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G = C24.32D6order 288 = 25·32

32nd non-split extension by C24 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C24.32D6, C24⋊S34C2, (C3×SD16)⋊2S3, (C3×D4).18D6, C6.123(S3×D4), (C3×Q8).36D6, SD162(C3⋊S3), C35(D4.D6), C3⋊Dic3.67D4, C327Q165C2, C325Q1610C2, C329SD168C2, (C3×C12).96C23, C12.92(C22×S3), (C3×C24).33C22, (C32×SD16)⋊4C2, C12.D6.3C2, C3219(C8.C22), C324C8.15C22, (D4×C32).19C22, (Q8×C32).16C22, C324Q8.17C22, C8.2(C2×C3⋊S3), (Q8×C3⋊S3)⋊4C2, D4.4(C2×C3⋊S3), C2.20(D4×C3⋊S3), Q8.6(C2×C3⋊S3), (C2×C3⋊S3).67D4, C4.6(C22×C3⋊S3), (C3×C6).244(C2×D4), (C4×C3⋊S3).25C22, SmallGroup(288,772)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C24.32D6
C1C3C32C3×C6C3×C12C4×C3⋊S3Q8×C3⋊S3 — C24.32D6
C32C3×C6C3×C12 — C24.32D6
C1C2C4SD16

Generators and relations for C24.32D6
 G = < a,b,c | a24=1, b6=c2=a12, bab-1=a19, cac-1=a-1, cbc-1=b5 >

Subgroups: 684 in 180 conjugacy classes, 55 normal (27 characteristic)
C1, C2, C2 [×2], C3 [×4], C4, C4 [×4], C22 [×2], S3 [×4], C6 [×4], C6 [×4], C8, C8, C2×C4 [×3], D4, D4, Q8, Q8 [×3], C32, Dic3 [×12], C12 [×4], C12 [×4], D6 [×4], C2×C6 [×4], M4(2), SD16, SD16, Q16 [×2], C2×Q8, C4○D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8 [×4], C24 [×4], Dic6 [×12], C4×S3 [×8], C2×Dic3 [×4], C3⋊D4 [×4], C3×D4 [×4], C3×Q8 [×4], C8.C22, C3⋊Dic3, C3⋊Dic3 [×2], C3×C12, C3×C12, C2×C3⋊S3, C62, C8⋊S3 [×4], Dic12 [×4], D4.S3 [×4], C3⋊Q16 [×4], C3×SD16 [×4], D42S3 [×4], S3×Q8 [×4], C324C8, C3×C24, C324Q8 [×2], C324Q8, C4×C3⋊S3, C4×C3⋊S3, C2×C3⋊Dic3, C327D4, D4×C32, Q8×C32, D4.D6 [×4], C24⋊S3, C325Q16, C329SD16, C327Q16, C32×SD16, C12.D6, Q8×C3⋊S3, C24.32D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, C22×S3 [×4], C8.C22, C2×C3⋊S3 [×3], S3×D4 [×4], C22×C3⋊S3, D4.D6 [×4], D4×C3⋊S3, C24.32D6

Smallest permutation representation of C24.32D6
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 58 116 128 25 76 13 70 104 140 37 88)(2 53 117 123 26 95 14 65 105 135 38 83)(3 72 118 142 27 90 15 60 106 130 39 78)(4 67 119 137 28 85 16 55 107 125 40 73)(5 62 120 132 29 80 17 50 108 144 41 92)(6 57 97 127 30 75 18 69 109 139 42 87)(7 52 98 122 31 94 19 64 110 134 43 82)(8 71 99 141 32 89 20 59 111 129 44 77)(9 66 100 136 33 84 21 54 112 124 45 96)(10 61 101 131 34 79 22 49 113 143 46 91)(11 56 102 126 35 74 23 68 114 138 47 86)(12 51 103 121 36 93 24 63 115 133 48 81)
(1 76 13 88)(2 75 14 87)(3 74 15 86)(4 73 16 85)(5 96 17 84)(6 95 18 83)(7 94 19 82)(8 93 20 81)(9 92 21 80)(10 91 22 79)(11 90 23 78)(12 89 24 77)(25 58 37 70)(26 57 38 69)(27 56 39 68)(28 55 40 67)(29 54 41 66)(30 53 42 65)(31 52 43 64)(32 51 44 63)(33 50 45 62)(34 49 46 61)(35 72 47 60)(36 71 48 59)(97 123 109 135)(98 122 110 134)(99 121 111 133)(100 144 112 132)(101 143 113 131)(102 142 114 130)(103 141 115 129)(104 140 116 128)(105 139 117 127)(106 138 118 126)(107 137 119 125)(108 136 120 124)

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

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

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

39 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D4E6A6B6C6D6E6F6G6H8A8B12A12B12C12D12E12F12G12H24A···24H
order12223333444446666666688121212121212121224···24
size1141822222418363622228888436444488884···4

39 irreducible representations

dim11111111222222444
type++++++++++++++-+-
imageC1C2C2C2C2C2C2C2S3D4D4D6D6D6C8.C22S3×D4D4.D6
kernelC24.32D6C24⋊S3C325Q16C329SD16C327Q16C32×SD16C12.D6Q8×C3⋊S3C3×SD16C3⋊Dic3C2×C3⋊S3C24C3×D4C3×Q8C32C6C3
# reps11111111411444148

Matrix representation of C24.32D6 in GL6(𝔽73)

100000
010000
0034393934
003468395
0034393439
0034683468
,
72720000
100000
00650640
00065064
0064080
0006408
,
100000
72720000
00650640
008899
0064080
00996565

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,34,34,34,0,0,39,68,39,68,0,0,39,39,34,34,0,0,34,5,39,68],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,65,0,64,0,0,0,0,65,0,64,0,0,64,0,8,0,0,0,0,64,0,8],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,65,8,64,9,0,0,0,8,0,9,0,0,64,9,8,65,0,0,0,9,0,65] >;

C24.32D6 in GAP, Magma, Sage, TeX

C_{24}._{32}D_6
% in TeX

G:=Group("C24.32D6");
// GroupNames label

G:=SmallGroup(288,772);
// by ID

G=gap.SmallGroup(288,772);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,135,346,185,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^24=1,b^6=c^2=a^12,b*a*b^-1=a^19,c*a*c^-1=a^-1,c*b*c^-1=b^5>;
// generators/relations

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