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## G = C5×C42⋊4S3order 480 = 25·3·5

### Direct product of C5 and C42⋊4S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×C42⋊4S3
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C60 — C5×C4○D12 — C5×C42⋊4S3
 Lower central C3 — C6 — C12 — C5×C42⋊4S3
 Upper central C1 — C20 — C2×C20 — C4×C20

Generators and relations for C5×C424S3
G = < a,b,c,d,e | a5=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, ebe=bc=cb, bd=db, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 212 in 88 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C42, M4(2), C4○D4, C20, C20, C2×C10, C2×C10, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C5×S3, C30, C30, C4≀C2, C40, C2×C20, C2×C20, C5×D4, C5×Q8, C4.Dic3, C4×C12, C4○D12, C5×Dic3, C60, C60, S3×C10, C2×C30, C4×C20, C5×M4(2), C5×C4○D4, C424S3, C5×C3⋊C8, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, C2×C60, C5×C4≀C2, C5×C4.Dic3, C4×C60, C5×C4○D12, C5×C424S3
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, C10, D6, C22⋊C4, C20, C2×C10, C4×S3, D12, C3⋊D4, C5×S3, C4≀C2, C2×C20, C5×D4, D6⋊C4, S3×C10, C5×C22⋊C4, C424S3, S3×C20, C5×D12, C5×C3⋊D4, C5×C4≀C2, C5×D6⋊C4, C5×C424S3

Smallest permutation representation of C5×C424S3
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 58)(2 59)(3 60)(4 56)(5 57)(6 43)(7 44)(8 45)(9 41)(10 42)(11 81 48 111)(12 82 49 112)(13 83 50 113)(14 84 46 114)(15 85 47 115)(16 66)(17 67)(18 68)(19 69)(20 70)(21 120 105 100)(22 116 101 96)(23 117 102 97)(24 118 103 98)(25 119 104 99)(26 89)(27 90)(28 86)(29 87)(30 88)(31 51)(32 52)(33 53)(34 54)(35 55)(36 73 79 110)(37 74 80 106)(38 75 76 107)(39 71 77 108)(40 72 78 109)(61 91)(62 92)(63 93)(64 94)(65 95)
(1 16 58 66)(2 17 59 67)(3 18 60 68)(4 19 56 69)(5 20 57 70)(6 93 43 63)(7 94 44 64)(8 95 45 65)(9 91 41 61)(10 92 42 62)(11 81 48 111)(12 82 49 112)(13 83 50 113)(14 84 46 114)(15 85 47 115)(21 120 105 100)(22 116 101 96)(23 117 102 97)(24 118 103 98)(25 119 104 99)(26 31 89 51)(27 32 90 52)(28 33 86 53)(29 34 87 54)(30 35 88 55)(36 73 79 110)(37 74 80 106)(38 75 76 107)(39 71 77 108)(40 72 78 109)
(1 8 90)(2 9 86)(3 10 87)(4 6 88)(5 7 89)(11 24 73)(12 25 74)(13 21 75)(14 22 71)(15 23 72)(16 95 52)(17 91 53)(18 92 54)(19 93 55)(20 94 51)(26 57 44)(27 58 45)(28 59 41)(29 60 42)(30 56 43)(31 70 64)(32 66 65)(33 67 61)(34 68 62)(35 69 63)(36 111 98)(37 112 99)(38 113 100)(39 114 96)(40 115 97)(46 101 108)(47 102 109)(48 103 110)(49 104 106)(50 105 107)(76 83 120)(77 84 116)(78 85 117)(79 81 118)(80 82 119)
(1 98)(2 99)(3 100)(4 96)(5 97)(6 114)(7 115)(8 111)(9 112)(10 113)(11 65)(12 61)(13 62)(14 63)(15 64)(16 103)(17 104)(18 105)(19 101)(20 102)(21 68)(22 69)(23 70)(24 66)(25 67)(26 78)(27 79)(28 80)(29 76)(30 77)(31 72)(32 73)(33 74)(34 75)(35 71)(36 90)(37 86)(38 87)(39 88)(40 89)(41 82)(42 83)(43 84)(44 85)(45 81)(46 93)(47 94)(48 95)(49 91)(50 92)(51 109)(52 110)(53 106)(54 107)(55 108)(56 116)(57 117)(58 118)(59 119)(60 120)

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

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

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

150 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C ··· 4G 4H 5A 5B 5C 5D 6A 6B 6C 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 12A ··· 12L 15A 15B 15C 15D 20A ··· 20H 20I ··· 20AB 20AC 20AD 20AE 20AF 30A ··· 30L 40A ··· 40H 60A ··· 60AV order 1 2 2 2 3 4 4 4 ··· 4 4 5 5 5 5 6 6 6 8 8 10 10 10 10 10 10 10 10 10 10 10 10 12 ··· 12 15 15 15 15 20 ··· 20 20 ··· 20 20 20 20 20 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 2 12 2 1 1 2 ··· 2 12 1 1 1 1 2 2 2 12 12 1 1 1 1 2 2 2 2 12 12 12 12 2 ··· 2 2 2 2 2 1 ··· 1 2 ··· 2 12 12 12 12 2 ··· 2 12 ··· 12 2 ··· 2

150 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C4 C4 C5 C10 C10 C10 C20 C20 S3 D4 D4 D6 C4×S3 D12 C3⋊D4 C5×S3 C4≀C2 C5×D4 C5×D4 S3×C10 C42⋊4S3 S3×C20 C5×D12 C5×C3⋊D4 C5×C4≀C2 C5×C42⋊4S3 kernel C5×C42⋊4S3 C5×C4.Dic3 C4×C60 C5×C4○D12 C5×Dic6 C5×D12 C42⋊4S3 C4.Dic3 C4×C12 C4○D12 Dic6 D12 C4×C20 C60 C2×C30 C2×C20 C20 C20 C2×C10 C42 C15 C12 C2×C6 C2×C4 C5 C4 C4 C22 C3 C1 # reps 1 1 1 1 2 2 4 4 4 4 8 8 1 1 1 1 2 2 2 4 4 4 4 4 8 8 8 8 16 32

Matrix representation of C5×C424S3 in GL2(𝔽241) generated by

 205 0 0 205
,
 240 0 81 64
,
 177 0 204 64
,
 15 0 169 225
,
 3 172 56 238
G:=sub<GL(2,GF(241))| [205,0,0,205],[240,81,0,64],[177,204,0,64],[15,169,0,225],[3,56,172,238] >;

C5×C424S3 in GAP, Magma, Sage, TeX

C_5\times C_4^2\rtimes_4S_3
% in TeX

G:=Group("C5xC4^2:4S3");
// GroupNames label

G:=SmallGroup(480,124);
// by ID

G=gap.SmallGroup(480,124);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,589,148,2803,4204,102,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=c^4=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e=b*c=c*b,b*d=d*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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