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G = C5×S3×M4(2)  order 480 = 25·3·5

Direct product of C5, S3 and M4(2)

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

Aliases: C5×S3×M4(2), C4028D6, C12035C22, C60.286C23, C86(S3×C10), (S3×C8)⋊7C10, C247(C2×C10), (S3×C40)⋊16C2, C8⋊S35C10, (C4×S3).1C20, C4.15(S3×C20), D6.9(C2×C20), C32(C10×M4(2)), (S3×C20).12C4, C20.117(C4×S3), C60.182(C2×C4), C12.12(C2×C20), (C2×C20).356D6, C1528(C2×M4(2)), C4.Dic35C10, C22.7(S3×C20), (C3×M4(2))⋊5C10, (C22×S3).4C20, C6.15(C22×C20), Dic3.8(C2×C20), (C2×Dic3).6C20, (C15×M4(2))⋊13C2, (S3×C20).65C22, C20.244(C22×S3), C12.38(C22×C10), (C2×C60).353C22, C30.206(C22×C4), (C10×Dic3).23C4, C3⋊C811(C2×C10), (S3×C2×C4).4C10, C4.38(S3×C2×C10), C2.16(S3×C2×C20), (C5×C3⋊C8)⋊44C22, (S3×C2×C20).15C2, (S3×C2×C10).14C4, C10.142(S3×C2×C4), (C2×C6).5(C2×C20), (C5×C8⋊S3)⋊13C2, (C2×C10).66(C4×S3), (C2×C4).45(S3×C10), (S3×C10).45(C2×C4), (C4×S3).16(C2×C10), (C2×C12).26(C2×C10), (C2×C30).129(C2×C4), (C5×C4.Dic3)⋊17C2, (C5×Dic3).50(C2×C4), SmallGroup(480,785)

Series: Derived Chief Lower central Upper central

C1C6 — C5×S3×M4(2)
C1C3C6C12C60S3×C20S3×C2×C20 — C5×S3×M4(2)
C3C6 — C5×S3×M4(2)
C1C20C5×M4(2)

Generators and relations for C5×S3×M4(2)
 G = < a,b,c,d,e | a5=b3=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d5 >

Subgroups: 260 in 136 conjugacy classes, 78 normal (54 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, S3 [×2], S3, C6, C6, C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, C10, C10 [×4], Dic3 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C15, C2×C8 [×2], M4(2), M4(2) [×3], C22×C4, C20 [×2], C20 [×2], C2×C10, C2×C10 [×4], C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, C5×S3 [×2], C5×S3, C30, C30, C2×M4(2), C40 [×2], C40 [×2], C2×C20, C2×C20 [×5], C22×C10, S3×C8 [×2], C8⋊S3 [×2], C4.Dic3, C3×M4(2), S3×C2×C4, C5×Dic3 [×2], C60 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C2×C40 [×2], C5×M4(2), C5×M4(2) [×3], C22×C20, S3×M4(2), C5×C3⋊C8 [×2], C120 [×2], S3×C20 [×4], C10×Dic3, C2×C60, S3×C2×C10, C10×M4(2), S3×C40 [×2], C5×C8⋊S3 [×2], C5×C4.Dic3, C15×M4(2), S3×C2×C20, C5×S3×M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C2×C4 [×6], C23, C10 [×7], D6 [×3], M4(2) [×2], C22×C4, C20 [×4], C2×C10 [×7], C4×S3 [×2], C22×S3, C5×S3, C2×M4(2), C2×C20 [×6], C22×C10, S3×C2×C4, S3×C10 [×3], C5×M4(2) [×2], C22×C20, S3×M4(2), S3×C20 [×2], S3×C2×C10, C10×M4(2), S3×C2×C20, C5×S3×M4(2)

Smallest permutation representation of C5×S3×M4(2)
On 120 points
Generators in S120
(1 15 35 108 29)(2 16 36 109 30)(3 9 37 110 31)(4 10 38 111 32)(5 11 39 112 25)(6 12 40 105 26)(7 13 33 106 27)(8 14 34 107 28)(17 43 58 82 51)(18 44 59 83 52)(19 45 60 84 53)(20 46 61 85 54)(21 47 62 86 55)(22 48 63 87 56)(23 41 64 88 49)(24 42 57 81 50)(65 93 118 77 102)(66 94 119 78 103)(67 95 120 79 104)(68 96 113 80 97)(69 89 114 73 98)(70 90 115 74 99)(71 91 116 75 100)(72 92 117 76 101)
(1 19 99)(2 20 100)(3 21 101)(4 22 102)(5 23 103)(6 24 104)(7 17 97)(8 18 98)(9 47 72)(10 48 65)(11 41 66)(12 42 67)(13 43 68)(14 44 69)(15 45 70)(16 46 71)(25 49 78)(26 50 79)(27 51 80)(28 52 73)(29 53 74)(30 54 75)(31 55 76)(32 56 77)(33 58 96)(34 59 89)(35 60 90)(36 61 91)(37 62 92)(38 63 93)(39 64 94)(40 57 95)(81 120 105)(82 113 106)(83 114 107)(84 115 108)(85 116 109)(86 117 110)(87 118 111)(88 119 112)
(17 97)(18 98)(19 99)(20 100)(21 101)(22 102)(23 103)(24 104)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 65)(49 78)(50 79)(51 80)(52 73)(53 74)(54 75)(55 76)(56 77)(57 95)(58 96)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(81 120)(82 113)(83 114)(84 115)(85 116)(86 117)(87 118)(88 119)
(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)
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)(34 38)(36 40)(42 46)(44 48)(50 54)(52 56)(57 61)(59 63)(65 69)(67 71)(73 77)(75 79)(81 85)(83 87)(89 93)(91 95)(98 102)(100 104)(105 109)(107 111)(114 118)(116 120)

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

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

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

150 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F5A5B5C5D6A6B8A8B8C8D8E8F8G8H10A10B10C10D10E10F10G10H10I···10P10Q10R10S10T12A12B12C15A15B15C15D20A···20H20I20J20K20L20M···20T20U20V20W20X24A24B24C24D30A30B30C30D30E30F30G30H40A···40P40Q···40AF60A···60H60I60J60K60L120A···120P
order122222344444455556688888888101010101010101010···10101010101212121515151520···202020202020···202020202024242424303030303030303040···4040···4060···6060606060120···120
size112336211233611112422226666111122223···3666622422221···122223···366664444222244442···26···62···244444···4

150 irreducible representations

dim11111111111111111122222222222244
type+++++++++
imageC1C2C2C2C2C2C4C4C4C5C10C10C10C10C10C20C20C20S3D6D6M4(2)C4×S3C4×S3C5×S3S3×C10S3×C10C5×M4(2)S3×C20S3×C20S3×M4(2)C5×S3×M4(2)
kernelC5×S3×M4(2)S3×C40C5×C8⋊S3C5×C4.Dic3C15×M4(2)S3×C2×C20S3×C20C10×Dic3S3×C2×C10S3×M4(2)S3×C8C8⋊S3C4.Dic3C3×M4(2)S3×C2×C4C4×S3C2×Dic3C22×S3C5×M4(2)C40C2×C20C5×S3C20C2×C10M4(2)C8C2×C4S3C4C22C5C1
# reps1221114224884441688121422484168828

Matrix representation of C5×S3×M4(2) in GL4(𝔽241) generated by

205000
020500
00910
00091
,
24024000
1000
0010
0001
,
1000
24024000
0010
0001
,
1000
0100
0046239
00126195
,
1000
0100
0010
0046240
G:=sub<GL(4,GF(241))| [205,0,0,0,0,205,0,0,0,0,91,0,0,0,0,91],[240,1,0,0,240,0,0,0,0,0,1,0,0,0,0,1],[1,240,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,46,126,0,0,239,195],[1,0,0,0,0,1,0,0,0,0,1,46,0,0,0,240] >;

C5×S3×M4(2) in GAP, Magma, Sage, TeX

C_5\times S_3\times M_4(2)
% in TeX

G:=Group("C5xS3xM4(2)");
// GroupNames label

G:=SmallGroup(480,785);
// by ID

G=gap.SmallGroup(480,785);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,891,226,102,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^3=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^5>;
// generators/relations

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