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G = C62.20D6order 432 = 24·33

3rd non-split extension by C62 of D6 acting via D6/C2=S3

metabelian, supersoluble, monomial

Aliases: C62.20D6, (C3×C12)⋊1C12, C4⋊(C32⋊C12), He37(C4⋊C4), (C6×C12).9S3, (C6×C12).5C6, (C4×He3)⋊3C4, C12⋊Dic3⋊C3, (C3×C12)⋊1Dic3, (C3×C6).19D12, C6.12(C3×D12), C62.6(C2×C6), (C2×He3).5Q8, C6.6(C3×Dic6), (C3×C6).6Dic6, (C2×He3).20D4, C6.13(C6×Dic3), C12.4(C3×Dic3), C2.1(He34D4), C324(C4⋊Dic3), C2.2(He33Q8), (C22×He3).18C22, C324(C3×C4⋊C4), (C3×C6).9(C3×D4), (C2×C4×He3).6C2, (C3×C6).3(C3×Q8), (C3×C6).8(C2×C12), (C2×C6).40(S3×C6), C3.2(C3×C4⋊Dic3), (C2×C12).12(C3×S3), C2.4(C2×C32⋊C12), (C2×C3⋊Dic3).2C6, (C3×C6).9(C2×Dic3), (C2×C32⋊C12).2C2, (C2×C4).3(C32⋊C6), (C2×He3).29(C2×C4), C22.5(C2×C32⋊C6), SmallGroup(432,140)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C62.20D6
C1C3C32C3×C6C62C22×He3C2×C32⋊C12 — C62.20D6
C32C3×C6 — C62.20D6
C1C22C2×C4

Generators and relations for C62.20D6
 G = < a,b,c,d | a6=b6=1, c6=b3, d2=a3, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=b3c5 >

Subgroups: 397 in 107 conjugacy classes, 46 normal (32 characteristic)
C1, C2 [×3], C3, C3 [×3], C4 [×2], C4 [×2], C22, C6 [×3], C6 [×9], C2×C4, C2×C4 [×2], C32 [×2], C32, Dic3 [×4], C12 [×2], C12 [×8], C2×C6, C2×C6 [×3], C4⋊C4, C3×C6 [×6], C3×C6 [×3], C2×Dic3 [×4], C2×C12, C2×C12 [×5], He3, C3×Dic3 [×2], C3⋊Dic3 [×2], C3×C12 [×4], C3×C12 [×2], C62 [×2], C62, C4⋊Dic3 [×2], C3×C4⋊C4, C2×He3 [×3], C6×Dic3 [×2], C2×C3⋊Dic3 [×2], C6×C12 [×2], C6×C12, C32⋊C12 [×2], C4×He3 [×2], C22×He3, C3×C4⋊Dic3, C12⋊Dic3, C2×C32⋊C12 [×2], C2×C4×He3, C62.20D6
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4, Q8, Dic3 [×2], C12 [×2], D6, C2×C6, C4⋊C4, C3×S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C3×Dic3 [×2], S3×C6, C4⋊Dic3, C3×C4⋊C4, C32⋊C6, C3×Dic6, C3×D12, C6×Dic3, C32⋊C12 [×2], C2×C32⋊C6, C3×C4⋊Dic3, He33Q8, He34D4, C2×C32⋊C12, C62.20D6

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

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

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

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

62 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F4A4B4C4D4E4F6A6B6C6D···6I6J···6R12A12B12C12D12E···12T12U···12AB
order12223333334444446666···66···61212121212···1212···12
size111123366622181818182223···36···622226···618···18

62 irreducible representations

dim111111112222222222222266666
type+++++--+-++-+-+
imageC1C2C2C3C4C6C6C12S3D4Q8Dic3D6C3×S3Dic6D12C3×D4C3×Q8C3×Dic3S3×C6C3×Dic6C3×D12C32⋊C6C32⋊C12C2×C32⋊C6He33Q8He34D4
kernelC62.20D6C2×C32⋊C12C2×C4×He3C12⋊Dic3C4×He3C2×C3⋊Dic3C6×C12C3×C12C6×C12C2×He3C2×He3C3×C12C62C2×C12C3×C6C3×C6C3×C6C3×C6C12C2×C6C6C6C2×C4C4C22C2C2
# reps121244281112122222424412122

Matrix representation of C62.20D6 in GL8(𝔽13)

10000000
01000000
000000120
000000012
001200000
000120000
000012000
000001200
,
120000000
012000000
00010000
001210000
00000100
000012100
00000001
000000121
,
63000000
103000000
00370000
006100000
00007300
0000101000
00000033
000000106
,
36000000
310000000
00800000
00850000
00000500
00005000
00000058
00000008

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

C62.20D6 in GAP, Magma, Sage, TeX

C_6^2._{20}D_6
% in TeX

G:=Group("C6^2.20D6");
// GroupNames label

G:=SmallGroup(432,140);
// by ID

G=gap.SmallGroup(432,140);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,176,4037,2035,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=1,c^6=b^3,d^2=a^3,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^3*c^5>;
// generators/relations

׿
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