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G = C2×Dic15order 120 = 23·3·5

Direct product of C2 and Dic15

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

 Derived series C1 — C15 — C2×Dic15
 Chief series C1 — C5 — C15 — C30 — Dic15 — C2×Dic15
 Lower central C15 — C2×Dic15
 Upper central C1 — C22

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

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 2A 2B 2C 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 10A ··· 10F 15A 15B 15C 15D 30A ··· 30L order 1 2 2 2 3 4 4 4 4 5 5 6 6 6 10 ··· 10 15 15 15 15 30 ··· 30 size 1 1 1 1 2 15 15 15 15 2 2 2 2 2 2 ··· 2 2 2 2 2 2 ··· 2

36 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + - + - + + - + image C1 C2 C2 C4 S3 D5 Dic3 D6 Dic5 D10 D15 Dic15 D30 kernel C2×Dic15 Dic15 C2×C30 C30 C2×C10 C2×C6 C10 C10 C6 C6 C22 C2 C2 # reps 1 2 1 4 1 2 2 1 4 2 4 8 4

Matrix representation of C2×Dic15 in GL3(𝔽61) generated by

 60 0 0 0 1 0 0 0 1
,
 1 0 0 0 8 25 0 13 56
,
 1 0 0 0 38 3 0 47 23
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

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