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G = C15×C4≀C2order 480 = 25·3·5

Direct product of C15 and C4≀C2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C15×C4≀C2
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — C2×C60 — C15×M4(2) — C15×C4≀C2
 Lower central C1 — C2 — C4 — C15×C4≀C2
 Upper central C1 — C60 — C2×C60 — C15×C4≀C2

Generators and relations for C15×C4≀C2
G = < a,b,c,d | a15=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >

Subgroups: 136 in 88 conjugacy classes, 48 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, C6, C6 [×2], C8, C2×C4, C2×C4 [×2], D4, D4, Q8, C10, C10 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6, C15, C42, M4(2), C4○D4, C20 [×2], C20 [×3], C2×C10, C2×C10, C24, C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C30, C30 [×2], C4≀C2, C40, C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C4×C12, C3×M4(2), C3×C4○D4, C60 [×2], C60 [×3], C2×C30, C2×C30, C4×C20, C5×M4(2), C5×C4○D4, C3×C4≀C2, C120, C2×C60, C2×C60 [×2], D4×C15, D4×C15, Q8×C15, C5×C4≀C2, C4×C60, C15×M4(2), C15×C4○D4, C15×C4≀C2
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], C4≀C2, C2×C20, C5×D4 [×2], C3×C22⋊C4, C60 [×2], C2×C30, C5×C22⋊C4, C3×C4≀C2, C2×C60, D4×C15 [×2], C5×C4≀C2, C15×C22⋊C4, C15×C4≀C2

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

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

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

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

210 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C ··· 4G 4H 5A 5B 5C 5D 6A 6B 6C 6D 6E 6F 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 12A 12B 12C 12D 12E ··· 12N 12O 12P 15A ··· 15H 20A ··· 20H 20I ··· 20AB 20AC 20AD 20AE 20AF 24A 24B 24C 24D 30A ··· 30H 30I ··· 30P 30Q ··· 30X 40A ··· 40H 60A ··· 60P 60Q ··· 60BD 60BE ··· 60BL 120A ··· 120P order 1 2 2 2 3 3 4 4 4 ··· 4 4 5 5 5 5 6 6 6 6 6 6 8 8 10 10 10 10 10 10 10 10 10 10 10 10 12 12 12 12 12 ··· 12 12 12 15 ··· 15 20 ··· 20 20 ··· 20 20 20 20 20 24 24 24 24 30 ··· 30 30 ··· 30 30 ··· 30 40 ··· 40 60 ··· 60 60 ··· 60 60 ··· 60 120 ··· 120 size 1 1 2 4 1 1 1 1 2 ··· 2 4 1 1 1 1 1 1 2 2 4 4 4 4 1 1 1 1 2 2 2 2 4 4 4 4 1 1 1 1 2 ··· 2 4 4 1 ··· 1 1 ··· 1 2 ··· 2 4 4 4 4 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4

210 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C3 C4 C4 C5 C6 C6 C6 C10 C10 C10 C12 C12 C15 C20 C20 C30 C30 C30 C60 C60 D4 D4 C3×D4 C3×D4 C4≀C2 C5×D4 C5×D4 C3×C4≀C2 D4×C15 D4×C15 C5×C4≀C2 C15×C4≀C2 kernel C15×C4≀C2 C4×C60 C15×M4(2) C15×C4○D4 C5×C4≀C2 D4×C15 Q8×C15 C3×C4≀C2 C4×C20 C5×M4(2) C5×C4○D4 C4×C12 C3×M4(2) C3×C4○D4 C5×D4 C5×Q8 C4≀C2 C3×D4 C3×Q8 C42 M4(2) C4○D4 D4 Q8 C60 C2×C30 C20 C2×C10 C15 C12 C2×C6 C5 C4 C22 C3 C1 # reps 1 1 1 1 2 2 2 4 2 2 2 4 4 4 4 4 8 8 8 8 8 8 16 16 1 1 2 2 4 4 4 8 8 8 16 32

Matrix representation of C15×C4≀C2 in GL2(𝔽241) generated by

 94 0 0 94
,
 64 0 152 177
,
 89 128 149 152
,
 177 0 209 1
G:=sub<GL(2,GF(241))| [94,0,0,94],[64,152,0,177],[89,149,128,152],[177,209,0,1] >;

C15×C4≀C2 in GAP, Magma, Sage, TeX

C_{15}\times C_4\wr C_2
% in TeX

G:=Group("C15xC4wrC2");
// GroupNames label

G:=SmallGroup(480,207);
// by ID

G=gap.SmallGroup(480,207);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,840,869,10504,5261,172,124]);
// Polycyclic

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

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