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G = C62.129D4order 288 = 25·32

34th non-split extension by C62 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial

Aliases: C62.129D4, C62.250C23, (C22×C12)⋊9S3, (C2×C12).361D6, C625C414C2, C6.11D122C2, C6.Dic63C2, (C22×C6).159D6, C6.108(C4○D12), (C6×C12).290C22, C35(C23.28D6), (C2×C62).111C22, C22.9(C327D4), C2.18(C12.59D6), C3222(C22.D4), (C2×C6×C12)⋊4C2, (C22×C4)⋊5(C3⋊S3), (C3×C6).276(C2×D4), C6.117(C2×C3⋊D4), C23.28(C2×C3⋊S3), C2.6(C2×C327D4), (C2×C6).97(C3⋊D4), (C3×C6).123(C4○D4), (C2×C6).267(C22×S3), (C2×C327D4).13C2, C22.55(C22×C3⋊S3), (C22×C3⋊S3).44C22, (C2×C3⋊Dic3).90C22, (C2×C4).68(C2×C3⋊S3), SmallGroup(288,786)

Series: Derived Chief Lower central Upper central

C1C62 — C62.129D4
C1C3C32C3×C6C62C22×C3⋊S3C2×C327D4 — C62.129D4
C32C62 — C62.129D4
C1C22C22×C4

Generators and relations for C62.129D4
 G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=dad=a-1b3, cbc-1=dbd=b-1, dcd=b3c-1 >

Subgroups: 844 in 234 conjugacy classes, 77 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×3], C3 [×4], C4 [×5], C22, C22 [×2], C22 [×5], S3 [×4], C6 [×12], C6 [×8], C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, C32, Dic3 [×12], C12 [×8], D6 [×12], C2×C6 [×12], C2×C6 [×8], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, C3⋊S3, C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3 [×12], C3⋊D4 [×8], C2×C12 [×8], C2×C12 [×8], C22×S3 [×4], C22×C6 [×4], C22.D4, C3⋊Dic3 [×3], C3×C12 [×2], C2×C3⋊S3 [×3], C62, C62 [×2], C62 [×2], Dic3⋊C4 [×8], D6⋊C4 [×8], C6.D4 [×4], C2×C3⋊D4 [×4], C22×C12 [×4], C2×C3⋊Dic3, C2×C3⋊Dic3 [×2], C327D4 [×2], C6×C12 [×2], C6×C12 [×2], C22×C3⋊S3, C2×C62, C23.28D6 [×4], C6.Dic6 [×2], C6.11D12 [×2], C625C4, C2×C327D4, C2×C6×C12, C62.129D4
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], C4○D12 [×8], C2×C3⋊D4 [×4], C327D4 [×2], C22×C3⋊S3, C23.28D6 [×4], C12.59D6 [×2], C2×C327D4, C62.129D4

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

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

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

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

78 conjugacy classes

class 1 2A2B2C2D2E2F3A3B3C3D4A4B4C4D4E4F4G6A···6AB12A···12AF
order1222222333344444446···612···12
size11112236222222223636362···22···2

78 irreducible representations

dim1111112222222
type++++++++++
imageC1C2C2C2C2C2S3D4D6D6C4○D4C3⋊D4C4○D12
kernelC62.129D4C6.Dic6C6.11D12C625C4C2×C327D4C2×C6×C12C22×C12C62C2×C12C22×C6C3×C6C2×C6C6
# reps122111428441632

Matrix representation of C62.129D4 in GL6(𝔽13)

630000
1070000
001100
0012000
00001212
000010
,
1200000
0120000
001000
000100
00001212
000010
,
500000
680000
001000
00121200
0000119
0000112
,
100000
9120000
001000
00121200
000010
00001212

G:=sub<GL(6,GF(13))| [6,10,0,0,0,0,3,7,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[5,6,0,0,0,0,0,8,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,11,11,0,0,0,0,9,2],[1,9,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12] >;

C62.129D4 in GAP, Magma, Sage, TeX

C_6^2._{129}D_4
% in TeX

G:=Group("C6^2.129D4");
// GroupNames label

G:=SmallGroup(288,786);
// by ID

G=gap.SmallGroup(288,786);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,254,100,2693,9414]);
// Polycyclic

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

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