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G = C15×C23⋊C4order 480 = 25·3·5

Direct product of C15 and C23⋊C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C15×C23⋊C4, C23⋊C60, (C2×C4)⋊C60, (C2×C60)⋊14C4, (C2×C12)⋊2C20, (C2×C20)⋊7C12, C22⋊C41C30, (C22×C30)⋊1C4, (C22×C6)⋊1C20, (C2×D4).1C30, (C6×D4).7C10, (D4×C10).7C6, (C22×C10)⋊3C12, (D4×C30).19C2, (C2×C30).128D4, C23.1(C2×C30), C22.2(C2×C60), C22.2(D4×C15), (C22×C30).1C22, C30.125(C22⋊C4), (C5×C22⋊C4)⋊2C6, (C2×C6).21(C5×D4), (C3×C22⋊C4)⋊2C10, (C15×C22⋊C4)⋊4C2, (C2×C6).19(C2×C20), (C2×C10).22(C3×D4), C6.21(C5×C22⋊C4), C2.3(C15×C22⋊C4), (C2×C30).164(C2×C4), (C2×C10).39(C2×C12), C10.32(C3×C22⋊C4), (C22×C6).1(C2×C10), (C22×C10).6(C2×C6), SmallGroup(480,202)

Series: Derived Chief Lower central Upper central

C1C22 — C15×C23⋊C4
C1C2C22C23C22×C10C22×C30C15×C22⋊C4 — C15×C23⋊C4
C1C2C22 — C15×C23⋊C4
C1C30C22×C30 — C15×C23⋊C4

Generators and relations for C15×C23⋊C4
 G = < a,b,c,d,e | a15=b2=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >

Subgroups: 200 in 104 conjugacy classes, 48 normal (32 characteristic)
C1, C2, C2 [×4], C3, C4 [×3], C22, C22 [×2], C22 [×3], C5, C6, C6 [×4], C2×C4, C2×C4 [×2], D4 [×2], C23 [×2], C10, C10 [×4], C12 [×3], C2×C6, C2×C6 [×2], C2×C6 [×3], C15, C22⋊C4 [×2], C2×D4, C20 [×3], C2×C10, C2×C10 [×2], C2×C10 [×3], C2×C12, C2×C12 [×2], C3×D4 [×2], C22×C6 [×2], C30, C30 [×4], C23⋊C4, C2×C20, C2×C20 [×2], C5×D4 [×2], C22×C10 [×2], C3×C22⋊C4 [×2], C6×D4, C60 [×3], C2×C30, C2×C30 [×2], C2×C30 [×3], C5×C22⋊C4 [×2], D4×C10, C3×C23⋊C4, C2×C60, C2×C60 [×2], D4×C15 [×2], C22×C30 [×2], C5×C23⋊C4, C15×C22⋊C4 [×2], D4×C30, C15×C23⋊C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C5, C6 [×3], C2×C4, D4 [×2], C10 [×3], C12 [×2], C2×C6, C15, C22⋊C4, C20 [×2], C2×C10, C2×C12, C3×D4 [×2], C30 [×3], C23⋊C4, C2×C20, C5×D4 [×2], C3×C22⋊C4, C60 [×2], C2×C30, C5×C22⋊C4, C3×C23⋊C4, C2×C60, D4×C15 [×2], C5×C23⋊C4, C15×C22⋊C4, C15×C23⋊C4

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

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

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

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

165 conjugacy classes

class 1 2A2B2C2D2E3A3B4A···4E5A5B5C5D6A6B6C···6H6I6J10A10B10C10D10E···10P10Q10R10S10T12A···12J15A···15H20A···20T30A···30H30I···30AF30AG···30AN60A···60AN
order122222334···45555666···6661010101010···101010101012···1215···1520···2030···3030···3030···3060···60
size112224114···41111112···24411112···244444···41···14···41···12···24···44···4

165 irreducible representations

dim1111111111111111111122224444
type+++++
imageC1C2C2C3C4C4C5C6C6C10C10C12C12C15C20C20C30C30C60C60D4C3×D4C5×D4D4×C15C23⋊C4C3×C23⋊C4C5×C23⋊C4C15×C23⋊C4
kernelC15×C23⋊C4C15×C22⋊C4D4×C30C5×C23⋊C4C2×C60C22×C30C3×C23⋊C4C5×C22⋊C4D4×C10C3×C22⋊C4C6×D4C2×C20C22×C10C23⋊C4C2×C12C22×C6C22⋊C4C2×D4C2×C4C23C2×C30C2×C10C2×C6C22C15C5C3C1
# reps12122244284448881681616248161248

Matrix representation of C15×C23⋊C4 in GL6(𝔽61)

1600000
0160000
0047000
0004700
0000470
0000047
,
6020000
010000
000010
000001
001000
000100
,
6000000
0600000
000100
001000
000001
000010
,
100000
010000
0060000
0006000
0000600
0000060
,
50220000
50110000
0000060
000010
001000
0006000

G:=sub<GL(6,GF(61))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,47,0,0,0,0,0,0,47,0,0,0,0,0,0,47,0,0,0,0,0,0,47],[60,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[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],[50,50,0,0,0,0,22,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,1,0,0,0,0,60,0,0,0] >;

C15×C23⋊C4 in GAP, Magma, Sage, TeX

C_{15}\times C_2^3\rtimes C_4
% in TeX

G:=Group("C15xC2^3:C4");
// GroupNames label

G:=SmallGroup(480,202);
// by ID

G=gap.SmallGroup(480,202);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,840,869,10504,7572]);
// Polycyclic

G:=Group<a,b,c,d,e|a^15=b^2=c^2=d^2=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,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