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## G = C2×D4×D15order 480 = 25·3·5

### Direct product of C2, D4 and D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×D4×D15
 Chief series C1 — C5 — C15 — C30 — D30 — C22×D15 — C23×D15 — C2×D4×D15
 Lower central C15 — C30 — C2×D4×D15
 Upper central C1 — C22 — C2×D4

Generators and relations for C2×D4×D15
G = < a,b,c,d,e | a2=b4=c2=d15=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 3092 in 472 conjugacy classes, 135 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, D4, D4, C23, C23, D5, C10, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, D15, D15, C30, C30, C30, C22×D4, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, Dic15, C60, D30, D30, C2×C30, C2×C30, C2×C30, C2×C4×D5, C2×D20, D4×D5, C2×C5⋊D4, D4×C10, C23×D5, C2×S3×D4, C4×D15, D60, C2×Dic15, C157D4, C2×C60, D4×C15, C22×D15, C22×D15, C22×D15, C22×C30, C2×D4×D5, C2×C4×D15, C2×D60, D4×D15, C2×C157D4, D4×C30, C23×D15, C2×D4×D15
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C24, D10, C22×S3, D15, C22×D4, C22×D5, S3×D4, S3×C23, D30, D4×D5, C23×D5, C2×S3×D4, C22×D15, C2×D4×D5, D4×D15, C23×D15, C2×D4×D15

Smallest permutation representation of C2×D4×D15
On 120 points
Generators in S120
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 73)(13 74)(14 75)(15 61)(16 83)(17 84)(18 85)(19 86)(20 87)(21 88)(22 89)(23 90)(24 76)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 98)(32 99)(33 100)(34 101)(35 102)(36 103)(37 104)(38 105)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 97)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)
(1 95 28 110)(2 96 29 111)(3 97 30 112)(4 98 16 113)(5 99 17 114)(6 100 18 115)(7 101 19 116)(8 102 20 117)(9 103 21 118)(10 104 22 119)(11 105 23 120)(12 91 24 106)(13 92 25 107)(14 93 26 108)(15 94 27 109)(31 83 53 65)(32 84 54 66)(33 85 55 67)(34 86 56 68)(35 87 57 69)(36 88 58 70)(37 89 59 71)(38 90 60 72)(39 76 46 73)(40 77 47 74)(41 78 48 75)(42 79 49 61)(43 80 50 62)(44 81 51 63)(45 82 52 64)
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 73)(13 74)(14 75)(15 61)(16 83)(17 84)(18 85)(19 86)(20 87)(21 88)(22 89)(23 90)(24 76)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 113)(32 114)(33 115)(34 116)(35 117)(36 118)(37 119)(38 120)(39 106)(40 107)(41 108)(42 109)(43 110)(44 111)(45 112)(46 91)(47 92)(48 93)(49 94)(50 95)(51 96)(52 97)(53 98)(54 99)(55 100)(56 101)(57 102)(58 103)(59 104)(60 105)
(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 46)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(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)(91 113)(92 112)(93 111)(94 110)(95 109)(96 108)(97 107)(98 106)(99 120)(100 119)(101 118)(102 117)(103 116)(104 115)(105 114)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A ··· 10F 10G ··· 10N 12A 12B 15A 15B 15C 15D 20A 20B 20C 20D 30A ··· 30L 30M ··· 30AB 60A ··· 60H order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 5 5 6 6 6 6 6 6 6 10 ··· 10 10 ··· 10 12 12 15 15 15 15 20 20 20 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 2 15 15 15 15 30 30 30 30 2 2 2 30 30 2 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 4 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

90 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D4 D5 D6 D6 D6 D10 D10 D10 D15 D30 D30 D30 S3×D4 D4×D5 D4×D15 kernel C2×D4×D15 C2×C4×D15 C2×D60 D4×D15 C2×C15⋊7D4 D4×C30 C23×D15 D4×C10 D30 C6×D4 C2×C20 C5×D4 C22×C10 C2×C12 C3×D4 C22×C6 C2×D4 C2×C4 D4 C23 C10 C6 C2 # reps 1 1 1 8 2 1 2 1 4 2 1 4 2 2 8 4 4 4 16 8 2 4 8

Matrix representation of C2×D4×D15 in GL4(𝔽61) generated by

 60 0 0 0 0 60 0 0 0 0 60 0 0 0 0 60
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 60 0
,
 60 0 0 0 0 60 0 0 0 0 60 0 0 0 0 1
,
 53 23 0 0 38 5 0 0 0 0 1 0 0 0 0 1
,
 53 23 0 0 45 8 0 0 0 0 60 0 0 0 0 60
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,0],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,1],[53,38,0,0,23,5,0,0,0,0,1,0,0,0,0,1],[53,45,0,0,23,8,0,0,0,0,60,0,0,0,0,60] >;

C2×D4×D15 in GAP, Magma, Sage, TeX

C_2\times D_4\times D_{15}
% in TeX

G:=Group("C2xD4xD15");
// GroupNames label

G:=SmallGroup(480,1169);
// by ID

G=gap.SmallGroup(480,1169);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,185,2693,18822]);
// Polycyclic

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

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