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## G = C2×C4×D15order 240 = 24·3·5

### Direct product of C2×C4 and D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C2×C4×D15
 Chief series C1 — C5 — C15 — C30 — D30 — C22×D15 — C2×C4×D15
 Lower central C15 — C2×C4×D15
 Upper central C1 — C2×C4

Generators and relations for C2×C4×D15
G = < a,b,c,d | a2=b4=c15=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 488 in 108 conjugacy classes, 51 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, 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, Dic15, C60, D30, C2×C30, C2×C4×D5, C4×D15, C2×Dic15, C2×C60, C22×D15, C2×C4×D15
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, D10, C4×S3, C22×S3, D15, C4×D5, C22×D5, S3×C2×C4, D30, C2×C4×D5, C4×D15, C22×D15, C2×C4×D15

Smallest permutation representation of C2×C4×D15
On 120 points
Generators in S120
(1 73)(2 74)(3 75)(4 61)(5 62)(6 63)(7 64)(8 65)(9 66)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 87)(17 88)(18 89)(19 90)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 102)(32 103)(33 104)(34 105)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 118)(47 119)(48 120)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 113)(57 114)(58 115)(59 116)(60 117)
(1 46 28 43)(2 47 29 44)(3 48 30 45)(4 49 16 31)(5 50 17 32)(6 51 18 33)(7 52 19 34)(8 53 20 35)(9 54 21 36)(10 55 22 37)(11 56 23 38)(12 57 24 39)(13 58 25 40)(14 59 26 41)(15 60 27 42)(61 106 87 102)(62 107 88 103)(63 108 89 104)(64 109 90 105)(65 110 76 91)(66 111 77 92)(67 112 78 93)(68 113 79 94)(69 114 80 95)(70 115 81 96)(71 116 82 97)(72 117 83 98)(73 118 84 99)(74 119 85 100)(75 120 86 101)
(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 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(13 30)(14 29)(15 28)(31 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)(41 47)(42 46)(43 60)(44 59)(45 58)(61 80)(62 79)(63 78)(64 77)(65 76)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(91 110)(92 109)(93 108)(94 107)(95 106)(96 120)(97 119)(98 118)(99 117)(100 116)(101 115)(102 114)(103 113)(104 112)(105 111)

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

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

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

72 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 15C 15D 20A ··· 20H 30A ··· 30L 60A ··· 60P 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 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 15 15 15 15 2 1 1 1 1 15 15 15 15 2 2 2 2 2 2 ··· 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D5 D6 D6 D10 D10 C4×S3 D15 C4×D5 D30 D30 C4×D15 kernel C2×C4×D15 C4×D15 C2×Dic15 C2×C60 C22×D15 D30 C2×C20 C2×C12 C20 C2×C10 C12 C2×C6 C10 C2×C4 C6 C4 C22 C2 # reps 1 4 1 1 1 8 1 2 2 1 4 2 4 4 8 8 4 16

Matrix representation of C2×C4×D15 in GL3(𝔽61) generated by

 60 0 0 0 60 0 0 0 60
,
 1 0 0 0 11 0 0 0 11
,
 1 0 0 0 37 14 0 47 31
,
 60 0 0 0 60 17 0 0 1
G:=sub<GL(3,GF(61))| [60,0,0,0,60,0,0,0,60],[1,0,0,0,11,0,0,0,11],[1,0,0,0,37,47,0,14,31],[60,0,0,0,60,0,0,17,1] >;

C2×C4×D15 in GAP, Magma, Sage, TeX

C_2\times C_4\times D_{15}
% in TeX

G:=Group("C2xC4xD15");
// GroupNames label

G:=SmallGroup(240,176);
// by ID

G=gap.SmallGroup(240,176);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,50,964,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^15=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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