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G = C2×Dic15order 120 = 23·3·5

Direct product of C2 and Dic15

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

Aliases: C2×Dic15, C6⋊Dic5, C303C4, C2.2D30, C22.D15, C102Dic3, C10.11D6, C6.11D10, C30.11C22, (C2×C6).D5, (C2×C10).S3, C1510(C2×C4), C53(C2×Dic3), C32(C2×Dic5), (C2×C30).1C2, SmallGroup(120,29)

Series: Derived Chief Lower central Upper central

C1C15 — C2×Dic15
C1C5C15C30Dic15 — C2×Dic15
C15 — C2×Dic15
C1C22

Generators and relations for C2×Dic15
 G = < a,b,c | a2=b30=1, c2=b15, ab=ba, ac=ca, cbc-1=b-1 >

15C4
15C4
15C2×C4
5Dic3
5Dic3
3Dic5
3Dic5
5C2×Dic3
3C2×Dic5

Smallest permutation representation of C2×Dic15
Regular action on 120 points
Generators in S120
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(61 101)(62 102)(63 103)(64 104)(65 105)(66 106)(67 107)(68 108)(69 109)(70 110)(71 111)(72 112)(73 113)(74 114)(75 115)(76 116)(77 117)(78 118)(79 119)(80 120)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)
(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)
(1 106 16 91)(2 105 17 120)(3 104 18 119)(4 103 19 118)(5 102 20 117)(6 101 21 116)(7 100 22 115)(8 99 23 114)(9 98 24 113)(10 97 25 112)(11 96 26 111)(12 95 27 110)(13 94 28 109)(14 93 29 108)(15 92 30 107)(31 80 46 65)(32 79 47 64)(33 78 48 63)(34 77 49 62)(35 76 50 61)(36 75 51 90)(37 74 52 89)(38 73 53 88)(39 72 54 87)(40 71 55 86)(41 70 56 85)(42 69 57 84)(43 68 58 83)(44 67 59 82)(45 66 60 81)

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

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

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

C2×Dic15 is a maximal subgroup of
Dic3×Dic5  D10⋊Dic3  D6⋊Dic5  C30.Q8  Dic155C4  C6.Dic10  C30.4Q8  C605C4  D303C4  C30.38D4  C2×D5×Dic3  C30.C23  C2×S3×Dic5  C2×C4×D15  D42D15  Q8⋊Dic15
C2×Dic15 is a maximal quotient of
C60.7C4  C605C4  C30.38D4

36 conjugacy classes

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

36 irreducible representations

dim1111222222222
type+++++-+-++-+
imageC1C2C2C4S3D5Dic3D6Dic5D10D15Dic15D30
kernelC2×Dic15Dic15C2×C30C30C2×C10C2×C6C10C10C6C6C22C2C2
# reps1214122142484

Matrix representation of C2×Dic15 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] >;

C2×Dic15 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 C2×Dic15 in TeX

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