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G = C5×S3×C4○D4order 480 = 25·3·5

Direct product of C5, S3 and C4○D4

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

Aliases: C5×S3×C4○D4, C30.94C24, C60.241C23, (S3×D4)⋊6C10, D47(S3×C10), (C5×D4)⋊29D6, (C2×C20)⋊30D6, Q87(S3×C10), (C5×Q8)⋊28D6, (S3×Q8)⋊6C10, C4○D127C10, D42S36C10, D1210(C2×C10), (C2×C60)⋊29C22, Q83S36C10, (S3×C20)⋊26C22, Dic610(C2×C10), (C5×D12)⋊40C22, (D4×C15)⋊39C22, C6.11(C23×C10), C10.79(S3×C23), (Q8×C15)⋊34C22, (S3×C10).41C23, C20.214(C22×S3), C12.25(C22×C10), (C2×C30).259C23, (C5×Dic6)⋊37C22, D6.10(C22×C10), (C10×Dic3)⋊37C22, (C5×Dic3).43C23, Dic3.7(C22×C10), (S3×C2×C4)⋊6C10, (C5×S3×D4)⋊13C2, C34(C10×C4○D4), (C5×S3×Q8)⋊13C2, (S3×C2×C20)⋊16C2, (C2×C4)⋊7(S3×C10), C1531(C2×C4○D4), C4.25(S3×C2×C10), (C4×S3)⋊7(C2×C10), (C2×C12)⋊4(C2×C10), (C3×C4○D4)⋊3C10, (C3×D4)⋊8(C2×C10), C3⋊D44(C2×C10), (C3×Q8)⋊7(C2×C10), C22.3(S3×C2×C10), (C15×C4○D4)⋊13C2, (C5×C4○D12)⋊17C2, C2.12(S3×C22×C10), (C5×D42S3)⋊13C2, (C5×Q83S3)⋊13C2, (C5×C3⋊D4)⋊20C22, (C2×C6).3(C22×C10), (C2×Dic3)⋊10(C2×C10), (S3×C2×C10).123C22, (C2×C10).22(C22×S3), (C22×S3).32(C2×C10), SmallGroup(480,1160)

Series: Derived Chief Lower central Upper central

C1C6 — C5×S3×C4○D4
C1C3C6C30S3×C10S3×C2×C10S3×C2×C20 — C5×S3×C4○D4
C3C6 — C5×S3×C4○D4
C1C20C5×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 [×8], C3, C4, C4 [×3], C4 [×4], C22 [×3], C22 [×10], C5, S3 [×2], S3 [×3], C6, C6 [×3], C2×C4 [×3], C2×C4 [×13], D4 [×3], D4 [×9], Q8, Q8 [×3], C23 [×3], C10, C10 [×8], Dic3, Dic3 [×3], C12, C12 [×3], D6, D6 [×3], D6 [×6], C2×C6 [×3], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4, C4○D4 [×7], C20, C20 [×3], C20 [×4], C2×C10 [×3], C2×C10 [×10], Dic6 [×3], C4×S3, C4×S3 [×9], D12 [×3], C2×Dic3 [×3], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×3], C5×S3 [×2], C5×S3 [×3], C30, C30 [×3], C2×C4○D4, C2×C20 [×3], C2×C20 [×13], C5×D4 [×3], C5×D4 [×9], C5×Q8, C5×Q8 [×3], C22×C10 [×3], S3×C2×C4 [×3], C4○D12 [×3], S3×D4 [×3], D42S3 [×3], S3×Q8, Q83S3, C3×C4○D4, C5×Dic3, C5×Dic3 [×3], C60, C60 [×3], S3×C10, S3×C10 [×3], S3×C10 [×6], C2×C30 [×3], C22×C20 [×3], D4×C10 [×3], Q8×C10, C5×C4○D4, C5×C4○D4 [×7], S3×C4○D4, C5×Dic6 [×3], S3×C20, S3×C20 [×9], C5×D12 [×3], C10×Dic3 [×3], C5×C3⋊D4 [×6], C2×C60 [×3], D4×C15 [×3], Q8×C15, S3×C2×C10 [×3], C10×C4○D4, S3×C2×C20 [×3], C5×C4○D12 [×3], C5×S3×D4 [×3], C5×D42S3 [×3], C5×S3×Q8, C5×Q83S3, C15×C4○D4, C5×S3×C4○D4
Quotients: C1, C2 [×15], C22 [×35], C5, S3, C23 [×15], C10 [×15], D6 [×7], C4○D4 [×2], C24, C2×C10 [×35], C22×S3 [×7], C5×S3, C2×C4○D4, C22×C10 [×15], S3×C23, S3×C10 [×7], C5×C4○D4 [×2], C23×C10, S3×C4○D4, S3×C2×C10 [×7], 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 14 19)(7 15 20)(8 11 16)(9 12 17)(10 13 18)(21 118 111)(22 119 112)(23 120 113)(24 116 114)(25 117 115)(26 47 41)(27 48 42)(28 49 43)(29 50 44)(30 46 45)(51 73 58)(52 74 59)(53 75 60)(54 71 56)(55 72 57)(61 80 67)(62 76 68)(63 77 69)(64 78 70)(65 79 66)(81 103 88)(82 104 89)(83 105 90)(84 101 86)(85 102 87)(91 110 97)(92 106 98)(93 107 99)(94 108 100)(95 109 96)
(6 19)(7 20)(8 16)(9 17)(10 18)(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 73)(52 74)(53 75)(54 71)(55 72)(61 67)(62 68)(63 69)(64 70)(65 66)(81 103)(82 104)(83 105)(84 101)(85 102)(91 97)(92 98)(93 99)(94 100)(95 96)
(1 56 26 77)(2 57 27 78)(3 58 28 79)(4 59 29 80)(5 60 30 76)(6 95 21 103)(7 91 22 104)(8 92 23 105)(9 93 24 101)(10 94 25 102)(11 106 120 90)(12 107 116 86)(13 108 117 87)(14 109 118 88)(15 110 119 89)(16 98 113 83)(17 99 114 84)(18 100 115 85)(19 96 111 81)(20 97 112 82)(31 74 44 61)(32 75 45 62)(33 71 41 63)(34 72 42 64)(35 73 43 65)(36 51 49 66)(37 52 50 67)(38 53 46 68)(39 54 47 69)(40 55 48 70)
(1 107 26 86)(2 108 27 87)(3 109 28 88)(4 110 29 89)(5 106 30 90)(6 73 21 65)(7 74 22 61)(8 75 23 62)(9 71 24 63)(10 72 25 64)(11 60 120 76)(12 56 116 77)(13 57 117 78)(14 58 118 79)(15 59 119 80)(16 53 113 68)(17 54 114 69)(18 55 115 70)(19 51 111 66)(20 52 112 67)(31 91 44 104)(32 92 45 105)(33 93 41 101)(34 94 42 102)(35 95 43 103)(36 96 49 81)(37 97 50 82)(38 98 46 83)(39 99 47 84)(40 100 48 85)
(6 21)(7 22)(8 23)(9 24)(10 25)(11 120)(12 116)(13 117)(14 118)(15 119)(16 113)(17 114)(18 115)(19 111)(20 112)(81 96)(82 97)(83 98)(84 99)(85 100)(86 107)(87 108)(88 109)(89 110)(90 106)(91 104)(92 105)(93 101)(94 102)(95 103)

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

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

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

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++++++++++++
imageC1C2C2C2C2C2C2C2C5C10C10C10C10C10C10C10S3D6D6D6C4○D4C5×S3S3×C10S3×C10S3×C10C5×C4○D4S3×C4○D4C5×S3×C4○D4
kernelC5×S3×C4○D4S3×C2×C20C5×C4○D12C5×S3×D4C5×D42S3C5×S3×Q8C5×Q83S3C15×C4○D4S3×C4○D4S3×C2×C4C4○D12S3×D4D42S3S3×Q8Q83S3C3×C4○D4C5×C4○D4C2×C20C5×D4C5×Q8C5×S3C4○D4C2×C4D4Q8S3C5C1
# reps13333111412121212444133144121241628

Matrix representation of C5×S3×C4○D4 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] >;

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

׿
×
𝔽