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## G = C5×S3×C4○D4order 480 = 25·3·5

### Direct product of C5, S3 and C4○D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×S3×C4○D4
 Chief series C1 — C3 — C6 — C30 — S3×C10 — S3×C2×C10 — S3×C2×C20 — C5×S3×C4○D4
 Lower central C3 — C6 — C5×S3×C4○D4
 Upper central C1 — C20 — C5×C4○D4

Generators and relations for C5×S3×C4○D4
G = < a,b,c,d,e,f | a5=b3=c2=d4=f2=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=d2e >

Subgroups: 660 in 328 conjugacy classes, 174 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C5, S3, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C10, C10, Dic3, Dic3, C12, C12, D6, D6, D6, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C20, C20, C20, C2×C10, C2×C10, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, C5×S3, C5×S3, C30, C30, C2×C4○D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, S3×C2×C4, C4○D12, S3×D4, D42S3, S3×Q8, Q83S3, C3×C4○D4, C5×Dic3, C5×Dic3, C60, C60, S3×C10, S3×C10, S3×C10, C2×C30, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C5×C4○D4, S3×C4○D4, C5×Dic6, S3×C20, S3×C20, C5×D12, C10×Dic3, C5×C3⋊D4, C2×C60, D4×C15, Q8×C15, S3×C2×C10, C10×C4○D4, S3×C2×C20, C5×C4○D12, C5×S3×D4, C5×D42S3, C5×S3×Q8, C5×Q83S3, C15×C4○D4, C5×S3×C4○D4
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, C4○D4, C24, C2×C10, C22×S3, C5×S3, C2×C4○D4, C22×C10, S3×C23, S3×C10, C5×C4○D4, C23×C10, S3×C4○D4, S3×C2×C10, C10×C4○D4, S3×C22×C10, C5×S3×C4○D4

Smallest permutation representation of C5×S3×C4○D4
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 39 33)(2 40 34)(3 36 35)(4 37 31)(5 38 32)(6 20 13)(7 16 14)(8 17 15)(9 18 11)(10 19 12)(21 118 111)(22 119 112)(23 120 113)(24 116 114)(25 117 115)(26 41 47)(27 42 48)(28 43 49)(29 44 50)(30 45 46)(51 79 58)(52 80 59)(53 76 60)(54 77 56)(55 78 57)(61 74 67)(62 75 68)(63 71 69)(64 72 70)(65 73 66)(81 109 88)(82 110 89)(83 106 90)(84 107 86)(85 108 87)(91 104 97)(92 105 98)(93 101 99)(94 102 100)(95 103 96)
(6 13)(7 14)(8 15)(9 11)(10 12)(21 111)(22 112)(23 113)(24 114)(25 115)(31 37)(32 38)(33 39)(34 40)(35 36)(41 47)(42 48)(43 49)(44 50)(45 46)(51 79)(52 80)(53 76)(54 77)(55 78)(61 67)(62 68)(63 69)(64 70)(65 66)(81 109)(82 110)(83 106)(84 107)(85 108)(91 97)(92 98)(93 99)(94 100)(95 96)
(1 56 26 71)(2 57 27 72)(3 58 28 73)(4 59 29 74)(5 60 30 75)(6 95 21 109)(7 91 22 110)(8 92 23 106)(9 93 24 107)(10 94 25 108)(11 99 114 84)(12 100 115 85)(13 96 111 81)(14 97 112 82)(15 98 113 83)(16 104 119 89)(17 105 120 90)(18 101 116 86)(19 102 117 87)(20 103 118 88)(31 80 50 61)(32 76 46 62)(33 77 47 63)(34 78 48 64)(35 79 49 65)(36 51 43 66)(37 52 44 67)(38 53 45 68)(39 54 41 69)(40 55 42 70)
(1 101 26 86)(2 102 27 87)(3 103 28 88)(4 104 29 89)(5 105 30 90)(6 79 21 65)(7 80 22 61)(8 76 23 62)(9 77 24 63)(10 78 25 64)(11 54 114 69)(12 55 115 70)(13 51 111 66)(14 52 112 67)(15 53 113 68)(16 59 119 74)(17 60 120 75)(18 56 116 71)(19 57 117 72)(20 58 118 73)(31 91 50 110)(32 92 46 106)(33 93 47 107)(34 94 48 108)(35 95 49 109)(36 96 43 81)(37 97 44 82)(38 98 45 83)(39 99 41 84)(40 100 42 85)
(6 21)(7 22)(8 23)(9 24)(10 25)(11 114)(12 115)(13 111)(14 112)(15 113)(16 119)(17 120)(18 116)(19 117)(20 118)(81 96)(82 97)(83 98)(84 99)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 110)(92 106)(93 107)(94 108)(95 109)

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

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

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

150 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 5C 5D 6A 6B 6C 6D 10A 10B 10C 10D 10E ··· 10P 10Q ··· 10X 10Y ··· 10AJ 12A 12B 12C 12D 12E 15A 15B 15C 15D 20A ··· 20H 20I ··· 20T 20U ··· 20AB 20AC ··· 20AN 30A 30B 30C 30D 30E ··· 30P 60A ··· 60H 60I ··· 60T order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 6 6 6 6 10 10 10 10 10 ··· 10 10 ··· 10 10 ··· 10 12 12 12 12 12 15 15 15 15 20 ··· 20 20 ··· 20 20 ··· 20 20 ··· 20 30 30 30 30 30 ··· 30 60 ··· 60 60 ··· 60 size 1 1 2 2 2 3 3 6 6 6 2 1 1 2 2 2 3 3 6 6 6 1 1 1 1 2 4 4 4 1 1 1 1 2 ··· 2 3 ··· 3 6 ··· 6 2 2 4 4 4 2 2 2 2 1 ··· 1 2 ··· 2 3 ··· 3 6 ··· 6 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

150 irreducible representations

 dim 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 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 C10 C10 S3 D6 D6 D6 C4○D4 C5×S3 S3×C10 S3×C10 S3×C10 C5×C4○D4 S3×C4○D4 C5×S3×C4○D4 kernel C5×S3×C4○D4 S3×C2×C20 C5×C4○D12 C5×S3×D4 C5×D4⋊2S3 C5×S3×Q8 C5×Q8⋊3S3 C15×C4○D4 S3×C4○D4 S3×C2×C4 C4○D12 S3×D4 D4⋊2S3 S3×Q8 Q8⋊3S3 C3×C4○D4 C5×C4○D4 C2×C20 C5×D4 C5×Q8 C5×S3 C4○D4 C2×C4 D4 Q8 S3 C5 C1 # reps 1 3 3 3 3 1 1 1 4 12 12 12 12 4 4 4 1 3 3 1 4 4 12 12 4 16 2 8

Matrix representation of C5×S3×C4○D4 in GL4(𝔽61) generated by

 9 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 60 0 0 1 60
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
,
 11 0 0 0 0 11 0 0 0 0 60 0 0 0 0 60
,
 60 1 0 0 59 1 0 0 0 0 1 0 0 0 0 1
,
 1 60 0 0 0 60 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(61))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,60,60],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[11,0,0,0,0,11,0,0,0,0,60,0,0,0,0,60],[60,59,0,0,1,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,60,60,0,0,0,0,1,0,0,0,0,1] >;

C5×S3×C4○D4 in GAP, Magma, Sage, TeX

C_5\times S_3\times C_4\circ D_4
% in TeX

G:=Group("C5xS3xC4oD4");
// GroupNames label

G:=SmallGroup(480,1160);
// by ID

G=gap.SmallGroup(480,1160);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,436,1242,15686]);
// Polycyclic

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

׿
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