Copied to
clipboard

G = D60⋊11C2  order 240 = 24·3·5

The semidirect product of D60 and C2 acting through Inn(D60)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D60⋊11C2, C4.16D30, C20.51D6, C12.47D10, C22.2D30, Dic30⋊11C2, C30.31C23, C60.58C22, D30.5C22, Dic15.8C22, (C2×C20)⋊4S3, (C2×C60)⋊6C2, (C2×C4)⋊3D15, (C2×C12)⋊4D5, (C4×D15)⋊4C2, C5⋊5(C4○D12), C3⋊5(C4○D20), C15⋊7D4⋊7C2, C15⋊11(C4○D4), (C2×C10).29D6, (C2×C6).29D10, C6.31(C22×D5), C2.5(C22×D15), C10.31(C22×S3), (C2×C30).30C22, SmallGroup(240,178)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C30 — D60⋊11C2
Chief series
C1 — C5 — C15 — C30 — D30 — C4×D15 — D60⋊11C2
Lower central
C15 — C30 — D60⋊11C2
Upper central
C1 — C4 — C2×C4

Generators and relations for D60⋊11C2
 G = < a,b,c | a60=b2=c2=1, bab=a-1, ac=ca, cbc=a30b >

Subgroups: 392 in 80 conjugacy classes, 35 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C4○D4, Dic5, C20, D10, C2×C10, Dic6, C4×S3, D12, C3⋊D4, C2×C12, D15, C30, C30, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C4○D12, Dic15, C60, D30, C2×C30, C4○D20, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, D60⋊11C2
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, D15, C22×D5, C4○D12, D30, C4○D20, C22×D15, D60⋊11C2

Smallest permutation representation of D60⋊11C2
►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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 60)(17 59)(18 58)(19 57)(20 56)(21 55)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 48)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)(62 120)(63 119)(64 118)(65 117)(66 116)(67 115)(68 114)(69 113)(70 112)(71 111)(72 110)(73 109)(74 108)(75 107)(76 106)(77 105)(78 104)(79 103)(80 102)(81 101)(82 100)(83 99)(84 98)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)

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

D60⋊11C2 is a maximal subgroup of
 D60⋊13C4  D60⋊16C4  D60⋊7C4  D60⋊10C4  D60.5C4  D60.4C4  C20.60D12  D60⋊36C22  D20.31D6  D60⋊30C22  C12.D20  C20.D12  D60.6C4  C40.69D6  D60.3C4  C8⋊D30  C8.D30  D4.D30  Q8.11D30  D4.8D30  D20.38D6  C30.C24  D5×C4○D12  S3×C4○D20  D20⋊25D6  D20⋊26D6  D4⋊6D30  Q8.15D30  C4○D4×D15  D4⋊8D30  D4.10D30
D60⋊11C2 is a maximal quotient of
 C4×Dic30  C60.24Q8  C42⋊2D15  C4×D60  C42⋊7D15  C42⋊3D15  C23.8D30  D30.28D4  D30⋊9D4  C23.11D30  Dic15.3Q8  D30.29D4  D30⋊5Q8  C4⋊C4⋊D15  C60.205D4  C23.26D30  C4×C15⋊7D4  C23.28D30  C60⋊29D4

66 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B6A6B6C10A···10F12A12B12C12D15A15B15C15D20A···20H30A···30L60A···60P
order122223444445566610···10121212121515151520···2030···3060···60
size112303021123030222222···2222222222···22···22···2

66 irreducible representations

dim1111112222222222222
type+++++++++++++++
imageC1C2C2C2C2C2S3D5D6D6C4○D4D10D10D15C4○D12D30D30C4○D20D60⋊11C2
kernelD60⋊11C2Dic30C4×D15D60C15⋊7D4C2×C60C2×C20C2×C12C20C2×C10C15C12C2×C6C2×C4C5C4C22C3C1
# reps11212112212424484816

Matrix representation of D60⋊11C2 ►in GL2(𝔽61) generated by

4955
68
,
3714
3324
,
1445
1647
G:=sub<GL(2,GF(61))| [49,6,55,8],[37,33,14,24],[14,16,45,47] >;
 

D60⋊11C2 in GAP, Magma, Sage, TeX 

D_{60}\rtimes_{11}C_2
 
% in TeX
 
G:=Group("D60:11C2");
 
// GroupNames label
 
G:=SmallGroup(240,178);
 
// by ID
 
G=gap.SmallGroup(240,178);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,55,218,964,6917]);
 
// Polycyclic
 
G:=Group<a,b,c|a^60=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^30*b>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁