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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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C30)⋊9D4, C157C22≀C2, (S3×C10)⋊17D4, (C2×C10)⋊11D12, C55(D6⋊D4), D68(C5⋊D4), (C2×Dic5)⋊4D6, (S3×C23)⋊2D5, D6⋊Dic536C2, C10.165(S3×D4), C10.70(C2×D12), C30.250(C2×D4), C31(C242D5), C23.D513S3, C23.34(S3×D5), C223(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×C157D4)⋊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

C1C2×C30 — (C2×C10)⋊11D12
C1C5C15C30C2×C30C6×Dic5C2×C5⋊D12 — (C2×C10)⋊11D12
C15C2×C30 — (C2×C10)⋊11D12
C1C22C23

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 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C5A5B6A6B6C6D6E10A···10N10O···10AD12A12B12C12D15A15B30A···30N
order122222222223444556666610···1010···1012121212151530···30
size111122666660220206022222442···26···620202020444···4

72 irreducible representations

dim1111112222222222244444
type+++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D5D6D6D10D10D12C5⋊D4C5⋊D4S3×D4S3×D5C5⋊D12C2×S3×D5S3×C5⋊D4
kernel(C2×C10)⋊11D12D6⋊Dic5C3×C23.D5C2×C5⋊D12C2×C157D4S3×C22×C10C23.D5S3×C10C2×C30S3×C23C2×Dic5C22×C10C22×S3C22×C6C2×C10D6C2×C6C10C23C22C22C2
# reps12121114222142416822428

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

6000000
0600000
001000
0006000
000010
000001
,
100000
010000
0041000
000300
000010
000001
,
60250000
1710000
000100
001000
00005919
0000481
,
60250000
010000
0006000
0060000
00005919
0000482

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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