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G = C5xS3xC4oD4order 480 = 25·3·5

Direct product of C5, S3 and C4oD4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5xS3xC4oD4, C30.94C24, C60.241C23, (S3xD4):6C10, D4:7(S3xC10), (C5xD4):29D6, (C2xC20):30D6, Q8:7(S3xC10), (C5xQ8):28D6, (S3xQ8):6C10, C4oD12:7C10, D4:2S3:6C10, D12:10(C2xC10), (C2xC60):29C22, Q8:3S3:6C10, (S3xC20):26C22, Dic6:10(C2xC10), (C5xD12):40C22, (D4xC15):39C22, C6.11(C23xC10), C10.79(S3xC23), (Q8xC15):34C22, (S3xC10).41C23, C20.214(C22xS3), C12.25(C22xC10), (C2xC30).259C23, (C5xDic6):37C22, D6.10(C22xC10), (C10xDic3):37C22, (C5xDic3).43C23, Dic3.7(C22xC10), (S3xC2xC4):6C10, (C5xS3xD4):13C2, C3:4(C10xC4oD4), (C5xS3xQ8):13C2, (S3xC2xC20):16C2, (C2xC4):7(S3xC10), C15:31(C2xC4oD4), C4.25(S3xC2xC10), (C4xS3):7(C2xC10), (C2xC12):4(C2xC10), (C3xC4oD4):3C10, (C3xD4):8(C2xC10), C3:D4:4(C2xC10), (C3xQ8):7(C2xC10), C22.3(S3xC2xC10), (C15xC4oD4):13C2, (C5xC4oD12):17C2, C2.12(S3xC22xC10), (C5xD4:2S3):13C2, (C5xQ8:3S3):13C2, (C5xC3:D4):20C22, (C2xC6).3(C22xC10), (C2xDic3):10(C2xC10), (S3xC2xC10).123C22, (C2xC10).22(C22xS3), (C22xS3).32(C2xC10), SmallGroup(480,1160)

Series: Derived Chief Lower central Upper central

C1C6 — C5xS3xC4oD4
C1C3C6C30S3xC10S3xC2xC10S3xC2xC20 — C5xS3xC4oD4
C3C6 — C5xS3xC4oD4
C1C20C5xC4oD4

Generators and relations for C5xS3xC4oD4
 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, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C10, C10, Dic3, Dic3, C12, C12, D6, D6, D6, C2xC6, C15, C22xC4, C2xD4, C2xQ8, C4oD4, C4oD4, C20, C20, C20, C2xC10, C2xC10, Dic6, C4xS3, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C3xD4, C3xQ8, C22xS3, C5xS3, C5xS3, C30, C30, C2xC4oD4, C2xC20, C2xC20, C5xD4, C5xD4, C5xQ8, C5xQ8, C22xC10, S3xC2xC4, C4oD12, S3xD4, D4:2S3, S3xQ8, Q8:3S3, C3xC4oD4, C5xDic3, C5xDic3, C60, C60, S3xC10, S3xC10, S3xC10, C2xC30, C22xC20, D4xC10, Q8xC10, C5xC4oD4, C5xC4oD4, S3xC4oD4, C5xDic6, S3xC20, S3xC20, C5xD12, C10xDic3, C5xC3:D4, C2xC60, D4xC15, Q8xC15, S3xC2xC10, C10xC4oD4, S3xC2xC20, C5xC4oD12, C5xS3xD4, C5xD4:2S3, C5xS3xQ8, C5xQ8:3S3, C15xC4oD4, C5xS3xC4oD4
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, C4oD4, C24, C2xC10, C22xS3, C5xS3, C2xC4oD4, C22xC10, S3xC23, S3xC10, C5xC4oD4, C23xC10, S3xC4oD4, S3xC2xC10, C10xC4oD4, S3xC22xC10, C5xS3xC4oD4

Smallest permutation representation of C5xS3xC4oD4
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 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J5A5B5C5D6A6B6C6D10A10B10C10D10E···10P10Q···10X10Y···10AJ12A12B12C12D12E15A15B15C15D20A···20H20I···20T20U···20AB20AC···20AN30A30B30C30D30E···30P60A···60H60I···60T
order122222222234444444444555566661010101010···1010···1010···1012121212121515151520···2020···2020···2020···203030303030···3060···6060···60
size1122233666211222336661111244411112···23···36···62244422221···12···23···36···622224···42···24···4

150 irreducible representations

dim1111111111111111222222222244
type++++++++++++
imageC1C2C2C2C2C2C2C2C5C10C10C10C10C10C10C10S3D6D6D6C4oD4C5xS3S3xC10S3xC10S3xC10C5xC4oD4S3xC4oD4C5xS3xC4oD4
kernelC5xS3xC4oD4S3xC2xC20C5xC4oD12C5xS3xD4C5xD4:2S3C5xS3xQ8C5xQ8:3S3C15xC4oD4S3xC4oD4S3xC2xC4C4oD12S3xD4D4:2S3S3xQ8Q8:3S3C3xC4oD4C5xC4oD4C2xC20C5xD4C5xQ8C5xS3C4oD4C2xC4D4Q8S3C5C1
# reps13333111412121212444133144121241628

Matrix representation of C5xS3xC4oD4 in GL4(F61) generated by

9000
0900
0010
0001
,
1000
0100
00060
00160
,
1000
0100
0001
0010
,
11000
01100
00600
00060
,
60100
59100
0010
0001
,
16000
06000
0010
0001
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] >;

C5xS3xC4oD4 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

׿
x
:
Z
F
o
wr
Q
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