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G = C2×C4×D15order 240 = 24·3·5

Direct product of C2×C4 and D15

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

Aliases: C2×C4×D15, C208D6, C128D10, C609C22, C22.9D30, C30.29C23, D30.15C22, Dic1510C22, C62(C4×D5), C103(C4×S3), C307(C2×C4), (C2×C60)⋊7C2, (C2×C20)⋊5S3, (C2×C12)⋊5D5, C158(C22×C4), (C2×C6).27D10, (C2×C10).27D6, C2.1(C22×D15), C6.29(C22×D5), (C2×Dic15)⋊11C2, (C2×C30).28C22, C10.29(C22×S3), (C22×D15).4C2, C54(S3×C2×C4), C33(C2×C4×D5), SmallGroup(240,176)

Series: Derived Chief Lower central Upper central

C1C15 — C2×C4×D15
C1C5C15C30D30C22×D15 — C2×C4×D15
C15 — C2×C4×D15
C1C2×C4

Generators and relations for C2×C4×D15
 G = < a,b,c,d | a2=b4=c15=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 488 in 108 conjugacy classes, 51 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×6], C5, S3 [×4], C6, C6 [×2], C2×C4, C2×C4 [×5], C23, D5 [×4], C10, C10 [×2], Dic3 [×2], C12 [×2], D6 [×6], C2×C6, C15, C22×C4, Dic5 [×2], C20 [×2], D10 [×6], C2×C10, C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, D15 [×4], C30, C30 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, S3×C2×C4, Dic15 [×2], C60 [×2], D30 [×6], C2×C30, C2×C4×D5, C4×D15 [×4], C2×Dic15, C2×C60, C22×D15, C2×C4×D15
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], C22×C4, D10 [×3], C4×S3 [×2], C22×S3, D15, C4×D5 [×2], C22×D5, S3×C2×C4, D30 [×3], C2×C4×D5, C4×D15 [×2], C22×D15, C2×C4×D15

Smallest permutation representation of C2×C4×D15
On 120 points
Generators in S120
(1 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 61)(9 62)(10 63)(11 64)(12 65)(13 66)(14 67)(15 68)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 85)(23 86)(24 87)(25 88)(26 89)(27 90)(28 76)(29 77)(30 78)(31 98)(32 99)(33 100)(34 101)(35 102)(36 103)(37 104)(38 105)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 97)(46 120)(47 106)(48 107)(49 108)(50 109)(51 110)(52 111)(53 112)(54 113)(55 114)(56 115)(57 116)(58 117)(59 118)(60 119)
(1 47 28 43)(2 48 29 44)(3 49 30 45)(4 50 16 31)(5 51 17 32)(6 52 18 33)(7 53 19 34)(8 54 20 35)(9 55 21 36)(10 56 22 37)(11 57 23 38)(12 58 24 39)(13 59 25 40)(14 60 26 41)(15 46 27 42)(61 113 83 102)(62 114 84 103)(63 115 85 104)(64 116 86 105)(65 117 87 91)(66 118 88 92)(67 119 89 93)(68 120 90 94)(69 106 76 95)(70 107 77 96)(71 108 78 97)(72 109 79 98)(73 110 80 99)(74 111 81 100)(75 112 82 101)
(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 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(13 30)(14 29)(15 28)(31 58)(32 57)(33 56)(34 55)(35 54)(36 53)(37 52)(38 51)(39 50)(40 49)(41 48)(42 47)(43 46)(44 60)(45 59)(61 83)(62 82)(63 81)(64 80)(65 79)(66 78)(67 77)(68 76)(69 90)(70 89)(71 88)(72 87)(73 86)(74 85)(75 84)(91 109)(92 108)(93 107)(94 106)(95 120)(96 119)(97 118)(98 117)(99 116)(100 115)(101 114)(102 113)(103 112)(104 111)(105 110)

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

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

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

C2×C4×D15 is a maximal subgroup of
D304C8  D303C8  D154M4(2)  (C4×D15)⋊8C4  D30.34D4  D30.35D4  D308Q8  D309Q8  (C4×D15)⋊10C4  D3010Q8  Dic1513D4  D30.Q8  Dic1514D4  D30.2Q8  D3012D4  C122D20  C202D12  D30.27D4  C422D15  Dic1519D4  D30.28D4  D309D4  C4⋊C47D15  D6011C4  D30.29D4  C4⋊D60  D305Q8  D306Q8  C602D4  D307Q8  S3×C2×C4×D5  D2024D6
C2×C4×D15 is a maximal quotient of
C422D15  C23.15D30  Dic1519D4  Dic1510Q8  C4⋊C47D15  D6011C4  D60.6C4  D60.3C4

72 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H5A5B6A6B6C10A···10F12A12B12C12D15A15B15C15D20A···20H30A···30L60A···60P
order122222223444444445566610···10121212121515151520···2030···3060···60
size1111151515152111115151515222222···2222222222···22···22···2

72 irreducible representations

dim111111222222222222
type++++++++++++++
imageC1C2C2C2C2C4S3D5D6D6D10D10C4×S3D15C4×D5D30D30C4×D15
kernelC2×C4×D15C4×D15C2×Dic15C2×C60C22×D15D30C2×C20C2×C12C20C2×C10C12C2×C6C10C2×C4C6C4C22C2
# reps1411181221424488416

Matrix representation of C2×C4×D15 in GL3(𝔽61) generated by

6000
0600
0060
,
100
0110
0011
,
100
03714
04731
,
6000
06017
001
G:=sub<GL(3,GF(61))| [60,0,0,0,60,0,0,0,60],[1,0,0,0,11,0,0,0,11],[1,0,0,0,37,47,0,14,31],[60,0,0,0,60,0,0,17,1] >;

C2×C4×D15 in GAP, Magma, Sage, TeX

C_2\times C_4\times D_{15}
% in TeX

G:=Group("C2xC4xD15");
// GroupNames label

G:=SmallGroup(240,176);
// by ID

G=gap.SmallGroup(240,176);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,50,964,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^15=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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