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## G = C2×D30.C2order 240 = 24·3·5

### Direct product of C2 and D30.C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C2×D30.C2
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — D30.C2 — C2×D30.C2
 Lower central C15 — C2×D30.C2
 Upper central C1 — C22

Generators and relations for C2×D30.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, C2×C4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C22×C4, Dic5, C20, D10, C2×C10, C4×S3, C2×Dic3, C2×C12, C22×S3, D15, C30, C30, C4×D5, C2×Dic5, C2×C20, C22×D5, S3×C2×C4, C5×Dic3, C3×Dic5, D30, C2×C30, C2×C4×D5, D30.C2, C6×Dic5, C10×Dic3, C22×D15, C2×D30.C2
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, D10, C4×S3, C22×S3, C4×D5, C22×D5, S3×C2×C4, S3×D5, C2×C4×D5, D30.C2, C2×S3×D5, C2×D30.C2

Smallest permutation representation of C2×D30.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)]])

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A 6B 6C 10A ··· 10F 12A 12B 12C 12D 15A 15B 20A ··· 20H 30A ··· 30F order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 5 5 6 6 6 10 ··· 10 12 12 12 12 15 15 20 ··· 20 30 ··· 30 size 1 1 1 1 15 15 15 15 2 3 3 3 3 5 5 5 5 2 2 2 2 2 2 ··· 2 10 10 10 10 4 4 6 ··· 6 4 ··· 4

48 irreducible representations

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

Matrix representation of C2×D30.C2 in GL5(𝔽61)

 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 42 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 60 0
,
 1 0 0 0 0 0 60 42 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 60 60
,
 60 0 0 0 0 0 15 15 0 0 0 50 46 0 0 0 0 0 50 0 0 0 0 0 50

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] >;

C2×D30.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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