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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, C3, C4, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×D4, C4○D4, C4○D4, C20, C20, C20, C2×C10, C2×C10, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, C5×S3, C30, C30, 2+ 1+4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, C2×D12, C4○D12, S3×D4, Q83S3, C3×C4○D4, C5×Dic3, C60, C60, S3×C10, S3×C10, C2×C30, D4×C10, C5×C4○D4, C5×C4○D4, D4○D12, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, D4×C15, Q8×C15, S3×C2×C10, C5×2+ 1+4, C10×D12, C5×C4○D12, C5×S3×D4, C5×Q83S3, C15×C4○D4, C5×D4○D12
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, C24, C2×C10, C22×S3, C5×S3, 2+ 1+4, C22×C10, S3×C23, S3×C10, C23×C10, D4○D12, S3×C2×C10, C5×2+ 1+4, S3×C22×C10, C5×D4○D12

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

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

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

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

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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