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## G = C5×D4⋊6D6order 480 = 25·3·5

### Direct product of C5 and D4⋊6D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×D4⋊6D6
 Chief series C1 — C3 — C6 — C30 — S3×C10 — S3×C2×C10 — C5×S3×D4 — C5×D4⋊6D6
 Lower central C3 — C6 — C5×D4⋊6D6
 Upper central C1 — C10 — D4×C10

Generators and relations for C5×D46D6
G = < a,b,c,d,e | a5=b4=c2=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=b-1, be=eb, dcd-1=ece=b2c, ede=d-1 >

Subgroups: 740 in 332 conjugacy classes, 170 normal (22 characteristic)
C1, C2, C2 [×9], C3, C4 [×2], C4 [×4], C22, C22 [×4], C22 [×10], C5, S3 [×4], C6, C6 [×5], C2×C4, C2×C4 [×8], D4 [×4], D4 [×14], Q8 [×2], C23 [×2], C23 [×4], C10, C10 [×9], Dic3 [×4], C12 [×2], D6 [×4], D6 [×4], C2×C6, C2×C6 [×4], C2×C6 [×2], C15, C2×D4, C2×D4 [×8], C4○D4 [×6], C20 [×2], C20 [×4], C2×C10, C2×C10 [×4], C2×C10 [×10], Dic6 [×2], C4×S3 [×4], D12 [×2], C2×Dic3 [×4], C3⋊D4 [×12], C2×C12, C3×D4 [×4], C22×S3 [×4], C22×C6 [×2], C5×S3 [×4], C30, C30 [×5], 2+ 1+4, C2×C20, C2×C20 [×8], C5×D4 [×4], C5×D4 [×14], C5×Q8 [×2], C22×C10 [×2], C22×C10 [×4], C4○D12 [×2], S3×D4 [×4], D42S3 [×4], C2×C3⋊D4 [×4], C6×D4, C5×Dic3 [×4], C60 [×2], S3×C10 [×4], S3×C10 [×4], C2×C30, C2×C30 [×4], C2×C30 [×2], D4×C10, D4×C10 [×8], C5×C4○D4 [×6], D46D6, C5×Dic6 [×2], S3×C20 [×4], C5×D12 [×2], C10×Dic3 [×4], C5×C3⋊D4 [×12], C2×C60, D4×C15 [×4], S3×C2×C10 [×4], C22×C30 [×2], C5×2+ 1+4, C5×C4○D12 [×2], C5×S3×D4 [×4], C5×D42S3 [×4], C10×C3⋊D4 [×4], D4×C30, C5×D46D6
Quotients: C1, C2 [×15], C22 [×35], C5, S3, C23 [×15], C10 [×15], D6 [×7], C24, C2×C10 [×35], C22×S3 [×7], C5×S3, 2+ 1+4, C22×C10 [×15], S3×C23, S3×C10 [×7], C23×C10, D46D6, S3×C2×C10 [×7], C5×2+ 1+4, S3×C22×C10, C5×D46D6

Smallest permutation representation of C5×D46D6
On 120 points
Generators in S120
(1 26 20 14 8)(2 27 21 15 9)(3 25 19 13 7)(4 30 22 16 10)(5 28 23 17 11)(6 29 24 18 12)(31 56 50 44 38)(32 57 51 45 39)(33 55 49 43 37)(34 58 52 46 40)(35 59 53 47 41)(36 60 54 48 42)(61 89 83 77 71)(62 90 84 78 72)(63 85 79 73 67)(64 86 80 74 68)(65 87 81 75 69)(66 88 82 76 70)(91 115 109 103 97)(92 116 110 104 98)(93 117 111 105 99)(94 118 112 106 100)(95 119 113 107 101)(96 120 114 108 102)
(1 92 5 95)(2 96 6 93)(3 94 4 91)(7 100 10 97)(8 98 11 101)(9 102 12 99)(13 106 16 103)(14 104 17 107)(15 108 18 105)(19 112 22 109)(20 110 23 113)(21 114 24 111)(25 118 30 115)(26 116 28 119)(27 120 29 117)(31 62 35 65)(32 66 36 63)(33 64 34 61)(37 68 40 71)(38 72 41 69)(39 70 42 67)(43 74 46 77)(44 78 47 75)(45 76 48 73)(49 80 52 83)(50 84 53 81)(51 82 54 79)(55 86 58 89)(56 90 59 87)(57 88 60 85)
(1 61)(2 65)(3 63)(4 66)(5 64)(6 62)(7 67)(8 71)(9 69)(10 70)(11 68)(12 72)(13 73)(14 77)(15 75)(16 76)(17 74)(18 78)(19 79)(20 83)(21 81)(22 82)(23 80)(24 84)(25 85)(26 89)(27 87)(28 86)(29 90)(30 88)(31 93)(32 91)(33 95)(34 92)(35 96)(36 94)(37 101)(38 99)(39 97)(40 98)(41 102)(42 100)(43 107)(44 105)(45 103)(46 104)(47 108)(48 106)(49 113)(50 111)(51 109)(52 110)(53 114)(54 112)(55 119)(56 117)(57 115)(58 116)(59 120)(60 118)
(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 3)(4 5)(7 8)(10 11)(13 14)(16 17)(19 20)(22 23)(25 26)(28 30)(31 35)(32 34)(33 36)(37 42)(38 41)(39 40)(43 48)(44 47)(45 46)(49 54)(50 53)(51 52)(55 60)(56 59)(57 58)(61 66)(62 65)(63 64)(67 68)(69 72)(70 71)(73 74)(75 78)(76 77)(79 80)(81 84)(82 83)(85 86)(87 90)(88 89)(91 95)(92 94)(97 101)(98 100)(103 107)(104 106)(109 113)(110 112)(115 119)(116 118)

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

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

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

135 conjugacy classes

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

135 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 S3 D6 D6 D6 C5×S3 S3×C10 S3×C10 S3×C10 2+ 1+4 D4⋊6D6 C5×2+ 1+4 C5×D4⋊6D6 kernel C5×D4⋊6D6 C5×C4○D12 C5×S3×D4 C5×D4⋊2S3 C10×C3⋊D4 D4×C30 D4⋊6D6 C4○D12 S3×D4 D4⋊2S3 C2×C3⋊D4 C6×D4 D4×C10 C2×C20 C5×D4 C22×C10 C2×D4 C2×C4 D4 C23 C15 C5 C3 C1 # reps 1 2 4 4 4 1 4 8 16 16 16 4 1 1 4 2 4 4 16 8 1 2 4 8

Matrix representation of C5×D46D6 in GL4(𝔽61) generated by

 58 0 0 0 0 58 0 0 0 0 58 0 0 0 0 58
,
 52 43 18 36 18 9 25 43 52 43 9 18 18 9 43 52
,
 1 0 59 0 0 1 0 59 0 0 60 0 0 0 0 60
,
 0 1 0 0 60 60 0 0 0 1 0 60 60 60 1 1
,
 0 1 0 0 1 0 0 0 0 1 0 60 1 0 60 0
G:=sub<GL(4,GF(61))| [58,0,0,0,0,58,0,0,0,0,58,0,0,0,0,58],[52,18,52,18,43,9,43,9,18,25,9,43,36,43,18,52],[1,0,0,0,0,1,0,0,59,0,60,0,0,59,0,60],[0,60,0,60,1,60,1,60,0,0,0,1,0,0,60,1],[0,1,0,1,1,0,1,0,0,0,0,60,0,0,60,0] >;

C5×D46D6 in GAP, Magma, Sage, TeX

C_5\times D_4\rtimes_6D_6
% in TeX

G:=Group("C5xD4:6D6");
// GroupNames label

G:=SmallGroup(480,1156);
// by ID

G=gap.SmallGroup(480,1156);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,891,2467,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=c^2=d^6=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations

׿
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