91)(72 92)(73 93)(74 94)(75 95)(76 118)(77 119)(78 120)(79 106)(80 107)(81 108)(82 109)(83 110)(84 111)(85 112)(86 113)(87 114)(88 115)(89 116)(90 117)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 16)(13 17)(14 18)(15 19)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)(41 60)(42 46)(43 47)(44 48)(45 49)(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)(91 113)(92 114)(93 115)(94 116)(95 117)(96 118)(97 119)(98 120)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)
(1 98 39 63)(2 94 40 74)(3 105 41 70)(4 101 42 66)(5 97 43 62)(6 93 44 73)(7 104 45 69)(8 100 31 65)(9 96 32 61)(10 92 33 72)(11 103 34 68)(12 99 35 64)(13 95 36 75)(14 91 37 71)(15 102 38 67)(16 106 54 79)(17 117 55 90)(18 113 56 86)(19 109 57 82)(20 120 58 78)(21 116 59 89)(22 112 60 85)(23 108 46 81)(24 119 47 77)(25 115 48 88)(26 111 49 84)(27 107 50 80)(28 118 51 76)(29 114 52 87)(30 110 53 83)

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

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

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

57 conjugacy classes

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

57 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C4 C4 S3 D4 D5 D6 D10 C4×S3 D12 C3⋊D4 C4×D5 D20 C5⋊D4 C23⋊C4 S3×D5 C23.6D6 D30.C2 C3⋊D20 C5⋊D12 C23.1D10 C15⋊9(C23⋊C4) kernel C15⋊9(C23⋊C4) C3×C23.D5 C5×C6.D4 C2×C15⋊7D4 C2×Dic15 C22×D15 C23.D5 C2×C30 C6.D4 C22×C10 C22×C6 C2×C10 C2×C10 C2×C10 C2×C6 C2×C6 C2×C6 C15 C23 C5 C22 C22 C22 C3 C1 # reps 1 1 1 1 2 2 1 2 2 1 2 2 2 2 4 4 4 1 2 2 2 2 2 4 8

Matrix representation of C159(C23⋊C4) in GL6(𝔽61)

 47 6 0 0 0 0 0 13 0 0 0 0 0 0 18 43 0 0 0 0 18 60 0 0 0 0 0 0 18 43 0 0 0 0 18 60
,
 14 55 0 0 0 0 2 47 0 0 0 0 0 0 0 0 1 43 0 0 0 0 0 60 0 0 1 43 0 0 0 0 0 60 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 31 44 0 0 0 0 17 30 0 0 0 0 0 0 30 17 0 0 0 0 44 31
,
 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
,
 32 56 0 0 0 0 22 29 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 30 17 0 0 0 0 44 31 0 0

G:=sub<GL(6,GF(61))| [47,0,0,0,0,0,6,13,0,0,0,0,0,0,18,18,0,0,0,0,43,60,0,0,0,0,0,0,18,18,0,0,0,0,43,60],[14,2,0,0,0,0,55,47,0,0,0,0,0,0,0,0,1,0,0,0,0,0,43,60,0,0,1,0,0,0,0,0,43,60,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,17,0,0,0,0,44,30,0,0,0,0,0,0,30,44,0,0,0,0,17,31],[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],[32,22,0,0,0,0,56,29,0,0,0,0,0,0,0,0,30,44,0,0,0,0,17,31,0,0,1,0,0,0,0,0,0,1,0,0] >;

C159(C23⋊C4) in GAP, Magma, Sage, TeX

C_{15}\rtimes_9(C_2^3\rtimes C_4)
% in TeX

G:=Group("C15:9(C2^3:C4)");
// GroupNames label

G:=SmallGroup(480,73);
// by ID

G=gap.SmallGroup(480,73);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,92,219,675,346,1356,18822]);
// Polycyclic

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

׿
×
𝔽