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G = C62.77D6order 432 = 24·33

25th non-split extension by C62 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C62.77D6
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C3×C62 — S3×C62 — C62.77D6
 Lower central C33 — C32×C6 — C62.77D6
 Upper central C1 — C22

Generators and relations for C62.77D6
G = < a,b,c,d | a6=b6=c6=1, d2=a3b3, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=b3c-1 >

Subgroups: 1176 in 268 conjugacy classes, 82 normal (26 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C23, C32, C32, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C3×S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×S3, C22×C6, C33, C3×Dic3, C3⋊Dic3, S3×C6, S3×C6, C62, C62, C62, D6⋊C4, C6.D4, S3×C32, C32×C6, C6×Dic3, C2×C3⋊Dic3, C2×C3⋊Dic3, S3×C2×C6, C2×C62, C3×C3⋊Dic3, C335C4, S3×C3×C6, S3×C3×C6, C3×C62, D6⋊Dic3, C625C4, C6×C3⋊Dic3, C2×C335C4, S3×C62, C62.77D6
Quotients:

Smallest permutation representation of C62.77D6
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 48 31 18 38 57)(2 43 32 13 39 58)(3 44 33 14 40 59)(4 45 34 15 41 60)(5 46 35 16 42 55)(6 47 36 17 37 56)(7 138 123 139 19 29)(8 133 124 140 20 30)(9 134 125 141 21 25)(10 135 126 142 22 26)(11 136 121 143 23 27)(12 137 122 144 24 28)(49 64 93 74 90 71)(50 65 94 75 85 72)(51 66 95 76 86 67)(52 61 96 77 87 68)(53 62 91 78 88 69)(54 63 92 73 89 70)(79 103 120 114 129 98)(80 104 115 109 130 99)(81 105 116 110 131 100)(82 106 117 111 132 101)(83 107 118 112 127 102)(84 108 119 113 128 97)
(1 102 33 114 42 105)(2 97 34 109 37 106)(3 98 35 110 38 107)(4 99 36 111 39 108)(5 100 31 112 40 103)(6 101 32 113 41 104)(7 90 125 95 23 53)(8 85 126 96 24 54)(9 86 121 91 19 49)(10 87 122 92 20 50)(11 88 123 93 21 51)(12 89 124 94 22 52)(13 119 60 80 47 132)(14 120 55 81 48 127)(15 115 56 82 43 128)(16 116 57 83 44 129)(17 117 58 84 45 130)(18 118 59 79 46 131)(25 67 136 78 139 64)(26 68 137 73 140 65)(27 69 138 74 141 66)(28 70 133 75 142 61)(29 71 134 76 143 62)(30 72 135 77 144 63)
(1 135 15 19)(2 134 16 24)(3 133 17 23)(4 138 18 22)(5 137 13 21)(6 136 14 20)(7 38 142 45)(8 37 143 44)(9 42 144 43)(10 41 139 48)(11 40 140 47)(12 39 141 46)(25 55 122 32)(26 60 123 31)(27 59 124 36)(28 58 125 35)(29 57 126 34)(30 56 121 33)(49 131 77 108)(50 130 78 107)(51 129 73 106)(52 128 74 105)(53 127 75 104)(54 132 76 103)(61 113 90 81)(62 112 85 80)(63 111 86 79)(64 110 87 84)(65 109 88 83)(66 114 89 82)(67 98 92 117)(68 97 93 116)(69 102 94 115)(70 101 95 120)(71 100 96 119)(72 99 91 118)```

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A ··· 3E 3F 3G 3H 3I 4A 4B 4C 4D 6A ··· 6O 6P ··· 6AA 6AB ··· 6AQ 12A 12B 12C 12D order 1 2 2 2 2 2 3 ··· 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 12 12 12 size 1 1 1 1 6 6 2 ··· 2 4 4 4 4 18 18 54 54 2 ··· 2 4 ··· 4 6 ··· 6 18 18 18 18

66 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - + + + - - + image C1 C2 C2 C2 C4 S3 S3 D4 Dic3 D6 C4×S3 D12 C3⋊D4 S32 S3×Dic3 D6⋊S3 C3⋊D12 kernel C62.77D6 C6×C3⋊Dic3 C2×C33⋊5C4 S3×C62 S3×C3×C6 C2×C3⋊Dic3 S3×C2×C6 C32×C6 S3×C6 C62 C3×C6 C3×C6 C3×C6 C2×C6 C6 C6 C6 # reps 1 1 1 1 4 1 4 2 8 5 2 2 18 4 4 4 4

Matrix representation of C62.77D6 in GL8(𝔽13)

 0 1 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0
,
 0 12 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 6 2 0 0 0 0 0 0 2 7 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 10 7 0 0 0 0 0 0 10 3 0 0 0 0 0 0 0 0 11 6 0 0 0 0 0 0 6 2 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

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

C62.77D6 in GAP, Magma, Sage, TeX

`C_6^2._{77}D_6`
`% in TeX`

`G:=Group("C6^2.77D6");`
`// GroupNames label`

`G:=SmallGroup(432,449);`
`// by ID`

`G=gap.SmallGroup(432,449);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,571,2028,14118]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=c^6=1,d^2=a^3*b^3,a*b=b*a,a*c=c*a,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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