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G = C62.74D4order 288 = 25·32

58th non-split extension by C62 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.74D4, (C3×D4).43D6, (C3×Q8).67D6, (C3×C12).175D4, (C2×C12).162D6, C3225(C4○D8), C327D811C2, C12.59D66C2, C37(Q8.13D6), C327Q1611C2, C3211SD1611C2, C329SD1611C2, C12.135(C3⋊D4), (C6×C12).154C22, (C3×C12).108C23, C12.104(C22×S3), C4.32(C327D4), C12⋊S3.32C22, C324C8.31C22, (D4×C32).28C22, C22.1(C327D4), (Q8×C32).29C22, C324Q8.32C22, (C3×C4○D4)⋊6S3, D4.8(C2×C3⋊S3), C4○D44(C3⋊S3), Q8.13(C2×C3⋊S3), (C3×C6).294(C2×D4), (C32×C4○D4)⋊4C2, C6.135(C2×C3⋊D4), C4.18(C22×C3⋊S3), (C2×C324C8)⋊13C2, (C2×C6).27(C3⋊D4), C2.24(C2×C327D4), (C2×C4).59(C2×C3⋊S3), SmallGroup(288,807)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C62.74D4
C1C3C32C3×C6C3×C12C12⋊S3C12.59D6 — C62.74D4
C32C3×C6C3×C12 — C62.74D4
C1C4C2×C4C4○D4

Generators and relations for C62.74D4
 G = < a,b,c,d | a6=b6=d2=1, c4=b3, ab=ba, cac-1=a-1, dad=a-1b3, cbc-1=dbd=b-1, dcd=b3c3 >

Subgroups: 644 in 186 conjugacy classes, 65 normal (29 characteristic)
C1, C2, C2 [×3], C3 [×4], C4 [×2], C4 [×2], C22, C22 [×2], S3 [×4], C6 [×4], C6 [×8], C8 [×2], C2×C4, C2×C4 [×2], D4, D4 [×3], Q8, Q8, C32, Dic3 [×4], C12 [×8], C12 [×4], D6 [×4], C2×C6 [×4], C2×C6 [×4], C2×C8, D8, SD16 [×2], Q16, C4○D4, C4○D4, C3⋊S3, C3×C6, C3×C6 [×2], C3⋊C8 [×8], Dic6 [×4], C4×S3 [×4], D12 [×4], C3⋊D4 [×4], C2×C12 [×4], C2×C12 [×4], C3×D4 [×4], C3×D4 [×4], C3×Q8 [×4], C4○D8, C3⋊Dic3, C3×C12 [×2], C3×C12, C2×C3⋊S3, C62, C62, C2×C3⋊C8 [×4], D4⋊S3 [×4], D4.S3 [×4], Q82S3 [×4], C3⋊Q16 [×4], C4○D12 [×4], C3×C4○D4 [×4], C324C8 [×2], C324Q8, C4×C3⋊S3, C12⋊S3, C327D4, C6×C12, C6×C12, D4×C32, D4×C32, Q8×C32, Q8.13D6 [×4], C2×C324C8, C327D8, C329SD16, C3211SD16, C327Q16, C12.59D6, C32×C4○D4, C62.74D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, C3⋊D4 [×8], C22×S3 [×4], C4○D8, C2×C3⋊S3 [×3], C2×C3⋊D4 [×4], C327D4 [×2], C22×C3⋊S3, Q8.13D6 [×4], C2×C327D4, C62.74D4

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

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

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

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

54 conjugacy classes

class 1 2A2B2C2D3A3B3C3D4A4B4C4D4E6A6B6C6D6E···6P8A8B8C8D12A···12H12I···12T
order1222233334444466666···6888812···1212···12
size112436222211243622224···4181818182···24···4

54 irreducible representations

dim111111112222222224
type++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D6D6D6C3⋊D4C3⋊D4C4○D8Q8.13D6
kernelC62.74D4C2×C324C8C327D8C329SD16C3211SD16C327Q16C12.59D6C32×C4○D4C3×C4○D4C3×C12C62C2×C12C3×D4C3×Q8C12C2×C6C32C3
# reps111111114114448848

Matrix representation of C62.74D4 in GL6(𝔽73)

7210000
7200000
00727200
001000
00004619
00002727
,
0720000
1720000
00727200
001000
0000720
0000072
,
60430000
30130000
00306000
00304300
00003232
0000570
,
010000
100000
0072000
001100
000010
00007272

G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,46,27,0,0,0,0,19,27],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[60,30,0,0,0,0,43,13,0,0,0,0,0,0,30,30,0,0,0,0,60,43,0,0,0,0,0,0,32,57,0,0,0,0,32,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;

C62.74D4 in GAP, Magma, Sage, TeX

C_6^2._{74}D_4
% in TeX

G:=Group("C6^2.74D4");
// GroupNames label

G:=SmallGroup(288,807);
// by ID

G=gap.SmallGroup(288,807);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,675,185,80,2693,9414]);
// Polycyclic

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

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