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## G = C15×2+ 1+4order 480 = 25·3·5

### Direct product of C15 and 2+ 1+4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C15×2+ 1+4
 Chief series C1 — C2 — C10 — C30 — C2×C30 — D4×C15 — D4×C30 — C15×2+ 1+4
 Lower central C1 — C2 — C15×2+ 1+4
 Upper central C1 — C30 — C15×2+ 1+4

Generators and relations for C15×2+ 1+4
G = < a,b,c,d,e | a15=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 440 in 332 conjugacy classes, 272 normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, D4, Q8, C23, C10, C10, C12, C2×C6, C2×C6, C15, C2×D4, C4○D4, C20, C2×C10, C2×C10, C2×C12, C3×D4, C3×Q8, C22×C6, C30, C30, 2+ 1+4, C2×C20, C5×D4, C5×Q8, C22×C10, C6×D4, C3×C4○D4, C60, C2×C30, C2×C30, D4×C10, C5×C4○D4, C3×2+ 1+4, C2×C60, D4×C15, Q8×C15, C22×C30, C5×2+ 1+4, D4×C30, C15×C4○D4, C15×2+ 1+4
Quotients: C1, C2, C3, C22, C5, C6, C23, C10, C2×C6, C15, C24, C2×C10, C22×C6, C30, 2+ 1+4, C22×C10, C23×C6, C2×C30, C23×C10, C3×2+ 1+4, C22×C30, C5×2+ 1+4, C23×C30, C15×2+ 1+4

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

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

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

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

255 conjugacy classes

 class 1 2A 2B ··· 2J 3A 3B 4A ··· 4F 5A 5B 5C 5D 6A 6B 6C ··· 6T 10A 10B 10C 10D 10E ··· 10AN 12A ··· 12L 15A ··· 15H 20A ··· 20X 30A ··· 30H 30I ··· 30CB 60A ··· 60AV order 1 2 2 ··· 2 3 3 4 ··· 4 5 5 5 5 6 6 6 ··· 6 10 10 10 10 10 ··· 10 12 ··· 12 15 ··· 15 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 2 ··· 2 1 1 2 ··· 2 1 1 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

255 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + image C1 C2 C2 C3 C5 C6 C6 C10 C10 C15 C30 C30 2+ 1+4 C3×2+ 1+4 C5×2+ 1+4 C15×2+ 1+4 kernel C15×2+ 1+4 D4×C30 C15×C4○D4 C5×2+ 1+4 C3×2+ 1+4 D4×C10 C5×C4○D4 C6×D4 C3×C4○D4 2+ 1+4 C2×D4 C4○D4 C15 C5 C3 C1 # reps 1 9 6 2 4 18 12 36 24 8 72 48 1 2 4 8

Matrix representation of C15×2+ 1+4 in GL5(𝔽61)

 47 0 0 0 0 0 34 0 0 0 0 0 34 0 0 0 0 0 34 0 0 0 0 0 34
,
 1 0 0 0 0 0 60 1 0 1 0 0 0 1 0 0 0 60 0 0 0 59 1 60 1
,
 1 0 0 0 0 0 60 0 0 0 0 59 1 60 1 0 0 0 0 1 0 0 0 1 0
,
 60 0 0 0 0 0 1 0 1 60 0 0 0 1 0 0 0 60 0 0 0 2 60 1 60
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 59 1 60 1 0 0 1 0 0

G:=sub<GL(5,GF(61))| [47,0,0,0,0,0,34,0,0,0,0,0,34,0,0,0,0,0,34,0,0,0,0,0,34],[1,0,0,0,0,0,60,0,0,59,0,1,0,60,1,0,0,1,0,60,0,1,0,0,1],[1,0,0,0,0,0,60,59,0,0,0,0,1,0,0,0,0,60,0,1,0,0,1,1,0],[60,0,0,0,0,0,1,0,0,2,0,0,0,60,60,0,1,1,0,1,0,60,0,0,60],[1,0,0,0,0,0,1,0,59,0,0,0,0,1,1,0,0,0,60,0,0,0,1,1,0] >;

C15×2+ 1+4 in GAP, Magma, Sage, TeX

C_{15}\times 2_+^{1+4}
% in TeX

G:=Group("C15xES+(2,2)");
// GroupNames label

G:=SmallGroup(480,1184);
// by ID

G=gap.SmallGroup(480,1184);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-5,-2,3389,2571,6947]);
// Polycyclic

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

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