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## G = C5×D6⋊D4order 480 = 25·3·5

### Direct product of C5 and D6⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C5×D6⋊D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — S3×C2×C10 — S3×C22×C10 — C5×D6⋊D4
 Lower central C3 — C2×C6 — C5×D6⋊D4
 Upper central C1 — C2×C10 — C5×C22⋊C4

Generators and relations for C5×D6⋊D4
G = < a,b,c,d,e | a5=b6=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b3c, ece=bc, ede=d-1 >

Subgroups: 756 in 260 conjugacy classes, 74 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C10, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22⋊C4, C2×D4, C24, C20, C2×C10, C2×C10, C2×C10, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×S3, C22×S3, C22×C6, C5×S3, C30, C30, C30, C22≀C2, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, D6⋊C4, C3×C22⋊C4, C2×D12, C2×C3⋊D4, S3×C23, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C2×C30, C2×C30, C5×C22⋊C4, C5×C22⋊C4, D4×C10, C23×C10, D6⋊D4, C5×D12, C10×Dic3, C5×C3⋊D4, C2×C60, S3×C2×C10, S3×C2×C10, S3×C2×C10, C22×C30, C5×C22≀C2, C5×D6⋊C4, C15×C22⋊C4, C10×D12, C10×C3⋊D4, S3×C22×C10, C5×D6⋊D4
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, D12, C22×S3, C5×S3, C22≀C2, C5×D4, C22×C10, C2×D12, S3×D4, S3×C10, D4×C10, D6⋊D4, C5×D12, S3×C2×C10, C5×C22≀C2, C10×D12, C5×S3×D4, C5×D6⋊D4

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

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 3 4A 4B 4C 5A 5B 5C 5D 6A 6B 6C 6D 6E 10A ··· 10L 10M ··· 10T 10U ··· 10AJ 10AK 10AL 10AM 10AN 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 20I 20J 20K 20L 30A ··· 30L 30M ··· 30T 60A ··· 60P order 1 2 2 2 2 2 2 2 2 2 2 3 4 4 4 5 5 5 5 6 6 6 6 6 10 ··· 10 10 ··· 10 10 ··· 10 10 10 10 10 12 12 12 12 15 15 15 15 20 ··· 20 20 20 20 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 6 6 6 6 12 2 4 4 12 1 1 1 1 2 2 2 4 4 1 ··· 1 2 ··· 2 6 ··· 6 12 12 12 12 4 4 4 4 2 2 2 2 4 ··· 4 12 12 12 12 2 ··· 2 4 ··· 4 4 ··· 4

120 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 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 S3 D4 D4 D6 D6 D12 C5×S3 C5×D4 C5×D4 S3×C10 S3×C10 C5×D12 S3×D4 C5×S3×D4 kernel C5×D6⋊D4 C5×D6⋊C4 C15×C22⋊C4 C10×D12 C10×C3⋊D4 S3×C22×C10 D6⋊D4 D6⋊C4 C3×C22⋊C4 C2×D12 C2×C3⋊D4 S3×C23 C5×C22⋊C4 S3×C10 C2×C30 C2×C20 C22×C10 C2×C10 C22⋊C4 D6 C2×C6 C2×C4 C23 C22 C10 C2 # reps 1 2 1 2 1 1 4 8 4 8 4 4 1 4 2 2 1 4 4 16 8 8 4 16 2 8

Matrix representation of C5×D6⋊D4 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 58 0 0 0 0 58
,
 60 0 0 0 0 60 0 0 0 0 1 60 0 0 1 0
,
 60 0 0 0 0 1 0 0 0 0 1 0 0 0 1 60
,
 0 1 0 0 1 0 0 0 0 0 23 15 0 0 46 38
,
 0 1 0 0 1 0 0 0 0 0 46 38 0 0 23 15
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,58,0,0,0,0,58],[60,0,0,0,0,60,0,0,0,0,1,1,0,0,60,0],[60,0,0,0,0,1,0,0,0,0,1,1,0,0,0,60],[0,1,0,0,1,0,0,0,0,0,23,46,0,0,15,38],[0,1,0,0,1,0,0,0,0,0,46,23,0,0,38,15] >;

C5×D6⋊D4 in GAP, Magma, Sage, TeX

C_5\times D_6\rtimes D_4
% in TeX

G:=Group("C5xD6:D4");
// GroupNames label

G:=SmallGroup(480,761);
// by ID

G=gap.SmallGroup(480,761);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,926,891,226,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^6=c^2=d^4=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^3*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations

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