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## G = C5×D4⋊D6order 480 = 25·3·5

### Direct product of C5 and D4⋊D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×D4⋊D6
 Chief series C1 — C3 — C6 — C12 — C60 — C5×D12 — C10×D12 — C5×D4⋊D6
 Lower central C3 — C6 — C12 — C5×D4⋊D6
 Upper central C1 — C10 — C2×C20 — C5×C4○D4

Generators and relations for C5×D4⋊D6
G = < a,b,c,d,e | a5=b4=c2=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b2c, ece=b-1c, ede=d-1 >

Subgroups: 372 in 136 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4, D4, D4 [×4], Q8, C23, C10, C10 [×4], C12 [×2], C12, D6 [×4], C2×C6, C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C20 [×2], C20, C2×C10, C2×C10 [×5], C3⋊C8 [×2], D12 [×2], D12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C5×S3 [×2], C30, C30 [×2], C8⋊C22, C40 [×2], C2×C20, C2×C20, C5×D4, C5×D4 [×4], C5×Q8, C22×C10, C4.Dic3, D4⋊S3 [×2], Q82S3 [×2], C2×D12, C3×C4○D4, C60 [×2], C60, S3×C10 [×4], C2×C30, C2×C30, C5×M4(2), C5×D8 [×2], C5×SD16 [×2], D4×C10, C5×C4○D4, D4⋊D6, C5×C3⋊C8 [×2], C5×D12 [×2], C5×D12, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, S3×C2×C10, C5×C8⋊C22, C5×C4.Dic3, C5×D4⋊S3 [×2], C5×Q82S3 [×2], C10×D12, C15×C4○D4, C5×D4⋊D6
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C2×C10 [×7], C3⋊D4 [×2], C22×S3, C5×S3, C8⋊C22, C5×D4 [×2], C22×C10, C2×C3⋊D4, S3×C10 [×3], D4×C10, D4⋊D6, C5×C3⋊D4 [×2], S3×C2×C10, C5×C8⋊C22, C10×C3⋊D4, C5×D4⋊D6

Smallest permutation representation of C5×D4⋊D6
On 120 points
Generators in S120
(1 49 37 25 14)(2 50 38 26 15)(3 51 39 27 13)(4 54 42 28 16)(5 52 40 29 17)(6 53 41 30 18)(7 57 45 31 19)(8 55 43 32 20)(9 56 44 33 21)(10 60 48 34 22)(11 58 46 35 23)(12 59 47 36 24)(61 113 101 89 78)(62 114 102 90 73)(63 109 97 85 74)(64 110 98 86 75)(65 111 99 87 76)(66 112 100 88 77)(67 115 103 91 79)(68 116 104 92 80)(69 117 105 93 81)(70 118 106 94 82)(71 119 107 95 83)(72 120 108 96 84)
(1 11 5 8)(2 12 6 9)(3 10 4 7)(13 22 16 19)(14 23 17 20)(15 24 18 21)(25 35 29 32)(26 36 30 33)(27 34 28 31)(37 46 40 43)(38 47 41 44)(39 48 42 45)(49 58 52 55)(50 59 53 56)(51 60 54 57)(61 71 64 68)(62 72 65 69)(63 67 66 70)(73 84 76 81)(74 79 77 82)(75 80 78 83)(85 91 88 94)(86 92 89 95)(87 93 90 96)(97 103 100 106)(98 104 101 107)(99 105 102 108)(109 115 112 118)(110 116 113 119)(111 117 114 120)
(1 68)(2 72)(3 70)(4 67)(5 71)(6 69)(7 63)(8 61)(9 65)(10 66)(11 64)(12 62)(13 82)(14 80)(15 84)(16 79)(17 83)(18 81)(19 74)(20 78)(21 76)(22 77)(23 75)(24 73)(25 92)(26 96)(27 94)(28 91)(29 95)(30 93)(31 85)(32 89)(33 87)(34 88)(35 86)(36 90)(37 104)(38 108)(39 106)(40 107)(41 105)(42 103)(43 101)(44 99)(45 97)(46 98)(47 102)(48 100)(49 116)(50 120)(51 118)(52 119)(53 117)(54 115)(55 113)(56 111)(57 109)(58 110)(59 114)(60 112)
(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 11)(8 10)(9 12)(13 14)(16 17)(19 23)(20 22)(21 24)(25 27)(28 29)(31 35)(32 34)(33 36)(37 39)(40 42)(43 48)(44 47)(45 46)(49 51)(52 54)(55 60)(56 59)(57 58)(61 70)(62 69)(63 68)(64 67)(65 72)(66 71)(73 81)(74 80)(75 79)(76 84)(77 83)(78 82)(85 92)(86 91)(87 96)(88 95)(89 94)(90 93)(97 104)(98 103)(99 108)(100 107)(101 106)(102 105)(109 116)(110 115)(111 120)(112 119)(113 118)(114 117)

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

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

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

105 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 5A 5B 5C 5D 6A 6B 6C 6D 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 10M ··· 10T 12A 12B 12C 12D 12E 15A 15B 15C 15D 20A ··· 20H 20I 20J 20K 20L 30A 30B 30C 30D 30E ··· 30P 40A ··· 40H 60A ··· 60H 60I ··· 60T order 1 2 2 2 2 2 3 4 4 4 5 5 5 5 6 6 6 6 8 8 10 10 10 10 10 10 10 10 10 10 10 10 10 ··· 10 12 12 12 12 12 15 15 15 15 20 ··· 20 20 20 20 20 30 30 30 30 30 ··· 30 40 ··· 40 60 ··· 60 60 ··· 60 size 1 1 2 4 12 12 2 2 2 4 1 1 1 1 2 4 4 4 12 12 1 1 1 1 2 2 2 2 4 4 4 4 12 ··· 12 2 2 4 4 4 2 2 2 2 2 ··· 2 4 4 4 4 2 2 2 2 4 ··· 4 12 ··· 12 2 ··· 2 4 ··· 4

105 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 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 D4 D4 D6 D6 D6 C3⋊D4 C3⋊D4 C5×S3 C5×D4 C5×D4 S3×C10 S3×C10 S3×C10 C5×C3⋊D4 C5×C3⋊D4 C8⋊C22 D4⋊D6 C5×C8⋊C22 C5×D4⋊D6 kernel C5×D4⋊D6 C5×C4.Dic3 C5×D4⋊S3 C5×Q8⋊2S3 C10×D12 C15×C4○D4 D4⋊D6 C4.Dic3 D4⋊S3 Q8⋊2S3 C2×D12 C3×C4○D4 C5×C4○D4 C60 C2×C30 C2×C20 C5×D4 C5×Q8 C20 C2×C10 C4○D4 C12 C2×C6 C2×C4 D4 Q8 C4 C22 C15 C5 C3 C1 # reps 1 1 2 2 1 1 4 4 8 8 4 4 1 1 1 1 1 1 2 2 4 4 4 4 4 4 8 8 1 2 4 8

Matrix representation of C5×D4⋊D6 in GL4(𝔽241) generated by

 87 0 0 0 0 87 0 0 0 0 87 0 0 0 0 87
,
 99 198 0 0 43 142 0 0 0 0 142 43 0 0 198 99
,
 0 0 142 43 0 0 198 99 99 198 0 0 43 142 0 0
,
 0 1 0 0 240 240 0 0 0 0 0 240 0 0 1 1
,
 240 240 0 0 0 1 0 0 0 0 142 99 0 0 198 99
G:=sub<GL(4,GF(241))| [87,0,0,0,0,87,0,0,0,0,87,0,0,0,0,87],[99,43,0,0,198,142,0,0,0,0,142,198,0,0,43,99],[0,0,99,43,0,0,198,142,142,198,0,0,43,99,0,0],[0,240,0,0,1,240,0,0,0,0,0,1,0,0,240,1],[240,0,0,0,240,1,0,0,0,0,142,198,0,0,99,99] >;

C5×D4⋊D6 in GAP, Magma, Sage, TeX

C_5\times D_4\rtimes D_6
% in TeX

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

G:=SmallGroup(480,828);
// by ID

G=gap.SmallGroup(480,828);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,926,891,4204,1068,102,15686]);
// 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=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations

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