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G = C62.80D6order 432 = 24·33

28th non-split extension by C62 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C62.80D6, C3312(C4⋊C4), (C3×C6).43D12, Dic3⋊(C3⋊Dic3), (C32×C6).6Q8, C6.31(S3×Dic3), (C3×C6).16Dic6, (C32×C6).44D4, (C3×Dic3)⋊1Dic3, C327(C4⋊Dic3), (C6×Dic3).11S3, (C32×Dic3)⋊3C4, C6.5(C322Q8), C6.18(C12⋊S3), C2.3(C338D4), C31(Dic3⋊Dic3), C6.13(C3⋊D12), C31(C12⋊Dic3), C2.1(C334Q8), C6.1(C324Q8), (C3×C62).10C22, C3213(Dic3⋊C4), (C2×C6).34S32, (C3×C6).92(C4×S3), C22.8(S3×C3⋊S3), C6.5(C2×C3⋊Dic3), C2.5(S3×C3⋊Dic3), (Dic3×C3×C6).7C2, (C6×C3⋊Dic3).3C2, (C2×C3⋊Dic3).5S3, (C3×C6).81(C3⋊D4), (C32×C6).41(C2×C4), (C2×C335C4).2C2, (C3×C6).39(C2×Dic3), (C2×Dic3).1(C3⋊S3), (C2×C6).16(C2×C3⋊S3), SmallGroup(432,452)

Series: Derived Chief Lower central Upper central

C1C32×C6 — C62.80D6
C1C3C32C33C32×C6C3×C62Dic3×C3×C6 — C62.80D6
C33C32×C6 — C62.80D6
C1C22

Generators and relations for C62.80D6
 G = < a,b,c,d | a6=b6=1, c6=b3, d2=a3, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=b3c5 >

Subgroups: 984 in 220 conjugacy classes, 82 normal (26 characteristic)
C1, C2 [×3], C3, C3 [×4], C3 [×4], C4 [×4], C22, C6 [×3], C6 [×12], C6 [×12], C2×C4 [×3], C32, C32 [×4], C32 [×4], Dic3 [×2], Dic3 [×17], C12 [×9], C2×C6, C2×C6 [×4], C2×C6 [×4], C4⋊C4, C3×C6 [×3], C3×C6 [×12], C3×C6 [×12], C2×Dic3, C2×Dic3 [×13], C2×C12 [×5], C33, C3×Dic3 [×8], C3×Dic3 [×4], C3⋊Dic3 [×14], C3×C12 [×2], C62, C62 [×4], C62 [×4], Dic3⋊C4, C4⋊Dic3 [×4], C32×C6 [×3], C6×Dic3 [×4], C6×Dic3 [×4], C2×C3⋊Dic3, C2×C3⋊Dic3 [×9], C6×C12, C32×Dic3 [×2], C3×C3⋊Dic3, C335C4, C3×C62, Dic3⋊Dic3 [×4], C12⋊Dic3, Dic3×C3×C6, C6×C3⋊Dic3, C2×C335C4, C62.80D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×5], C2×C4, D4, Q8, Dic3 [×8], D6 [×5], C4⋊C4, C3⋊S3, Dic6 [×5], C4×S3, D12 [×4], C2×Dic3 [×4], C3⋊D4, C3⋊Dic3 [×2], S32 [×4], C2×C3⋊S3, Dic3⋊C4, C4⋊Dic3 [×4], S3×Dic3 [×4], C3⋊D12 [×4], C322Q8 [×4], C324Q8, C12⋊S3, C2×C3⋊Dic3, S3×C3⋊S3, Dic3⋊Dic3 [×4], C12⋊Dic3, S3×C3⋊Dic3, C338D4, C334Q8, C62.80D6

Smallest permutation representation of C62.80D6
On 144 points
Generators in S144
(1 31 23 98 90 73)(2 32 24 99 91 74)(3 33 13 100 92 75)(4 34 14 101 93 76)(5 35 15 102 94 77)(6 36 16 103 95 78)(7 25 17 104 96 79)(8 26 18 105 85 80)(9 27 19 106 86 81)(10 28 20 107 87 82)(11 29 21 108 88 83)(12 30 22 97 89 84)(37 56 67 140 111 129)(38 57 68 141 112 130)(39 58 69 142 113 131)(40 59 70 143 114 132)(41 60 71 144 115 121)(42 49 72 133 116 122)(43 50 61 134 117 123)(44 51 62 135 118 124)(45 52 63 136 119 125)(46 53 64 137 120 126)(47 54 65 138 109 127)(48 55 66 139 110 128)
(1 21 86 7 15 92)(2 93 16 8 87 22)(3 23 88 9 17 94)(4 95 18 10 89 24)(5 13 90 11 19 96)(6 85 20 12 91 14)(25 102 75 31 108 81)(26 82 97 32 76 103)(27 104 77 33 98 83)(28 84 99 34 78 105)(29 106 79 35 100 73)(30 74 101 36 80 107)(37 113 71 43 119 65)(38 66 120 44 72 114)(39 115 61 45 109 67)(40 68 110 46 62 116)(41 117 63 47 111 69)(42 70 112 48 64 118)(49 143 130 55 137 124)(50 125 138 56 131 144)(51 133 132 57 139 126)(52 127 140 58 121 134)(53 135 122 59 141 128)(54 129 142 60 123 136)
(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)(121 122 123 124 125 126 127 128 129 130 131 132)(133 134 135 136 137 138 139 140 141 142 143 144)
(1 137 98 46)(2 136 99 45)(3 135 100 44)(4 134 101 43)(5 133 102 42)(6 144 103 41)(7 143 104 40)(8 142 105 39)(9 141 106 38)(10 140 107 37)(11 139 108 48)(12 138 97 47)(13 51 75 118)(14 50 76 117)(15 49 77 116)(16 60 78 115)(17 59 79 114)(18 58 80 113)(19 57 81 112)(20 56 82 111)(21 55 83 110)(22 54 84 109)(23 53 73 120)(24 52 74 119)(25 70 96 132)(26 69 85 131)(27 68 86 130)(28 67 87 129)(29 66 88 128)(30 65 89 127)(31 64 90 126)(32 63 91 125)(33 62 92 124)(34 61 93 123)(35 72 94 122)(36 71 95 121)

G:=sub<Sym(144)| (1,31,23,98,90,73)(2,32,24,99,91,74)(3,33,13,100,92,75)(4,34,14,101,93,76)(5,35,15,102,94,77)(6,36,16,103,95,78)(7,25,17,104,96,79)(8,26,18,105,85,80)(9,27,19,106,86,81)(10,28,20,107,87,82)(11,29,21,108,88,83)(12,30,22,97,89,84)(37,56,67,140,111,129)(38,57,68,141,112,130)(39,58,69,142,113,131)(40,59,70,143,114,132)(41,60,71,144,115,121)(42,49,72,133,116,122)(43,50,61,134,117,123)(44,51,62,135,118,124)(45,52,63,136,119,125)(46,53,64,137,120,126)(47,54,65,138,109,127)(48,55,66,139,110,128), (1,21,86,7,15,92)(2,93,16,8,87,22)(3,23,88,9,17,94)(4,95,18,10,89,24)(5,13,90,11,19,96)(6,85,20,12,91,14)(25,102,75,31,108,81)(26,82,97,32,76,103)(27,104,77,33,98,83)(28,84,99,34,78,105)(29,106,79,35,100,73)(30,74,101,36,80,107)(37,113,71,43,119,65)(38,66,120,44,72,114)(39,115,61,45,109,67)(40,68,110,46,62,116)(41,117,63,47,111,69)(42,70,112,48,64,118)(49,143,130,55,137,124)(50,125,138,56,131,144)(51,133,132,57,139,126)(52,127,140,58,121,134)(53,135,122,59,141,128)(54,129,142,60,123,136), (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)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144), (1,137,98,46)(2,136,99,45)(3,135,100,44)(4,134,101,43)(5,133,102,42)(6,144,103,41)(7,143,104,40)(8,142,105,39)(9,141,106,38)(10,140,107,37)(11,139,108,48)(12,138,97,47)(13,51,75,118)(14,50,76,117)(15,49,77,116)(16,60,78,115)(17,59,79,114)(18,58,80,113)(19,57,81,112)(20,56,82,111)(21,55,83,110)(22,54,84,109)(23,53,73,120)(24,52,74,119)(25,70,96,132)(26,69,85,131)(27,68,86,130)(28,67,87,129)(29,66,88,128)(30,65,89,127)(31,64,90,126)(32,63,91,125)(33,62,92,124)(34,61,93,123)(35,72,94,122)(36,71,95,121)>;

G:=Group( (1,31,23,98,90,73)(2,32,24,99,91,74)(3,33,13,100,92,75)(4,34,14,101,93,76)(5,35,15,102,94,77)(6,36,16,103,95,78)(7,25,17,104,96,79)(8,26,18,105,85,80)(9,27,19,106,86,81)(10,28,20,107,87,82)(11,29,21,108,88,83)(12,30,22,97,89,84)(37,56,67,140,111,129)(38,57,68,141,112,130)(39,58,69,142,113,131)(40,59,70,143,114,132)(41,60,71,144,115,121)(42,49,72,133,116,122)(43,50,61,134,117,123)(44,51,62,135,118,124)(45,52,63,136,119,125)(46,53,64,137,120,126)(47,54,65,138,109,127)(48,55,66,139,110,128), (1,21,86,7,15,92)(2,93,16,8,87,22)(3,23,88,9,17,94)(4,95,18,10,89,24)(5,13,90,11,19,96)(6,85,20,12,91,14)(25,102,75,31,108,81)(26,82,97,32,76,103)(27,104,77,33,98,83)(28,84,99,34,78,105)(29,106,79,35,100,73)(30,74,101,36,80,107)(37,113,71,43,119,65)(38,66,120,44,72,114)(39,115,61,45,109,67)(40,68,110,46,62,116)(41,117,63,47,111,69)(42,70,112,48,64,118)(49,143,130,55,137,124)(50,125,138,56,131,144)(51,133,132,57,139,126)(52,127,140,58,121,134)(53,135,122,59,141,128)(54,129,142,60,123,136), (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)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144), (1,137,98,46)(2,136,99,45)(3,135,100,44)(4,134,101,43)(5,133,102,42)(6,144,103,41)(7,143,104,40)(8,142,105,39)(9,141,106,38)(10,140,107,37)(11,139,108,48)(12,138,97,47)(13,51,75,118)(14,50,76,117)(15,49,77,116)(16,60,78,115)(17,59,79,114)(18,58,80,113)(19,57,81,112)(20,56,82,111)(21,55,83,110)(22,54,84,109)(23,53,73,120)(24,52,74,119)(25,70,96,132)(26,69,85,131)(27,68,86,130)(28,67,87,129)(29,66,88,128)(30,65,89,127)(31,64,90,126)(32,63,91,125)(33,62,92,124)(34,61,93,123)(35,72,94,122)(36,71,95,121) );

G=PermutationGroup([(1,31,23,98,90,73),(2,32,24,99,91,74),(3,33,13,100,92,75),(4,34,14,101,93,76),(5,35,15,102,94,77),(6,36,16,103,95,78),(7,25,17,104,96,79),(8,26,18,105,85,80),(9,27,19,106,86,81),(10,28,20,107,87,82),(11,29,21,108,88,83),(12,30,22,97,89,84),(37,56,67,140,111,129),(38,57,68,141,112,130),(39,58,69,142,113,131),(40,59,70,143,114,132),(41,60,71,144,115,121),(42,49,72,133,116,122),(43,50,61,134,117,123),(44,51,62,135,118,124),(45,52,63,136,119,125),(46,53,64,137,120,126),(47,54,65,138,109,127),(48,55,66,139,110,128)], [(1,21,86,7,15,92),(2,93,16,8,87,22),(3,23,88,9,17,94),(4,95,18,10,89,24),(5,13,90,11,19,96),(6,85,20,12,91,14),(25,102,75,31,108,81),(26,82,97,32,76,103),(27,104,77,33,98,83),(28,84,99,34,78,105),(29,106,79,35,100,73),(30,74,101,36,80,107),(37,113,71,43,119,65),(38,66,120,44,72,114),(39,115,61,45,109,67),(40,68,110,46,62,116),(41,117,63,47,111,69),(42,70,112,48,64,118),(49,143,130,55,137,124),(50,125,138,56,131,144),(51,133,132,57,139,126),(52,127,140,58,121,134),(53,135,122,59,141,128),(54,129,142,60,123,136)], [(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),(121,122,123,124,125,126,127,128,129,130,131,132),(133,134,135,136,137,138,139,140,141,142,143,144)], [(1,137,98,46),(2,136,99,45),(3,135,100,44),(4,134,101,43),(5,133,102,42),(6,144,103,41),(7,143,104,40),(8,142,105,39),(9,141,106,38),(10,140,107,37),(11,139,108,48),(12,138,97,47),(13,51,75,118),(14,50,76,117),(15,49,77,116),(16,60,78,115),(17,59,79,114),(18,58,80,113),(19,57,81,112),(20,56,82,111),(21,55,83,110),(22,54,84,109),(23,53,73,120),(24,52,74,119),(25,70,96,132),(26,69,85,131),(27,68,86,130),(28,67,87,129),(29,66,88,128),(30,65,89,127),(31,64,90,126),(32,63,91,125),(33,62,92,124),(34,61,93,123),(35,72,94,122),(36,71,95,121)])

66 conjugacy classes

class 1 2A2B2C3A···3E3F3G3H3I4A4B4C4D4E4F6A···6O6P···6AA12A···12P12Q12R12S12T
order12223···333334444446···66···612···1212121212
size11112···2444466181854542···24···46···618181818

66 irreducible representations

dim1111122222222224444
type+++++++--+-++-+-
imageC1C2C2C2C4S3S3D4Q8Dic3D6Dic6C4×S3D12C3⋊D4S32S3×Dic3C3⋊D12C322Q8
kernelC62.80D6Dic3×C3×C6C6×C3⋊Dic3C2×C335C4C32×Dic3C6×Dic3C2×C3⋊Dic3C32×C6C32×C6C3×Dic3C62C3×C6C3×C6C3×C6C3×C6C2×C6C6C6C6
# reps11114411185102824444

Matrix representation of C62.80D6 in GL6(𝔽13)

1000000
040000
0011200
001000
000010
000001
,
1200000
0120000
001000
000100
0000121
0000120
,
1100000
060000
001000
000100
000001
000010
,
070000
1100000
000500
005000
000001
000010

G:=sub<GL(6,GF(13))| [10,0,0,0,0,0,0,4,0,0,0,0,0,0,1,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[11,0,0,0,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,11,0,0,0,0,7,0,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C62.80D6 in GAP, Magma, Sage, TeX

C_6^2._{80}D_6
% in TeX

G:=Group("C6^2.80D6");
// GroupNames label

G:=SmallGroup(432,452);
// by ID

G=gap.SmallGroup(432,452);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,36,571,2028,14118]);
// Polycyclic

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

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