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G = C62.72D4order 288 = 25·32

56th non-split extension by C62 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.72D4, C62.253C23, (C6×D4).29S3, (C2×C12).250D6, (C22×C6).94D6, C625C416C2, (C6×C12).266C22, C6.Dic625C2, (C2×C62).71C22, C6.105(D42S3), C35(C23.23D6), C22.4(C327D4), C2.15(C12.D6), C3224(C22.D4), (D4×C3×C6).17C2, (C2×D4).5(C3⋊S3), (C3×C6).281(C2×D4), C6.122(C2×C3⋊D4), C23.12(C2×C3⋊S3), (C2×C6).25(C3⋊D4), (C22×C3⋊Dic3)⋊9C2, C2.11(C2×C327D4), (C3×C6).151(C4○D4), (C2×C6).270(C22×S3), C22.57(C22×C3⋊S3), (C2×C3⋊Dic3).165C22, (C2×C4).18(C2×C3⋊S3), SmallGroup(288,792)

Series: Derived Chief Lower central Upper central

C1C62 — C62.72D4
C1C3C32C3×C6C62C2×C3⋊Dic3C22×C3⋊Dic3 — C62.72D4
C32C62 — C62.72D4
C1C22C2×D4

Generators and relations for C62.72D4
 G = < a,b,c,d | a6=b6=c4=1, d2=b3, ab=ba, cac-1=a-1b3, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=b3c-1 >

Subgroups: 716 in 234 conjugacy classes, 77 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×3], C3 [×4], C4 [×5], C22, C22 [×2], C22 [×5], C6 [×12], C6 [×12], C2×C4, C2×C4 [×6], D4 [×2], C23 [×2], C32, Dic3 [×16], C12 [×4], C2×C6 [×12], C2×C6 [×20], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, C3×C6, C3×C6 [×2], C3×C6 [×3], C2×Dic3 [×24], C2×C12 [×4], C3×D4 [×8], C22×C6 [×8], C22.D4, C3⋊Dic3 [×4], C3×C12, C62, C62 [×2], C62 [×5], Dic3⋊C4 [×8], C6.D4 [×12], C22×Dic3 [×4], C6×D4 [×4], C2×C3⋊Dic3 [×4], C2×C3⋊Dic3 [×2], C6×C12, D4×C32 [×2], C2×C62 [×2], C23.23D6 [×4], C6.Dic6 [×2], C625C4, C625C4 [×2], C22×C3⋊Dic3, D4×C3×C6, C62.72D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C4○D4 [×2], C3⋊S3, C3⋊D4 [×8], C22×S3 [×4], C22.D4, C2×C3⋊S3 [×3], D42S3 [×8], C2×C3⋊D4 [×4], C327D4 [×2], C22×C3⋊S3, C23.23D6 [×4], C12.D6 [×2], C2×C327D4, C62.72D4

Smallest permutation representation of C62.72D4
On 144 points
Generators in S144
(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 19 110 40 91 115)(2 20 111 41 92 116)(3 21 112 42 93 117)(4 22 113 37 94 118)(5 23 114 38 95 119)(6 24 109 39 96 120)(7 105 63 99 122 60)(8 106 64 100 123 55)(9 107 65 101 124 56)(10 108 66 102 125 57)(11 103 61 97 126 58)(12 104 62 98 121 59)(13 70 28 33 50 47)(14 71 29 34 51 48)(15 72 30 35 52 43)(16 67 25 36 53 44)(17 68 26 31 54 45)(18 69 27 32 49 46)(73 90 130 135 141 79)(74 85 131 136 142 80)(75 86 132 137 143 81)(76 87 127 138 144 82)(77 88 128 133 139 83)(78 89 129 134 140 84)
(1 75 44 125)(2 136 45 107)(3 73 46 123)(4 134 47 105)(5 77 48 121)(6 138 43 103)(7 22 129 13)(8 93 130 32)(9 20 131 17)(10 91 132 36)(11 24 127 15)(12 95 128 34)(14 98 23 83)(16 102 19 81)(18 100 21 79)(25 108 40 137)(26 124 41 74)(27 106 42 135)(28 122 37 78)(29 104 38 133)(30 126 39 76)(31 101 92 80)(33 99 94 84)(35 97 96 82)(49 55 117 90)(50 63 118 140)(51 59 119 88)(52 61 120 144)(53 57 115 86)(54 65 116 142)(56 111 85 68)(58 109 87 72)(60 113 89 70)(62 114 139 71)(64 112 141 69)(66 110 143 67)
(1 108 40 125)(2 107 41 124)(3 106 42 123)(4 105 37 122)(5 104 38 121)(6 103 39 126)(7 94 99 22)(8 93 100 21)(9 92 101 20)(10 91 102 19)(11 96 97 24)(12 95 98 23)(13 129 33 84)(14 128 34 83)(15 127 35 82)(16 132 36 81)(17 131 31 80)(18 130 32 79)(25 75 44 137)(26 74 45 136)(27 73 46 135)(28 78 47 134)(29 77 48 133)(30 76 43 138)(49 141 69 90)(50 140 70 89)(51 139 71 88)(52 144 72 87)(53 143 67 86)(54 142 68 85)(55 117 64 112)(56 116 65 111)(57 115 66 110)(58 120 61 109)(59 119 62 114)(60 118 63 113)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E2F3A3B3C3D4A4B4C4D4E4F4G6A···6L6M···6AB12A···12H
order1222222333344444446···66···612···12
size1111224222241818181836362···24···44···4

54 irreducible representations

dim111112222224
type+++++++++-
imageC1C2C2C2C2S3D4D6D6C4○D4C3⋊D4D42S3
kernelC62.72D4C6.Dic6C625C4C22×C3⋊Dic3D4×C3×C6C6×D4C62C2×C12C22×C6C3×C6C2×C6C6
# reps1231142484168

Matrix representation of C62.72D4 in GL6(𝔽13)

300000
090000
001000
0081200
000090
000003
,
300000
090000
0012000
0001200
000010
000001
,
010000
1200000
0081100
000500
000001
000010
,
010000
100000
008000
000800
000001
000010

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,9,0,0,0,0,0,0,1,8,0,0,0,0,0,12,0,0,0,0,0,0,9,0,0,0,0,0,0,3],[3,0,0,0,0,0,0,9,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,11,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C62.72D4 in GAP, Magma, Sage, TeX

C_6^2._{72}D_4
% in TeX

G:=Group("C6^2.72D4");
// GroupNames label

G:=SmallGroup(288,792);
// by ID

G=gap.SmallGroup(288,792);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,254,219,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=c^4=1,d^2=b^3,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^3*c^-1>;
// generators/relations

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