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G = C2xD30.C2order 240 = 24·3·5

Direct product of C2 and D30.C2

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

Aliases: C2xD30.C2, D30:5C4, Dic5:6D6, Dic3:6D10, C30.20C23, D30.14C22, C6:1(C4xD5), C10:2(C4xS3), C30:6(C2xC4), D15:3(C2xC4), C15:7(C22xC4), (C2xDic3):5D5, (C2xDic5):5S3, (C6xDic5):5C2, (C2xC6).15D10, (C2xC10).15D6, (C10xDic3):5C2, C6.20(C22xD5), C22.13(S3xD5), (C2xC30).14C22, C10.20(C22xS3), (C5xDic3):6C22, (C3xDic5):6C22, (C22xD15).3C2, C5:3(S3xC2xC4), C3:2(C2xC4xD5), C2.4(C2xS3xD5), SmallGroup(240,144)

Series: Derived Chief Lower central Upper central

C1C15 — C2xD30.C2
C1C5C15C30C3xDic5D30.C2 — C2xD30.C2
C15 — C2xD30.C2
C1C22

Generators and relations for C2xD30.C2
 G = < a,b,c,d | a2=b30=c2=1, d2=b15, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b19, dcd-1=b18c >

Subgroups: 464 in 108 conjugacy classes, 48 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2xC4, C23, D5, C10, C10, Dic3, C12, D6, C2xC6, C15, C22xC4, Dic5, C20, D10, C2xC10, C4xS3, C2xDic3, C2xC12, C22xS3, D15, C30, C30, C4xD5, C2xDic5, C2xC20, C22xD5, S3xC2xC4, C5xDic3, C3xDic5, D30, C2xC30, C2xC4xD5, D30.C2, C6xDic5, C10xDic3, C22xD15, C2xD30.C2
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, D5, D6, C22xC4, D10, C4xS3, C22xS3, C4xD5, C22xD5, S3xC2xC4, S3xD5, C2xC4xD5, D30.C2, C2xS3xD5, C2xD30.C2

Smallest permutation representation of C2xD30.C2
On 120 points
Generators in S120
(1 58)(2 59)(3 60)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(61 105)(62 106)(63 107)(64 108)(65 109)(66 110)(67 111)(68 112)(69 113)(70 114)(71 115)(72 116)(73 117)(74 118)(75 119)(76 120)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)
(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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 60)(14 59)(15 58)(16 57)(17 56)(18 55)(19 54)(20 53)(21 52)(22 51)(23 50)(24 49)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(61 103)(62 102)(63 101)(64 100)(65 99)(66 98)(67 97)(68 96)(69 95)(70 94)(71 93)(72 92)(73 91)(74 120)(75 119)(76 118)(77 117)(78 116)(79 115)(80 114)(81 113)(82 112)(83 111)(84 110)(85 109)(86 108)(87 107)(88 106)(89 105)(90 104)
(1 97 16 112)(2 116 17 101)(3 105 18 120)(4 94 19 109)(5 113 20 98)(6 102 21 117)(7 91 22 106)(8 110 23 95)(9 99 24 114)(10 118 25 103)(11 107 26 92)(12 96 27 111)(13 115 28 100)(14 104 29 119)(15 93 30 108)(31 80 46 65)(32 69 47 84)(33 88 48 73)(34 77 49 62)(35 66 50 81)(36 85 51 70)(37 74 52 89)(38 63 53 78)(39 82 54 67)(40 71 55 86)(41 90 56 75)(42 79 57 64)(43 68 58 83)(44 87 59 72)(45 76 60 61)

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

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

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

C2xD30.C2 is a maximal subgroup of
D30:C8  D30.D4  Dic3:4D20  D30.C2:C4  D30.23(C2xC4)  D30.Q8  Dic5:4D12  D30:Q8  D60:17C4  D30:2Q8  D30:D4  D30:3Q8  D60:14C4  D30:4Q8  D30.6D4  D30.2Q8  D30.7D4  C15:20(C4xD4)  C15:22(C4xD4)  D30:2D4  D30.27D4  D30:6D4  C15:26(C4xD4)  C15:28(C4xD4)  D30:7D4  D30.45D4  D30.16D4  D15:2M4(2)  S3xC2xC4xD5  D30.C23
C2xD30.C2 is a maximal quotient of
D60.5C4  D60.4C4  D15:4M4(2)  Dic30:17C4  Dic30:14C4  (C4xD15):8C4  (C4xD15):10C4  D60:17C4  D60:14C4  D30.2Q8  C2xDic3xDic5  C23.48(S3xD5)  C15:26(C4xD4)  C15:28(C4xD4)  D30.45D4

48 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H5A5B6A6B6C10A···10F12A12B12C12D15A15B20A···20H30A···30F
order122222223444444445566610···1012121212151520···2030···30
size111115151515233335555222222···210101010446···64···4

48 irreducible representations

dim11111122222222444
type++++++++++++++
imageC1C2C2C2C2C4S3D5D6D6D10D10C4xS3C4xD5S3xD5D30.C2C2xS3xD5
kernelC2xD30.C2D30.C2C6xDic5C10xDic3C22xD15D30C2xDic5C2xDic3Dic5C2xC10Dic3C2xC6C10C6C22C2C2
# reps14111812214248242

Matrix representation of C2xD30.C2 in GL5(F61)

600000
01000
00100
00010
00001
,
10000
0434200
01100
00011
000600
,
10000
0604200
00100
00010
0006060
,
600000
0151500
0504600
000500
000050

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,43,1,0,0,0,42,1,0,0,0,0,0,1,60,0,0,0,1,0],[1,0,0,0,0,0,60,0,0,0,0,42,1,0,0,0,0,0,1,60,0,0,0,0,60],[60,0,0,0,0,0,15,50,0,0,0,15,46,0,0,0,0,0,50,0,0,0,0,0,50] >;

C2xD30.C2 in GAP, Magma, Sage, TeX

C_2\times D_{30}.C_2
% in TeX

G:=Group("C2xD30.C2");
// GroupNames label

G:=SmallGroup(240,144);
// by ID

G=gap.SmallGroup(240,144);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,55,490,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^30=c^2=1,d^2=b^15,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^19,d*c*d^-1=b^18*c>;
// generators/relations

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