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G = C62.8Q8order 288 = 25·32

5th non-split extension by C62 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.8Q8, C12.28(C4×S3), (C2×C12).89D6, (C2×C6).5Dic6, (C3×C12).170D4, C324C8.3C4, (C6×C12).56C22, C33(C12.53D4), C12.131(C3⋊D4), C6.19(Dic3⋊C4), C3210(C8.C4), C12.58D6.4C2, M4(2).1(C3⋊S3), (C3×M4(2)).11S3, C4.28(C327D4), C2.5(C6.Dic6), (C32×M4(2)).3C2, C22.1(C324Q8), C4.13(C4×C3⋊S3), (C3×C6).43(C4⋊C4), (C3×C12).50(C2×C4), (C2×C324C8).8C2, (C2×C4).38(C2×C3⋊S3), SmallGroup(288,297)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C62.8Q8
C1C3C32C3×C6C3×C12C6×C12C2×C324C8 — C62.8Q8
C32C3×C6C3×C12 — C62.8Q8
C1C4C2×C4M4(2)

Generators and relations for C62.8Q8
 G = < a,b,c,d | a6=b6=1, c4=b3, d2=a3b3c2, ab=ba, cac-1=ab3, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=a3b3c3 >

Subgroups: 220 in 90 conjugacy classes, 47 normal (19 characteristic)
C1, C2, C2, C3 [×4], C4 [×2], C22, C6 [×4], C6 [×4], C8 [×4], C2×C4, C32, C12 [×8], C2×C6 [×4], C2×C8, M4(2), M4(2), C3×C6, C3×C6, C3⋊C8 [×12], C24 [×4], C2×C12 [×4], C8.C4, C3×C12 [×2], C62, C2×C3⋊C8 [×4], C4.Dic3 [×4], C3×M4(2) [×4], C324C8 [×2], C324C8, C3×C24, C6×C12, C12.53D4 [×4], C2×C324C8, C12.58D6, C32×M4(2), C62.8Q8
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4, Q8, D6 [×4], C4⋊C4, C3⋊S3, Dic6 [×4], C4×S3 [×4], C3⋊D4 [×4], C8.C4, C2×C3⋊S3, Dic3⋊C4 [×4], C324Q8, C4×C3⋊S3, C327D4, C12.53D4 [×4], C6.Dic6, C62.8Q8

Smallest permutation representation of C62.8Q8
On 144 points
Generators in S144
(1 100 105)(2 97 106 6 101 110)(3 102 107)(4 99 108 8 103 112)(5 104 109)(7 98 111)(9 32 53 13 28 49)(10 29 54)(11 26 55 15 30 51)(12 31 56)(14 25 50)(16 27 52)(17 91 36 21 95 40)(18 96 37)(19 93 38 23 89 34)(20 90 39)(22 92 33)(24 94 35)(41 61 140)(42 58 141 46 62 137)(43 63 142)(44 60 143 48 64 139)(45 57 144)(47 59 138)(65 121 133)(66 126 134 70 122 130)(67 123 135)(68 128 136 72 124 132)(69 125 129)(71 127 131)(73 86 114)(74 83 115 78 87 119)(75 88 116)(76 85 117 80 81 113)(77 82 118)(79 84 120)
(1 121 120 5 125 116)(2 122 113 6 126 117)(3 123 114 7 127 118)(4 124 115 8 128 119)(9 36 42 13 40 46)(10 37 43 14 33 47)(11 38 44 15 34 48)(12 39 45 16 35 41)(17 62 32 21 58 28)(18 63 25 22 59 29)(19 64 26 23 60 30)(20 57 27 24 61 31)(49 91 137 53 95 141)(50 92 138 54 96 142)(51 93 139 55 89 143)(52 94 140 56 90 144)(65 84 109 69 88 105)(66 85 110 70 81 106)(67 86 111 71 82 107)(68 87 112 72 83 108)(73 98 131 77 102 135)(74 99 132 78 103 136)(75 100 133 79 104 129)(76 101 134 80 97 130)
(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 59 7 57 5 63 3 61)(2 62 4 64 6 58 8 60)(9 74 11 76 13 78 15 80)(10 73 16 79 14 77 12 75)(17 124 19 126 21 128 23 122)(18 123 24 121 22 127 20 125)(25 118 31 116 29 114 27 120)(26 113 28 115 30 117 32 119)(33 131 39 129 37 135 35 133)(34 134 36 136 38 130 40 132)(41 100 47 98 45 104 43 102)(42 103 44 97 46 99 48 101)(49 83 51 85 53 87 55 81)(50 82 56 88 54 86 52 84)(65 92 71 90 69 96 67 94)(66 95 68 89 70 91 72 93)(105 138 111 144 109 142 107 140)(106 141 108 143 110 137 112 139)

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

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

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

54 conjugacy classes

class 1 2A2B3A3B3C3D4A4B4C6A6B6C6D6E6F6G6H8A8B8C8D8E8F8G8H12A···12H12I12J12K12L24A···24P
order1223333444666666668888888812···121212121224···24
size112222211222224444441818181836362···244444···4

54 irreducible representations

dim11111222222224
type++++++-+-
imageC1C2C2C2C4S3D4Q8D6C4×S3C3⋊D4Dic6C8.C4C12.53D4
kernelC62.8Q8C2×C324C8C12.58D6C32×M4(2)C324C8C3×M4(2)C3×C12C62C2×C12C12C12C2×C6C32C3
# reps11114411488848

Matrix representation of C62.8Q8 in GL6(𝔽73)

0720000
1720000
00727200
001000
000010
00005272
,
0720000
1720000
001000
000100
0000720
0000072
,
7200000
0720000
001000
000100
0000683
000075
,
1720000
0720000
00103200
00226300
0000630
00002051

G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,52,0,0,0,0,0,72],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,68,7,0,0,0,0,3,5],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,10,22,0,0,0,0,32,63,0,0,0,0,0,0,63,20,0,0,0,0,0,51] >;

C62.8Q8 in GAP, Magma, Sage, TeX

C_6^2._8Q_8
% in TeX

G:=Group("C6^2.8Q8");
// GroupNames label

G:=SmallGroup(288,297);
// by ID

G=gap.SmallGroup(288,297);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,36,100,346,80,2693,9414]);
// Polycyclic

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

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