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G = C62.78D6order 432 = 24·33

26th non-split extension by C62 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C62.78D6, (C6×Dic3)⋊3S3, (C3×C6).41D12, C6.9(S3×Dic3), C31(D6⋊Dic3), C328(D6⋊C4), (C32×C6).42D4, C3311(C22⋊C4), C6.16(C12⋊S3), C2.2(C336D4), C2.1(C338D4), (C3×C62).8C22, C6.12(C3⋊D12), C6.25(D6⋊S3), C33(C6.11D12), C6.12(C327D4), C329(C6.D4), (C6×C3⋊S3)⋊2C4, (C2×C6).32S32, C6.19(C4×C3⋊S3), (Dic3×C3×C6)⋊3C2, (C3×C6).49(C4×S3), (C2×C3⋊S3)⋊3Dic3, C22.6(S3×C3⋊S3), C2.4(Dic3×C3⋊S3), (C2×C335C4)⋊2C2, (C22×C3⋊S3).4S3, (C2×Dic3)⋊1(C3⋊S3), (C3×C6).61(C3⋊D4), (C32×C6).39(C2×C4), (C3×C6).52(C2×Dic3), (C2×C6×C3⋊S3).2C2, (C2×C6).14(C2×C3⋊S3), SmallGroup(432,450)

Series: Derived Chief Lower central Upper central

C1C32×C6 — C62.78D6
C1C3C32C33C32×C6C3×C62Dic3×C3×C6 — C62.78D6
C33C32×C6 — C62.78D6
C1C22

Generators and relations for C62.78D6
 G = < a,b,c,d | a6=b6=1, c6=a3, d2=a3b3, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=b3c5 >

Subgroups: 1336 in 268 conjugacy classes, 74 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C3, C3 [×4], C3 [×4], C4 [×2], C22, C22 [×4], S3 [×8], C6 [×3], C6 [×12], C6 [×14], C2×C4 [×2], C23, C32, C32 [×4], C32 [×4], Dic3 [×14], C12 [×4], D6 [×16], C2×C6, C2×C6 [×4], C2×C6 [×8], C22⋊C4, C3×S3 [×8], C3⋊S3 [×2], C3×C6 [×3], C3×C6 [×12], C3×C6 [×12], C2×Dic3, C2×Dic3 [×9], C2×C12 [×4], C22×S3 [×4], C22×C6, C33, C3×Dic3 [×4], C3⋊Dic3 [×13], C3×C12, S3×C6 [×16], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C62 [×4], C62 [×4], D6⋊C4 [×4], C6.D4, C3×C3⋊S3 [×2], C32×C6 [×3], C6×Dic3 [×4], C2×C3⋊Dic3 [×9], C6×C12, S3×C2×C6 [×4], C22×C3⋊S3, C32×Dic3, C335C4, C6×C3⋊S3 [×2], C6×C3⋊S3 [×2], C3×C62, D6⋊Dic3 [×4], C6.11D12, Dic3×C3×C6, C2×C335C4, C2×C6×C3⋊S3, C62.78D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×5], C2×C4, D4 [×2], Dic3 [×2], D6 [×5], C22⋊C4, C3⋊S3, C4×S3 [×4], D12 [×4], C2×Dic3, C3⋊D4 [×6], S32 [×4], C2×C3⋊S3, D6⋊C4 [×4], C6.D4, S3×Dic3 [×4], D6⋊S3 [×4], C3⋊D12 [×4], C4×C3⋊S3, C12⋊S3, C327D4, S3×C3⋊S3, D6⋊Dic3 [×4], C6.11D12, Dic3×C3⋊S3, C336D4, C338D4, C62.78D6

Smallest permutation representation of C62.78D6
On 144 points
Generators in S144
(1 14 26 7 20 32)(2 15 27 8 21 33)(3 16 28 9 22 34)(4 17 29 10 23 35)(5 18 30 11 24 36)(6 19 31 12 13 25)(37 90 140 43 96 134)(38 91 141 44 85 135)(39 92 142 45 86 136)(40 93 143 46 87 137)(41 94 144 47 88 138)(42 95 133 48 89 139)(49 111 105 55 117 99)(50 112 106 56 118 100)(51 113 107 57 119 101)(52 114 108 58 120 102)(53 115 97 59 109 103)(54 116 98 60 110 104)(61 83 122 67 77 128)(62 84 123 68 78 129)(63 73 124 69 79 130)(64 74 125 70 80 131)(65 75 126 71 81 132)(66 76 127 72 82 121)
(1 108 16 60 30 112)(2 113 31 49 17 97)(3 98 18 50 32 114)(4 115 33 51 19 99)(5 100 20 52 34 116)(6 117 35 53 21 101)(7 102 22 54 36 118)(8 119 25 55 23 103)(9 104 24 56 26 120)(10 109 27 57 13 105)(11 106 14 58 28 110)(12 111 29 59 15 107)(37 73 92 126 144 61)(38 62 133 127 93 74)(39 75 94 128 134 63)(40 64 135 129 95 76)(41 77 96 130 136 65)(42 66 137 131 85 78)(43 79 86 132 138 67)(44 68 139 121 87 80)(45 81 88 122 140 69)(46 70 141 123 89 82)(47 83 90 124 142 71)(48 72 143 125 91 84)
(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 77 54 142)(2 141 55 76)(3 75 56 140)(4 139 57 74)(5 73 58 138)(6 137 59 84)(7 83 60 136)(8 135 49 82)(9 81 50 134)(10 133 51 80)(11 79 52 144)(12 143 53 78)(13 93 115 68)(14 67 116 92)(15 91 117 66)(16 65 118 90)(17 89 119 64)(18 63 120 88)(19 87 109 62)(20 61 110 86)(21 85 111 72)(22 71 112 96)(23 95 113 70)(24 69 114 94)(25 40 97 123)(26 122 98 39)(27 38 99 121)(28 132 100 37)(29 48 101 131)(30 130 102 47)(31 46 103 129)(32 128 104 45)(33 44 105 127)(34 126 106 43)(35 42 107 125)(36 124 108 41)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E3A···3E3F3G3H3I4A4B4C4D6A···6O6P···6AA6AB6AC6AD6AE12A···12P
order1222223···3333344446···66···6666612···12
size111118182···244446654542···24···4181818186···6

66 irreducible representations

dim11111222222224444
type+++++++-+++--+
imageC1C2C2C2C4S3S3D4Dic3D6C4×S3D12C3⋊D4S32S3×Dic3D6⋊S3C3⋊D12
kernelC62.78D6Dic3×C3×C6C2×C335C4C2×C6×C3⋊S3C6×C3⋊S3C6×Dic3C22×C3⋊S3C32×C6C2×C3⋊S3C62C3×C6C3×C6C3×C6C2×C6C6C6C6
# reps111144122588124444

Matrix representation of C62.78D6 in GL8(𝔽13)

120000000
012000000
00300000
00090000
000001200
00001100
00000010
00000001
,
120000000
012000000
00100000
00010000
00001000
00000100
000000121
000000120
,
01000000
120000000
00300000
00090000
00008000
00000800
00000001
00000010
,
10000000
012000000
00090000
00300000
00005000
00008800
00000001
00000010

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

C62.78D6 in GAP, Magma, Sage, TeX

C_6^2._{78}D_6
% in TeX

G:=Group("C6^2.78D6");
// GroupNames label

G:=SmallGroup(432,450);
// by ID

G=gap.SmallGroup(432,450);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,571,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=1,c^6=a^3,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^5>;
// generators/relations

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