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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C15×C23⋊C4
 Chief series C1 — C2 — C22 — C23 — C22×C10 — C22×C30 — C15×C22⋊C4 — C15×C23⋊C4
 Lower central C1 — C2 — C22 — C15×C23⋊C4
 Upper central C1 — C30 — C22×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, C3, C4, C22, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, C23, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, C20, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C3×D4, C22×C6, C30, C30, C23⋊C4, C2×C20, C2×C20, C5×D4, C22×C10, C3×C22⋊C4, C6×D4, C60, C2×C30, C2×C30, C2×C30, C5×C22⋊C4, D4×C10, C3×C23⋊C4, C2×C60, C2×C60, D4×C15, C22×C30, C5×C23⋊C4, C15×C22⋊C4, D4×C30, C15×C23⋊C4
Quotients: C1, C2, C3, C4, C22, C5, C6, C2×C4, D4, C10, C12, C2×C6, C15, C22⋊C4, C20, C2×C10, C2×C12, C3×D4, C30, C23⋊C4, C2×C20, C5×D4, C3×C22⋊C4, C60, C2×C30, C5×C22⋊C4, C3×C23⋊C4, C2×C60, D4×C15, 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 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 16)(12 17)(13 18)(14 19)(15 20)(31 89)(32 90)(33 76)(34 77)(35 78)(36 79)(37 80)(38 81)(39 82)(40 83)(41 84)(42 85)(43 86)(44 87)(45 88)(46 97)(47 98)(48 99)(49 100)(50 101)(51 102)(52 103)(53 104)(54 105)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 116)(62 117)(63 118)(64 119)(65 120)(66 106)(67 107)(68 108)(69 109)(70 110)(71 111)(72 112)(73 113)(74 114)(75 115)
(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 55)(17 56)(18 57)(19 58)(20 59)(21 60)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 110)(32 111)(33 112)(34 113)(35 114)(36 115)(37 116)(38 117)(39 118)(40 119)(41 120)(42 106)(43 107)(44 108)(45 109)(61 80)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 76)(73 77)(74 78)(75 79)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 84)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 76)(24 77)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(46 71)(47 72)(48 73)(49 74)(50 75)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)(91 120)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(99 113)(100 114)(101 115)(102 116)(103 117)(104 118)(105 119)
(1 21 96 70)(2 22 97 71)(3 23 98 72)(4 24 99 73)(5 25 100 74)(6 26 101 75)(7 27 102 61)(8 28 103 62)(9 29 104 63)(10 30 105 64)(11 16 91 65)(12 17 92 66)(13 18 93 67)(14 19 94 68)(15 20 95 69)(31 89 110 60)(32 90 111 46)(33 76 112 47)(34 77 113 48)(35 78 114 49)(36 79 115 50)(37 80 116 51)(38 81 117 52)(39 82 118 53)(40 83 119 54)(41 84 120 55)(42 85 106 56)(43 86 107 57)(44 87 108 58)(45 88 109 59)

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

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

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

165 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A ··· 4E 5A 5B 5C 5D 6A 6B 6C ··· 6H 6I 6J 10A 10B 10C 10D 10E ··· 10P 10Q 10R 10S 10T 12A ··· 12J 15A ··· 15H 20A ··· 20T 30A ··· 30H 30I ··· 30AF 30AG ··· 30AN 60A ··· 60AN order 1 2 2 2 2 2 3 3 4 ··· 4 5 5 5 5 6 6 6 ··· 6 6 6 10 10 10 10 10 ··· 10 10 10 10 10 12 ··· 12 15 ··· 15 20 ··· 20 30 ··· 30 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 2 2 2 4 1 1 4 ··· 4 1 1 1 1 1 1 2 ··· 2 4 4 1 1 1 1 2 ··· 2 4 4 4 4 4 ··· 4 1 ··· 1 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4

165 irreducible representations

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

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

 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 2 0 0 0 0 0 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 22 0 0 0 0 50 11 0 0 0 0 0 0 0 0 0 60 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 60 0 0

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

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