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G = C5×D4○D12order 480 = 25·3·5

Direct product of C5 and D4○D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×D4○D12
 Chief series C1 — C3 — C6 — C30 — S3×C10 — S3×C2×C10 — C5×S3×D4 — C5×D4○D12
 Lower central C3 — C6 — C5×D4○D12
 Upper central C1 — C10 — C5×C4○D4

Generators and relations for C5×D4○D12
G = < a,b,c,d,e | a5=b4=c2=e2=1, d6=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d5 >

Subgroups: 788 in 332 conjugacy classes, 170 normal (24 characteristic)
C1, C2, C2 [×9], C3, C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], C5, S3 [×6], C6, C6 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×3], D4 [×15], Q8, Q8, C23 [×6], C10, C10 [×9], Dic3 [×2], C12, C12 [×3], D6 [×6], D6 [×6], C2×C6 [×3], C15, C2×D4 [×9], C4○D4, C4○D4 [×5], C20, C20 [×3], C20 [×2], C2×C10 [×3], C2×C10 [×12], Dic6, C4×S3 [×6], D12 [×9], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×6], C5×S3 [×6], C30, C30 [×3], 2+ 1+4, C2×C20 [×3], C2×C20 [×6], C5×D4 [×3], C5×D4 [×15], C5×Q8, C5×Q8, C22×C10 [×6], C2×D12 [×3], C4○D12 [×3], S3×D4 [×6], Q83S3 [×2], C3×C4○D4, C5×Dic3 [×2], C60, C60 [×3], S3×C10 [×6], S3×C10 [×6], C2×C30 [×3], D4×C10 [×9], C5×C4○D4, C5×C4○D4 [×5], D4○D12, C5×Dic6, S3×C20 [×6], C5×D12 [×9], C5×C3⋊D4 [×6], C2×C60 [×3], D4×C15 [×3], Q8×C15, S3×C2×C10 [×6], C5×2+ 1+4, C10×D12 [×3], C5×C4○D12 [×3], C5×S3×D4 [×6], C5×Q83S3 [×2], C15×C4○D4, C5×D4○D12
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, D4○D12, S3×C2×C10 [×7], C5×2+ 1+4, S3×C22×C10, C5×D4○D12

Smallest permutation representation of C5×D4○D12
On 120 points
Generators in S120
(1 64 43 59 16)(2 65 44 60 17)(3 66 45 49 18)(4 67 46 50 19)(5 68 47 51 20)(6 69 48 52 21)(7 70 37 53 22)(8 71 38 54 23)(9 72 39 55 24)(10 61 40 56 13)(11 62 41 57 14)(12 63 42 58 15)(25 100 109 81 90)(26 101 110 82 91)(27 102 111 83 92)(28 103 112 84 93)(29 104 113 73 94)(30 105 114 74 95)(31 106 115 75 96)(32 107 116 76 85)(33 108 117 77 86)(34 97 118 78 87)(35 98 119 79 88)(36 99 120 80 89)
(1 88 7 94)(2 89 8 95)(3 90 9 96)(4 91 10 85)(5 92 11 86)(6 93 12 87)(13 76 19 82)(14 77 20 83)(15 78 21 84)(16 79 22 73)(17 80 23 74)(18 81 24 75)(25 72 31 66)(26 61 32 67)(27 62 33 68)(28 63 34 69)(29 64 35 70)(30 65 36 71)(37 104 43 98)(38 105 44 99)(39 106 45 100)(40 107 46 101)(41 108 47 102)(42 97 48 103)(49 109 55 115)(50 110 56 116)(51 111 57 117)(52 112 58 118)(53 113 59 119)(54 114 60 120)
(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)(49 56)(50 55)(51 54)(52 53)(57 60)(58 59)(61 66)(62 65)(63 64)(67 72)(68 71)(69 70)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(85 90)(86 89)(87 88)(91 96)(92 95)(93 94)(97 98)(99 108)(100 107)(101 106)(102 105)(103 104)(109 116)(110 115)(111 114)(112 113)(117 120)(118 119)

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

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

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

135 conjugacy classes

 class 1 2A 2B 2C 2D 2E ··· 2J 3 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 6A 6B 6C 6D 10A 10B 10C 10D 10E ··· 10P 10Q ··· 10AN 12A 12B 12C 12D 12E 15A 15B 15C 15D 20A ··· 20P 20Q ··· 20X 30A 30B 30C 30D 30E ··· 30P 60A ··· 60H 60I ··· 60T order 1 2 2 2 2 2 ··· 2 3 4 4 4 4 4 4 5 5 5 5 6 6 6 6 10 10 10 10 10 ··· 10 10 ··· 10 12 12 12 12 12 15 15 15 15 20 ··· 20 20 ··· 20 30 30 30 30 30 ··· 30 60 ··· 60 60 ··· 60 size 1 1 2 2 2 6 ··· 6 2 2 2 2 2 6 6 1 1 1 1 2 4 4 4 1 1 1 1 2 ··· 2 6 ··· 6 2 2 4 4 4 2 2 2 2 2 ··· 2 6 ··· 6 2 2 2 2 4 ··· 4 2 ··· 2 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○D12 C5×2+ 1+4 C5×D4○D12 kernel C5×D4○D12 C10×D12 C5×C4○D12 C5×S3×D4 C5×Q8⋊3S3 C15×C4○D4 D4○D12 C2×D12 C4○D12 S3×D4 Q8⋊3S3 C3×C4○D4 C5×C4○D4 C2×C20 C5×D4 C5×Q8 C4○D4 C2×C4 D4 Q8 C15 C5 C3 C1 # reps 1 3 3 6 2 1 4 12 12 24 8 4 1 3 3 1 4 12 12 4 1 2 4 8

Matrix representation of C5×D4○D12 in GL4(𝔽61) generated by

 34 0 0 0 0 34 0 0 0 0 34 0 0 0 0 34
,
 1 0 2 0 0 1 0 2 60 0 60 0 0 60 0 60
,
 1 0 0 0 0 1 0 0 60 0 60 0 0 60 0 60
,
 46 23 0 0 38 23 0 0 0 0 46 23 0 0 38 23
,
 46 23 0 0 38 15 0 0 0 0 46 23 0 0 38 15
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,34,0,0,0,0,34],[1,0,60,0,0,1,0,60,2,0,60,0,0,2,0,60],[1,0,60,0,0,1,0,60,0,0,60,0,0,0,0,60],[46,38,0,0,23,23,0,0,0,0,46,38,0,0,23,23],[46,38,0,0,23,15,0,0,0,0,46,38,0,0,23,15] >;

C5×D4○D12 in GAP, Magma, Sage, TeX

C_5\times D_4\circ D_{12}
% in TeX

G:=Group("C5xD4oD12");
// GroupNames label

G:=SmallGroup(480,1161);
// by ID

G=gap.SmallGroup(480,1161);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,891,2467,304,15686]);
// Polycyclic

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

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