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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 [×2], C2 [×12], C3, C4 [×2], C4 [×2], C22, C22 [×4], C22 [×34], C5, S3 [×8], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×5], D4 [×4], D4 [×12], C23 [×2], C23 [×19], D5 [×8], C10, C10 [×2], C10 [×4], Dic3 [×2], C12 [×2], D6 [×30], C2×C6, C2×C6 [×4], C2×C6 [×4], C15, C22×C4, C2×D4, C2×D4 [×11], C24 [×2], Dic5 [×2], C20 [×2], D10 [×30], C2×C10, C2×C10 [×4], C2×C10 [×4], C4×S3 [×4], D12 [×4], C2×Dic3, C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3 [×19], C22×C6 [×2], D15 [×4], D15 [×4], C30, C30 [×2], C30 [×4], C22×D4, C4×D5 [×4], D20 [×4], C2×Dic5, C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5 [×19], C22×C10 [×2], S3×C2×C4, C2×D12, S3×D4 [×8], C2×C3⋊D4 [×2], C6×D4, S3×C23 [×2], Dic15 [×2], C60 [×2], D30 [×10], D30 [×20], C2×C30, C2×C30 [×4], C2×C30 [×4], C2×C4×D5, C2×D20, D4×D5 [×8], C2×C5⋊D4 [×2], D4×C10, C23×D5 [×2], C2×S3×D4, C4×D15 [×4], D60 [×4], C2×Dic15, C157D4 [×8], C2×C60, D4×C15 [×4], C22×D15, C22×D15 [×10], C22×D15 [×8], C22×C30 [×2], C2×D4×D5, C2×C4×D15, C2×D60, D4×D15 [×8], C2×C157D4 [×2], D4×C30, C23×D15 [×2], C2×D4×D15
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D5, D6 [×7], C2×D4 [×6], C24, D10 [×7], C22×S3 [×7], D15, C22×D4, C22×D5 [×7], S3×D4 [×2], S3×C23, D30 [×7], D4×D5 [×2], C23×D5, C2×S3×D4, C22×D15 [×7], C2×D4×D5, D4×D15 [×2], C23×D15, C2×D4×D15

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

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

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

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

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

׿
×
𝔽