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G = C2xDic15order 120 = 23·3·5

Direct product of C2 and Dic15

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

Aliases: C2xDic15, C6:Dic5, C30:3C4, C2.2D30, C22.D15, C10:2Dic3, C10.11D6, C6.11D10, C30.11C22, (C2xC6).D5, (C2xC10).S3, C15:10(C2xC4), C5:3(C2xDic3), C3:2(C2xDic5), (C2xC30).1C2, SmallGroup(120,29)

Series: Derived Chief Lower central Upper central

C1C15 — C2xDic15
C1C5C15C30Dic15 — C2xDic15
C15 — C2xDic15
C1C22

Generators and relations for C2xDic15
 G = < a,b,c | a2=b30=1, c2=b15, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 92 in 32 conjugacy classes, 23 normal (13 characteristic)
Quotients: C1, C2, C4, C22, S3, C2xC4, D5, Dic3, D6, Dic5, D10, C2xDic3, D15, C2xDic5, Dic15, D30, C2xDic15
15C4
15C4
15C2xC4
5Dic3
5Dic3
3Dic5
3Dic5
5C2xDic3
3C2xDic5

Smallest permutation representation of C2xDic15
Regular action on 120 points
Generators in S120
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(61 116)(62 117)(63 118)(64 119)(65 120)(66 91)(67 92)(68 93)(69 94)(70 95)(71 96)(72 97)(73 98)(74 99)(75 100)(76 101)(77 102)(78 103)(79 104)(80 105)(81 106)(82 107)(83 108)(84 109)(85 110)(86 111)(87 112)(88 113)(89 114)(90 115)
(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 91 16 106)(2 120 17 105)(3 119 18 104)(4 118 19 103)(5 117 20 102)(6 116 21 101)(7 115 22 100)(8 114 23 99)(9 113 24 98)(10 112 25 97)(11 111 26 96)(12 110 27 95)(13 109 28 94)(14 108 29 93)(15 107 30 92)(31 81 46 66)(32 80 47 65)(33 79 48 64)(34 78 49 63)(35 77 50 62)(36 76 51 61)(37 75 52 90)(38 74 53 89)(39 73 54 88)(40 72 55 87)(41 71 56 86)(42 70 57 85)(43 69 58 84)(44 68 59 83)(45 67 60 82)

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

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

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

C2xDic15 is a maximal subgroup of
Dic3xDic5  D10:Dic3  D6:Dic5  C30.Q8  Dic15:5C4  C6.Dic10  C30.4Q8  C60:5C4  D30:3C4  C30.38D4  C2xD5xDic3  C30.C23  C2xS3xDic5  C2xC4xD15  D4:2D15  Q8:Dic15
C2xDic15 is a maximal quotient of
C60.7C4  C60:5C4  C30.38D4

36 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D5A5B6A6B6C10A···10F15A15B15C15D30A···30L
order1222344445566610···101515151530···30
size1111215151515222222···222222···2

36 irreducible representations

dim1111222222222
type+++++-+-++-+
imageC1C2C2C4S3D5Dic3D6Dic5D10D15Dic15D30
kernelC2xDic15Dic15C2xC30C30C2xC10C2xC6C10C10C6C6C22C2C2
# reps1214122142484

Matrix representation of C2xDic15 in GL3(F61) generated by

6000
010
001
,
100
0825
01356
,
100
0383
04723
G:=sub<GL(3,GF(61))| [60,0,0,0,1,0,0,0,1],[1,0,0,0,8,13,0,25,56],[1,0,0,0,38,47,0,3,23] >;

C2xDic15 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_{15}
% in TeX

G:=Group("C2xDic15");
// GroupNames label

G:=SmallGroup(120,29);
// by ID

G=gap.SmallGroup(120,29);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-5,20,323,2404]);
// Polycyclic

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

Export

Subgroup lattice of C2xDic15 in TeX

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