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G = C5×D42S3order 240 = 24·3·5

Direct product of C5 and D42S3

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

Aliases: C5×D42S3, C20.40D6, Dic63C10, C60.47C22, C30.53C23, D42(C5×S3), (C5×D4)⋊5S3, (C4×S3)⋊2C10, (S3×C20)⋊7C2, (D4×C15)⋊9C2, (C3×D4)⋊3C10, C3⋊D42C10, C4.5(S3×C10), (C2×C10).4D6, C1516(C4○D4), C12.5(C2×C10), (C5×Dic6)⋊9C2, D6.2(C2×C10), (C10×Dic3)⋊9C2, (C2×Dic3)⋊3C10, C6.6(C22×C10), C22.1(S3×C10), (C2×C30).20C22, C10.43(C22×S3), Dic3.3(C2×C10), (S3×C10).13C22, (C5×Dic3).16C22, C32(C5×C4○D4), (C2×C6).(C2×C10), C2.7(S3×C2×C10), (C5×C3⋊D4)⋊6C2, (C5×D4)(C5×Dic3), SmallGroup(240,170)

Series: Derived Chief Lower central Upper central

C1C6 — C5×D42S3
C1C3C6C30S3×C10S3×C20 — C5×D42S3
C3C6 — C5×D42S3
C1C10C5×D4

Generators and relations for C5×D42S3
 G = < a,b,c,d,e | a5=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 144 in 80 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, D4, Q8, C10, C10, Dic3, Dic3, C12, D6, C2×C6, C15, C4○D4, C20, C20, C2×C10, C2×C10, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C5×S3, C30, C30, C2×C20, C5×D4, C5×D4, C5×Q8, D42S3, C5×Dic3, C5×Dic3, C60, S3×C10, C2×C30, C5×C4○D4, C5×Dic6, S3×C20, C10×Dic3, C5×C3⋊D4, D4×C15, C5×D42S3
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, C4○D4, C2×C10, C22×S3, C5×S3, C22×C10, D42S3, S3×C10, C5×C4○D4, S3×C2×C10, C5×D42S3

Smallest permutation representation of C5×D42S3
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 86 26 107)(2 87 27 108)(3 88 28 109)(4 89 29 110)(5 90 30 106)(6 77 116 56)(7 78 117 57)(8 79 118 58)(9 80 119 59)(10 76 120 60)(11 64 25 66)(12 65 21 67)(13 61 22 68)(14 62 23 69)(15 63 24 70)(16 74 113 53)(17 75 114 54)(18 71 115 55)(19 72 111 51)(20 73 112 52)(31 98 38 91)(32 99 39 92)(33 100 40 93)(34 96 36 94)(35 97 37 95)(41 85 48 101)(42 81 49 102)(43 82 50 103)(44 83 46 104)(45 84 47 105)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 34)(12 35)(13 31)(14 32)(15 33)(16 44)(17 45)(18 41)(19 42)(20 43)(21 37)(22 38)(23 39)(24 40)(25 36)(26 116)(27 117)(28 118)(29 119)(30 120)(46 113)(47 114)(48 115)(49 111)(50 112)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)
(1 33 45)(2 34 41)(3 35 42)(4 31 43)(5 32 44)(6 15 17)(7 11 18)(8 12 19)(9 13 20)(10 14 16)(21 111 118)(22 112 119)(23 113 120)(24 114 116)(25 115 117)(26 40 47)(27 36 48)(28 37 49)(29 38 50)(30 39 46)(51 58 67)(52 59 68)(53 60 69)(54 56 70)(55 57 66)(61 73 80)(62 74 76)(63 75 77)(64 71 78)(65 72 79)(81 88 97)(82 89 98)(83 90 99)(84 86 100)(85 87 96)(91 103 110)(92 104 106)(93 105 107)(94 101 108)(95 102 109)
(6 116)(7 117)(8 118)(9 119)(10 120)(11 115)(12 111)(13 112)(14 113)(15 114)(16 23)(17 24)(18 25)(19 21)(20 22)(31 43)(32 44)(33 45)(34 41)(35 42)(36 48)(37 49)(38 50)(39 46)(40 47)(51 65)(52 61)(53 62)(54 63)(55 64)(56 77)(57 78)(58 79)(59 80)(60 76)(66 71)(67 72)(68 73)(69 74)(70 75)(81 97)(82 98)(83 99)(84 100)(85 96)(91 103)(92 104)(93 105)(94 101)(95 102)

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

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

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

C5×D42S3 is a maximal subgroup of
D60.C22  C60.10C23  D20.24D6  C60.19C23  C15⋊2- 1+4  D30.C23  D2014D6  C5×S3×C4○D4
C5×D42S3 is a maximal quotient of
C5×D4×Dic3

75 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B5C5D6A6B6C10A10B10C10D10E···10L10M10N10O10P 12 15A15B15C15D20A20B20C20D20E···20L20M···20T30A30B30C30D30E···30L60A60B60C60D
order1222234444455556661010101010···101010101012151515152020202020···2020···203030303030···3060606060
size11226223366111124411112···266664222222223···36···622224···44444

75 irreducible representations

dim1111111111112222222244
type+++++++++-
imageC1C2C2C2C2C2C5C10C10C10C10C10S3D6D6C4○D4C5×S3S3×C10S3×C10C5×C4○D4D42S3C5×D42S3
kernelC5×D42S3C5×Dic6S3×C20C10×Dic3C5×C3⋊D4D4×C15D42S3Dic6C4×S3C2×Dic3C3⋊D4C3×D4C5×D4C20C2×C10C15D4C4C22C3C5C1
# reps1112214448841122448814

Matrix representation of C5×D42S3 in GL4(𝔽61) generated by

9000
0900
0010
0001
,
60000
06000
00500
005011
,
1000
0100
001139
001150
,
60100
60000
0010
0001
,
0100
1000
0010
00160
G:=sub<GL(4,GF(61))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,50,50,0,0,0,11],[1,0,0,0,0,1,0,0,0,0,11,11,0,0,39,50],[60,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,1,0,0,0,60] >;

C5×D42S3 in GAP, Magma, Sage, TeX

C_5\times D_4\rtimes_2S_3
% in TeX

G:=Group("C5xD4:2S3");
// GroupNames label

G:=SmallGroup(240,170);
// by ID

G=gap.SmallGroup(240,170);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-3,247,794,404,5765]);
// Polycyclic

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

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