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## G = (C2×C10)⋊11D12order 480 = 25·3·5

### 2nd semidirect product of C2×C10 and D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — (C2×C10)⋊11D12
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — C2×C5⋊D12 — (C2×C10)⋊11D12
 Lower central C15 — C2×C30 — (C2×C10)⋊11D12
 Upper central C1 — C22 — C23

Generators and relations for (C2×C10)⋊11D12
G = < a,b,c,d | a2=b10=c12=d2=1, ab=ba, cac-1=dad=ab5, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 1324 in 260 conjugacy classes, 60 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×7], C3, C4 [×3], C22, C22 [×2], C22 [×21], C5, S3 [×5], C6, C6 [×2], C6 [×2], C2×C4 [×3], D4 [×6], C23, C23 [×9], D5, C10, C10 [×2], C10 [×6], Dic3, C12 [×2], D6 [×4], D6 [×15], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22⋊C4 [×3], C2×D4 [×3], C24, Dic5 [×3], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×18], D12 [×4], C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C22×S3 [×2], C22×S3 [×7], C22×C6, C5×S3 [×4], D15, C30, C30 [×2], C30 [×2], C22≀C2, C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×6], C22×D5, C22×C10, C22×C10 [×8], D6⋊C4 [×2], C3×C22⋊C4, C2×D12 [×2], C2×C3⋊D4, S3×C23, C3×Dic5 [×2], Dic15, S3×C10 [×4], S3×C10 [×12], D30 [×3], C2×C30, C2×C30 [×2], C2×C30 [×2], C23.D5, C23.D5 [×2], C2×C5⋊D4 [×3], C23×C10, D6⋊D4, C5⋊D12 [×4], C6×Dic5 [×2], C2×Dic15, C157D4 [×2], S3×C2×C10 [×2], S3×C2×C10 [×6], C22×D15, C22×C30, C242D5, D6⋊Dic5 [×2], C3×C23.D5, C2×C5⋊D12 [×2], C2×C157D4, S3×C22×C10, (C2×C10)⋊11D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D5, D6 [×3], C2×D4 [×3], D10 [×3], D12 [×2], C22×S3, C22≀C2, C5⋊D4 [×6], C22×D5, C2×D12, S3×D4 [×2], S3×D5, C2×C5⋊D4 [×3], D6⋊D4, C5⋊D12 [×2], C2×S3×D5, C242D5, C2×C5⋊D12, S3×C5⋊D4 [×2], (C2×C10)⋊11D12

Smallest permutation representation of (C2×C10)⋊11D12
On 120 points
Generators in S120
(1 7)(2 23)(3 9)(4 13)(5 11)(6 15)(8 17)(10 19)(12 21)(14 20)(16 22)(18 24)(25 70)(26 32)(27 72)(28 34)(29 62)(30 36)(31 64)(33 66)(35 68)(37 89)(38 44)(39 91)(40 46)(41 93)(42 48)(43 95)(45 85)(47 87)(49 55)(50 119)(51 57)(52 109)(53 59)(54 111)(56 113)(58 115)(60 117)(61 67)(63 69)(65 71)(73 79)(74 107)(75 81)(76 97)(77 83)(78 99)(80 101)(82 103)(84 105)(86 92)(88 94)(90 96)(98 104)(100 106)(102 108)(110 116)(112 118)(114 120)
(1 94 57 98 30 16 48 120 83 69)(2 70 84 109 37 17 31 99 58 95)(3 96 59 100 32 18 38 110 73 71)(4 72 74 111 39 19 33 101 60 85)(5 86 49 102 34 20 40 112 75 61)(6 62 76 113 41 21 35 103 50 87)(7 88 51 104 36 22 42 114 77 63)(8 64 78 115 43 23 25 105 52 89)(9 90 53 106 26 24 44 116 79 65)(10 66 80 117 45 13 27 107 54 91)(11 92 55 108 28 14 46 118 81 67)(12 68 82 119 47 15 29 97 56 93)
(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 15)(2 14)(3 13)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(25 61)(26 72)(27 71)(28 70)(29 69)(30 68)(31 67)(32 66)(33 65)(34 64)(35 63)(36 62)(37 92)(38 91)(39 90)(40 89)(41 88)(42 87)(43 86)(44 85)(45 96)(46 95)(47 94)(48 93)(49 115)(50 114)(51 113)(52 112)(53 111)(54 110)(55 109)(56 120)(57 119)(58 118)(59 117)(60 116)(73 107)(74 106)(75 105)(76 104)(77 103)(78 102)(79 101)(80 100)(81 99)(82 98)(83 97)(84 108)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 3 4A 4B 4C 5A 5B 6A 6B 6C 6D 6E 10A ··· 10N 10O ··· 10AD 12A 12B 12C 12D 15A 15B 30A ··· 30N order 1 2 2 2 2 2 2 2 2 2 2 3 4 4 4 5 5 6 6 6 6 6 10 ··· 10 10 ··· 10 12 12 12 12 15 15 30 ··· 30 size 1 1 1 1 2 2 6 6 6 6 60 2 20 20 60 2 2 2 2 2 4 4 2 ··· 2 6 ··· 6 20 20 20 20 4 4 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D10 D10 D12 C5⋊D4 C5⋊D4 S3×D4 S3×D5 C5⋊D12 C2×S3×D5 S3×C5⋊D4 kernel (C2×C10)⋊11D12 D6⋊Dic5 C3×C23.D5 C2×C5⋊D12 C2×C15⋊7D4 S3×C22×C10 C23.D5 S3×C10 C2×C30 S3×C23 C2×Dic5 C22×C10 C22×S3 C22×C6 C2×C10 D6 C2×C6 C10 C23 C22 C22 C2 # reps 1 2 1 2 1 1 1 4 2 2 2 1 4 2 4 16 8 2 2 4 2 8

Matrix representation of (C2×C10)⋊11D12 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 41 0 0 0 0 0 0 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 25 0 0 0 0 17 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 59 19 0 0 0 0 48 1
,
 60 25 0 0 0 0 0 1 0 0 0 0 0 0 0 60 0 0 0 0 60 0 0 0 0 0 0 0 59 19 0 0 0 0 48 2

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,41,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,17,0,0,0,0,25,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,59,48,0,0,0,0,19,1],[60,0,0,0,0,0,25,1,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,0,0,59,48,0,0,0,0,19,2] >;

(C2×C10)⋊11D12 in GAP, Magma, Sage, TeX

(C_2\times C_{10})\rtimes_{11}D_{12}
% in TeX

G:=Group("(C2xC10):11D12");
// GroupNames label

G:=SmallGroup(480,646);
// by ID

G=gap.SmallGroup(480,646);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,422,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=c^12=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a*b^5,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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