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G = C62.77D6order 432 = 24·33

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

metabelian, supersoluble, monomial

Aliases: C62.77D6, D6⋊(C3⋊Dic3), (S3×C6)⋊2Dic3, (C3×C6).82D12, C33(D6⋊Dic3), (S3×C62).2C2, C6.30(S3×Dic3), (C32×C6).41D4, C31(C625C4), C3215(D6⋊C4), C3310(C22⋊C4), C2.1(C336D4), C6.5(C327D4), C2.1(C337D4), (C3×C62).7C22, C6.24(D6⋊S3), C6.26(C3⋊D12), C326(C6.D4), (S3×C3×C6)⋊4C4, (C2×C6).31S32, (S3×C2×C6).8S3, (C3×C6).91(C4×S3), (C6×C3⋊Dic3)⋊2C2, (C2×C3⋊Dic3)⋊7S3, C22.5(S3×C3⋊S3), C2.4(S3×C3⋊Dic3), C6.4(C2×C3⋊Dic3), (C2×C335C4)⋊1C2, (C22×S3).(C3⋊S3), (C3×C6).60(C3⋊D4), (C32×C6).38(C2×C4), (C3×C6).38(C2×Dic3), (C2×C6).13(C2×C3⋊S3), SmallGroup(432,449)

Series: Derived Chief Lower central Upper central

C1C32×C6 — C62.77D6
C1C3C32C33C32×C6C3×C62S3×C62 — C62.77D6
C33C32×C6 — C62.77D6
C1C22

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 [×3], C2 [×2], C3, C3 [×4], C3 [×4], C4 [×2], C22, C22 [×4], S3 [×2], C6 [×3], C6 [×12], C6 [×20], C2×C4 [×2], C23, C32, C32 [×4], C32 [×4], Dic3 [×17], C12, D6 [×2], D6 [×2], C2×C6, C2×C6 [×4], C2×C6 [×20], C22⋊C4, C3×S3 [×8], C3×C6 [×3], C3×C6 [×12], C3×C6 [×14], C2×Dic3 [×13], C2×C12, C22×S3, C22×C6 [×4], C33, C3×Dic3 [×4], C3⋊Dic3 [×14], S3×C6 [×8], S3×C6 [×8], C62, C62 [×4], C62 [×8], D6⋊C4, C6.D4 [×4], S3×C32 [×2], C32×C6 [×3], C6×Dic3 [×4], C2×C3⋊Dic3, C2×C3⋊Dic3 [×9], S3×C2×C6 [×4], C2×C62, C3×C3⋊Dic3, C335C4, S3×C3×C6 [×2], S3×C3×C6 [×2], C3×C62, D6⋊Dic3 [×4], C625C4, C6×C3⋊Dic3, C2×C335C4, S3×C62, C62.77D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×5], C2×C4, D4 [×2], Dic3 [×8], D6 [×5], C22⋊C4, C3⋊S3, C4×S3, D12, C2×Dic3 [×4], C3⋊D4 [×9], C3⋊Dic3 [×2], S32 [×4], C2×C3⋊S3, D6⋊C4, C6.D4 [×4], S3×Dic3 [×4], D6⋊S3 [×4], C3⋊D12 [×4], C2×C3⋊Dic3, C327D4 [×2], S3×C3⋊S3, D6⋊Dic3 [×4], C625C4, S3×C3⋊Dic3, C336D4, C337D4, C62.77D6

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

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E3A···3E3F3G3H3I4A4B4C4D6A···6O6P···6AA6AB···6AQ12A12B12C12D
order1222223···3333344446···66···66···612121212
size1111662···24444181854542···24···46···618181818

66 irreducible representations

dim11111222222224444
type+++++++-+++--+
imageC1C2C2C2C4S3S3D4Dic3D6C4×S3D12C3⋊D4S32S3×Dic3D6⋊S3C3⋊D12
kernelC62.77D6C6×C3⋊Dic3C2×C335C4S3×C62S3×C3×C6C2×C3⋊Dic3S3×C2×C6C32×C6S3×C6C62C3×C6C3×C6C3×C6C2×C6C6C6C6
# reps111141428522184444

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

01000000
1212000000
001200000
000120000
000011200
00001000
00000010
00000001
,
10000000
01000000
001200000
000120000
00001000
00000100
000000121
000000120
,
012000000
11000000
00620000
00270000
00001000
00000100
00000001
00000010
,
107000000
103000000
001160000
00620000
00000500
00005000
00000001
00000010

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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