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G = D6011C2order 240 = 24·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6011C2, C4.16D30, C20.51D6, C12.47D10, C22.2D30, Dic3011C2, C30.31C23, C60.58C22, D30.5C22, Dic15.8C22, (C2×C20)⋊4S3, (C2×C60)⋊6C2, (C2×C4)⋊3D15, (C2×C12)⋊4D5, (C4×D15)⋊4C2, C55(C4○D12), C35(C4○D20), C157D47C2, C1511(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

C1C30 — D6011C2
C1C5C15C30D30C4×D15 — D6011C2
C15C30 — D6011C2
C1C4C2×C4

Generators and relations for D6011C2
 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 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C5, S3 [×2], C6, C6, C2×C4, C2×C4 [×2], D4 [×3], Q8, D5 [×2], C10, C10, Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C15, C4○D4, Dic5 [×2], C20 [×2], D10 [×2], C2×C10, Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12, D15 [×2], C30, C30, Dic10, C4×D5 [×2], D20, C5⋊D4 [×2], C2×C20, C4○D12, Dic15 [×2], C60 [×2], D30 [×2], C2×C30, C4○D20, Dic30, C4×D15 [×2], D60, C157D4 [×2], C2×C60, D6011C2
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4, D10 [×3], C22×S3, D15, C22×D5, C4○D12, D30 [×3], C4○D20, C22×D15, D6011C2

Smallest permutation representation of D6011C2
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)(61 63)(64 120)(65 119)(66 118)(67 117)(68 116)(69 115)(70 114)(71 113)(72 112)(73 111)(74 110)(75 109)(76 108)(77 107)(78 106)(79 105)(80 104)(81 103)(82 102)(83 101)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)
(1 100)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 110)(12 111)(13 112)(14 113)(15 114)(16 115)(17 116)(18 117)(19 118)(20 119)(21 120)(22 61)(23 62)(24 63)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(33 72)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)(41 80)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 88)(50 89)(51 90)(52 91)(53 92)(54 93)(55 94)(56 95)(57 96)(58 97)(59 98)(60 99)

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

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

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

D6011C2 is a maximal subgroup of
D6013C4  D6016C4  D607C4  D6010C4  D60.5C4  D60.4C4  C20.60D12  D6036C22  D20.31D6  D6030C22  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  D2025D6  D2026D6  D46D30  Q8.15D30  C4○D4×D15  D48D30  D4.10D30
D6011C2 is a maximal quotient of
C4×Dic30  C60.24Q8  C422D15  C4×D60  C427D15  C423D15  C23.8D30  D30.28D4  D309D4  C23.11D30  Dic15.3Q8  D30.29D4  D305Q8  C4⋊C4⋊D15  C60.205D4  C23.26D30  C4×C157D4  C23.28D30  C6029D4

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○D20D6011C2
kernelD6011C2Dic30C4×D15D60C157D4C2×C60C2×C20C2×C12C20C2×C10C15C12C2×C6C2×C4C5C4C22C3C1
# reps11212112212424484816

Matrix representation of D6011C2 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] >;

D6011C2 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

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