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G = C10×C3⋊D4order 240 = 24·3·5

Direct product of C10 and C3⋊D4

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

Aliases: C10×C3⋊D4, C309D4, C30.57C23, C62(C5×D4), C33(D4×C10), C1518(C2×D4), D63(C2×C10), (C2×C10)⋊10D6, C232(C5×S3), C223(S3×C10), (C22×C10)⋊3S3, (C22×C30)⋊6C2, (C22×C6)⋊2C10, (C22×S3)⋊3C10, (C2×C30)⋊13C22, (C2×Dic3)⋊4C10, Dic32(C2×C10), (S3×C10)⋊11C22, (C10×Dic3)⋊10C2, C10.47(C22×S3), C6.10(C22×C10), (C5×Dic3)⋊9C22, (S3×C2×C10)⋊7C2, (C2×C6)⋊3(C2×C10), C2.10(S3×C2×C10), SmallGroup(240,174)

Series: Derived Chief Lower central Upper central

C1C6 — C10×C3⋊D4
C1C3C6C30S3×C10S3×C2×C10 — C10×C3⋊D4
C3C6 — C10×C3⋊D4
C1C2×C10C22×C10

Generators and relations for C10×C3⋊D4
 G = < a,b,c,d | a10=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 216 in 108 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×2], C22 [×6], C5, S3 [×2], C6, C6 [×2], C6 [×2], C2×C4, D4 [×4], C23, C23, C10, C10 [×2], C10 [×4], Dic3 [×2], D6 [×2], D6 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C2×D4, C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×6], C2×Dic3, C3⋊D4 [×4], C22×S3, C22×C6, C5×S3 [×2], C30, C30 [×2], C30 [×2], C2×C20, C5×D4 [×4], C22×C10, C22×C10, C2×C3⋊D4, C5×Dic3 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], D4×C10, C10×Dic3, C5×C3⋊D4 [×4], S3×C2×C10, C22×C30, C10×C3⋊D4
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C2×C10 [×7], C3⋊D4 [×2], C22×S3, C5×S3, C5×D4 [×2], C22×C10, C2×C3⋊D4, S3×C10 [×3], D4×C10, C5×C3⋊D4 [×2], S3×C2×C10, C10×C3⋊D4

Smallest permutation representation of C10×C3⋊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 56 28)(2 57 29)(3 58 30)(4 59 21)(5 60 22)(6 51 23)(7 52 24)(8 53 25)(9 54 26)(10 55 27)(11 112 80)(12 113 71)(13 114 72)(14 115 73)(15 116 74)(16 117 75)(17 118 76)(18 119 77)(19 120 78)(20 111 79)(31 108 83)(32 109 84)(33 110 85)(34 101 86)(35 102 87)(36 103 88)(37 104 89)(38 105 90)(39 106 81)(40 107 82)(41 69 92)(42 70 93)(43 61 94)(44 62 95)(45 63 96)(46 64 97)(47 65 98)(48 66 99)(49 67 100)(50 68 91)
(1 20 43 104)(2 11 44 105)(3 12 45 106)(4 13 46 107)(5 14 47 108)(6 15 48 109)(7 16 49 110)(8 17 50 101)(9 18 41 102)(10 19 42 103)(21 114 97 82)(22 115 98 83)(23 116 99 84)(24 117 100 85)(25 118 91 86)(26 119 92 87)(27 120 93 88)(28 111 94 89)(29 112 95 90)(30 113 96 81)(31 60 73 65)(32 51 74 66)(33 52 75 67)(34 53 76 68)(35 54 77 69)(36 55 78 70)(37 56 79 61)(38 57 80 62)(39 58 71 63)(40 59 72 64)
(11 105)(12 106)(13 107)(14 108)(15 109)(16 110)(17 101)(18 102)(19 103)(20 104)(21 59)(22 60)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 115)(32 116)(33 117)(34 118)(35 119)(36 120)(37 111)(38 112)(39 113)(40 114)(61 94)(62 95)(63 96)(64 97)(65 98)(66 99)(67 100)(68 91)(69 92)(70 93)(71 81)(72 82)(73 83)(74 84)(75 85)(76 86)(77 87)(78 88)(79 89)(80 90)

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

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

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

C10×C3⋊D4 is a maximal subgroup of
C158(C23⋊C4)  C23.D5⋊S3  C30.(C2×D4)  (C2×C10).D12  (C6×D5)⋊D4  (S3×C10).D4  D307D4  Dic154D4  Dic1517D4  (C2×C30)⋊D4  (S3×C10)⋊D4  (C2×C10)⋊4D12  Dic155D4  Dic1518D4  D3019D4  D308D4  C15⋊2+ 1+4  S3×D4×C10

90 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B5A5B5C5D6A···6G10A···10L10M···10T10U···10AB15A15B15C15D20A···20H30A···30AB
order1222222234455556···610···1010···1010···101515151520···2030···30
size1111226626611112···21···12···26···622226···62···2

90 irreducible representations

dim111111111122222222
type++++++++
imageC1C2C2C2C2C5C10C10C10C10S3D4D6C3⋊D4C5×S3C5×D4S3×C10C5×C3⋊D4
kernelC10×C3⋊D4C10×Dic3C5×C3⋊D4S3×C2×C10C22×C30C2×C3⋊D4C2×Dic3C3⋊D4C22×S3C22×C6C22×C10C30C2×C10C10C23C6C22C2
# reps114114416441234481216

Matrix representation of C10×C3⋊D4 in GL3(𝔽61) generated by

6000
0270
0027
,
100
06060
010
,
6000
0918
0952
,
6000
010
06060
G:=sub<GL(3,GF(61))| [60,0,0,0,27,0,0,0,27],[1,0,0,0,60,1,0,60,0],[60,0,0,0,9,9,0,18,52],[60,0,0,0,1,60,0,0,60] >;

C10×C3⋊D4 in GAP, Magma, Sage, TeX

C_{10}\times C_3\rtimes D_4
% in TeX

G:=Group("C10xC3:D4");
// GroupNames label

G:=SmallGroup(240,174);
// by ID

G=gap.SmallGroup(240,174);
# by ID

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

G:=Group<a,b,c,d|a^10=b^3=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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