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## G = C5×D12⋊6C22order 480 = 25·3·5

### Direct product of C5 and D12⋊6C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×D12⋊6C22
 Chief series C1 — C3 — C6 — C12 — C60 — C5×D12 — C5×C4○D12 — C5×D12⋊6C22
 Lower central C3 — C6 — C12 — C5×D12⋊6C22
 Upper central C1 — C10 — C2×C20 — D4×C10

Generators and relations for C5×D126C22
G = < a,b,c,d,e | a5=b12=c2=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b7, dcd=b6c, ece=b3c, de=ed >

Subgroups: 324 in 136 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, C20, C20, C2×C10, C2×C10, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×C6, C5×S3, C30, C30, C8⋊C22, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×C10, C4.Dic3, D4⋊S3, D4.S3, C4○D12, C6×D4, C5×Dic3, C60, S3×C10, C2×C30, C2×C30, C5×M4(2), C5×D8, C5×SD16, D4×C10, C5×C4○D4, D126C22, C5×C3⋊C8, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, D4×C15, D4×C15, C22×C30, C5×C8⋊C22, C5×C4.Dic3, C5×D4⋊S3, C5×D4.S3, C5×C4○D12, D4×C30, C5×D126C22
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, C3⋊D4, C22×S3, C5×S3, C8⋊C22, C5×D4, C22×C10, C2×C3⋊D4, S3×C10, D4×C10, D126C22, C5×C3⋊D4, S3×C2×C10, C5×C8⋊C22, C10×C3⋊D4, C5×D126C22

Smallest permutation representation of C5×D126C22
On 120 points
Generators in S120
(1 56 45 35 18)(2 57 46 36 19)(3 58 47 25 20)(4 59 48 26 21)(5 60 37 27 22)(6 49 38 28 23)(7 50 39 29 24)(8 51 40 30 13)(9 52 41 31 14)(10 53 42 32 15)(11 54 43 33 16)(12 55 44 34 17)(61 118 106 94 73)(62 119 107 95 74)(63 120 108 96 75)(64 109 97 85 76)(65 110 98 86 77)(66 111 99 87 78)(67 112 100 88 79)(68 113 101 89 80)(69 114 102 90 81)(70 115 103 91 82)(71 116 104 92 83)(72 117 105 93 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 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 72)(11 71)(12 70)(13 74)(14 73)(15 84)(16 83)(17 82)(18 81)(19 80)(20 79)(21 78)(22 77)(23 76)(24 75)(25 88)(26 87)(27 86)(28 85)(29 96)(30 95)(31 94)(32 93)(33 92)(34 91)(35 90)(36 89)(37 98)(38 97)(39 108)(40 107)(41 106)(42 105)(43 104)(44 103)(45 102)(46 101)(47 100)(48 99)(49 109)(50 120)(51 119)(52 118)(53 117)(54 116)(55 115)(56 114)(57 113)(58 112)(59 111)(60 110)
(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(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 4)(2 11)(3 6)(5 8)(7 10)(9 12)(13 22)(14 17)(15 24)(16 19)(18 21)(20 23)(25 28)(26 35)(27 30)(29 32)(31 34)(33 36)(37 40)(38 47)(39 42)(41 44)(43 46)(45 48)(49 58)(50 53)(51 60)(52 55)(54 57)(56 59)(61 67)(63 69)(65 71)(73 79)(75 81)(77 83)(86 92)(88 94)(90 96)(98 104)(100 106)(102 108)(110 116)(112 118)(114 120)

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

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

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

105 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 5A 5B 5C 5D 6A 6B 6C 6D 6E 6F 6G 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10P 10Q 10R 10S 10T 12A 12B 15A 15B 15C 15D 20A ··· 20H 20I 20J 20K 20L 30A ··· 30L 30M ··· 30AB 40A ··· 40H 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 5 5 5 5 6 6 6 6 6 6 6 8 8 10 10 10 10 10 10 10 10 10 ··· 10 10 10 10 10 12 12 15 15 15 15 20 ··· 20 20 20 20 20 30 ··· 30 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 2 4 4 12 2 2 2 12 1 1 1 1 2 2 2 4 4 4 4 12 12 1 1 1 1 2 2 2 2 4 ··· 4 12 12 12 12 4 4 2 2 2 2 2 ··· 2 12 12 12 12 2 ··· 2 4 ··· 4 12 ··· 12 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 4 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 S3 D4 D4 D6 D6 C3⋊D4 C3⋊D4 C5×S3 C5×D4 C5×D4 S3×C10 S3×C10 C5×C3⋊D4 C5×C3⋊D4 C8⋊C22 D12⋊6C22 C5×C8⋊C22 C5×D12⋊6C22 kernel C5×D12⋊6C22 C5×C4.Dic3 C5×D4⋊S3 C5×D4.S3 C5×C4○D12 D4×C30 D12⋊6C22 C4.Dic3 D4⋊S3 D4.S3 C4○D12 C6×D4 D4×C10 C60 C2×C30 C2×C20 C5×D4 C20 C2×C10 C2×D4 C12 C2×C6 C2×C4 D4 C4 C22 C15 C5 C3 C1 # reps 1 1 2 2 1 1 4 4 8 8 4 4 1 1 1 1 2 2 2 4 4 4 4 8 8 8 1 2 4 8

Matrix representation of C5×D126C22 in GL4(𝔽241) generated by

 91 0 0 0 0 91 0 0 0 0 91 0 0 0 0 91
,
 0 225 0 0 16 0 0 0 0 0 0 226 0 0 15 0
,
 0 0 0 226 0 0 15 0 0 225 0 0 16 0 0 0
,
 1 0 0 0 0 1 0 0 0 0 240 0 0 0 0 240
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 240
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,91,0,0,0,0,91],[0,16,0,0,225,0,0,0,0,0,0,15,0,0,226,0],[0,0,0,16,0,0,225,0,0,15,0,0,226,0,0,0],[1,0,0,0,0,1,0,0,0,0,240,0,0,0,0,240],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,240] >;

C5×D126C22 in GAP, Magma, Sage, TeX

C_5\times D_{12}\rtimes_6C_2^2
% in TeX

G:=Group("C5xD12:6C2^2");
// GroupNames label

G:=SmallGroup(480,811);
// by ID

G=gap.SmallGroup(480,811);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,926,891,4204,1068,102,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^12=c^2=d^2=e^2=1,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,e*b*e=b^7,d*c*d=b^6*c,e*c*e=b^3*c,d*e=e*d>;
// generators/relations

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