metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C30)⋊9D4, C15⋊7C22≀C2, (S3×C10)⋊17D4, (C2×C10)⋊11D12, C5⋊5(D6⋊D4), D6⋊8(C5⋊D4), (C2×Dic5)⋊4D6, (S3×C23)⋊2D5, D6⋊Dic5⋊36C2, C10.165(S3×D4), C10.70(C2×D12), C30.250(C2×D4), C3⋊1(C24⋊2D5), C23.D5⋊13S3, C23.34(S3×D5), C22⋊3(C5⋊D12), (C6×Dic5)⋊8C22, (C22×C6).46D10, (C2×C30).212C23, (C22×S3).79D10, (C22×C10).112D6, (C22×D15)⋊7C22, (C2×Dic15)⋊14C22, (C22×C30).74C22, (C2×C6)⋊2(C5⋊D4), (S3×C22×C10)⋊2C2, C2.45(S3×C5⋊D4), C6.24(C2×C5⋊D4), (C2×C5⋊D12)⋊15C2, (C2×C15⋊7D4)⋊20C2, C2.25(C2×C5⋊D12), C22.241(C2×S3×D5), (S3×C2×C10).96C22, (C3×C23.D5)⋊15C2, (C2×C6).224(C22×D5), (C2×C10).224(C22×S3), SmallGroup(480,646)
Series: Derived ►Chief ►Lower central ►Upper central
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, C15⋊7D4 [×2], S3×C2×C10 [×2], S3×C2×C10 [×6], C22×D15, C22×C30, C24⋊2D5, D6⋊Dic5 [×2], C3×C23.D5, C2×C5⋊D12 [×2], C2×C15⋊7D4, 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, C24⋊2D5, C2×C5⋊D12, S3×C5⋊D4 [×2], (C2×C10)⋊11D12
(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