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G = C6.D28order 336 = 24·3·7

2nd non-split extension by C6 of D28 acting via D28/D14=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.6D4, C6.7D28, C213SD16, D28.1S3, C28.23D6, C12.5D14, Dic425C2, C84.9C22, C3⋊C82D7, C4.2(S3×D7), C33(C56⋊C2), C71(D4.S3), (C3×D28).1C2, C14.2(C3⋊D4), C2.5(C3⋊D28), (C7×C3⋊C8)⋊2C2, SmallGroup(336,34)

Series: Derived Chief Lower central Upper central

C1C84 — C6.D28
C1C7C21C42C84C3×D28 — C6.D28
C21C42C84 — C6.D28
C1C2C4

Generators and relations for C6.D28
 G = < a,b,c | a6=1, b28=c2=a3, bab-1=cac-1=a-1, cbc-1=b27 >

28C2
14C22
42C4
28C6
4D7
3C8
7D4
21Q8
14Dic3
14C2×C6
2D14
6Dic7
4C3×D7
21SD16
7Dic6
7C3×D4
3C56
3Dic14
2C6×D7
2Dic21
7D4.S3
3C56⋊C2

Smallest permutation representation of C6.D28
On 168 points
Generators in S168
(1 68 126 29 96 154)(2 155 97 30 127 69)(3 70 128 31 98 156)(4 157 99 32 129 71)(5 72 130 33 100 158)(6 159 101 34 131 73)(7 74 132 35 102 160)(8 161 103 36 133 75)(9 76 134 37 104 162)(10 163 105 38 135 77)(11 78 136 39 106 164)(12 165 107 40 137 79)(13 80 138 41 108 166)(14 167 109 42 139 81)(15 82 140 43 110 168)(16 113 111 44 141 83)(17 84 142 45 112 114)(18 115 57 46 143 85)(19 86 144 47 58 116)(20 117 59 48 145 87)(21 88 146 49 60 118)(22 119 61 50 147 89)(23 90 148 51 62 120)(24 121 63 52 149 91)(25 92 150 53 64 122)(26 123 65 54 151 93)(27 94 152 55 66 124)(28 125 67 56 153 95)
(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)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)
(1 28 29 56)(2 55 30 27)(3 26 31 54)(4 53 32 25)(5 24 33 52)(6 51 34 23)(7 22 35 50)(8 49 36 21)(9 20 37 48)(10 47 38 19)(11 18 39 46)(12 45 40 17)(13 16 41 44)(14 43 42 15)(57 78 85 106)(58 105 86 77)(59 76 87 104)(60 103 88 75)(61 74 89 102)(62 101 90 73)(63 72 91 100)(64 99 92 71)(65 70 93 98)(66 97 94 69)(67 68 95 96)(79 112 107 84)(80 83 108 111)(81 110 109 82)(113 138 141 166)(114 165 142 137)(115 136 143 164)(116 163 144 135)(117 134 145 162)(118 161 146 133)(119 132 147 160)(120 159 148 131)(121 130 149 158)(122 157 150 129)(123 128 151 156)(124 155 152 127)(125 126 153 154)(139 168 167 140)

G:=sub<Sym(168)| (1,68,126,29,96,154)(2,155,97,30,127,69)(3,70,128,31,98,156)(4,157,99,32,129,71)(5,72,130,33,100,158)(6,159,101,34,131,73)(7,74,132,35,102,160)(8,161,103,36,133,75)(9,76,134,37,104,162)(10,163,105,38,135,77)(11,78,136,39,106,164)(12,165,107,40,137,79)(13,80,138,41,108,166)(14,167,109,42,139,81)(15,82,140,43,110,168)(16,113,111,44,141,83)(17,84,142,45,112,114)(18,115,57,46,143,85)(19,86,144,47,58,116)(20,117,59,48,145,87)(21,88,146,49,60,118)(22,119,61,50,147,89)(23,90,148,51,62,120)(24,121,63,52,149,91)(25,92,150,53,64,122)(26,123,65,54,151,93)(27,94,152,55,66,124)(28,125,67,56,153,95), (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)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,28,29,56)(2,55,30,27)(3,26,31,54)(4,53,32,25)(5,24,33,52)(6,51,34,23)(7,22,35,50)(8,49,36,21)(9,20,37,48)(10,47,38,19)(11,18,39,46)(12,45,40,17)(13,16,41,44)(14,43,42,15)(57,78,85,106)(58,105,86,77)(59,76,87,104)(60,103,88,75)(61,74,89,102)(62,101,90,73)(63,72,91,100)(64,99,92,71)(65,70,93,98)(66,97,94,69)(67,68,95,96)(79,112,107,84)(80,83,108,111)(81,110,109,82)(113,138,141,166)(114,165,142,137)(115,136,143,164)(116,163,144,135)(117,134,145,162)(118,161,146,133)(119,132,147,160)(120,159,148,131)(121,130,149,158)(122,157,150,129)(123,128,151,156)(124,155,152,127)(125,126,153,154)(139,168,167,140)>;

G:=Group( (1,68,126,29,96,154)(2,155,97,30,127,69)(3,70,128,31,98,156)(4,157,99,32,129,71)(5,72,130,33,100,158)(6,159,101,34,131,73)(7,74,132,35,102,160)(8,161,103,36,133,75)(9,76,134,37,104,162)(10,163,105,38,135,77)(11,78,136,39,106,164)(12,165,107,40,137,79)(13,80,138,41,108,166)(14,167,109,42,139,81)(15,82,140,43,110,168)(16,113,111,44,141,83)(17,84,142,45,112,114)(18,115,57,46,143,85)(19,86,144,47,58,116)(20,117,59,48,145,87)(21,88,146,49,60,118)(22,119,61,50,147,89)(23,90,148,51,62,120)(24,121,63,52,149,91)(25,92,150,53,64,122)(26,123,65,54,151,93)(27,94,152,55,66,124)(28,125,67,56,153,95), (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)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,28,29,56)(2,55,30,27)(3,26,31,54)(4,53,32,25)(5,24,33,52)(6,51,34,23)(7,22,35,50)(8,49,36,21)(9,20,37,48)(10,47,38,19)(11,18,39,46)(12,45,40,17)(13,16,41,44)(14,43,42,15)(57,78,85,106)(58,105,86,77)(59,76,87,104)(60,103,88,75)(61,74,89,102)(62,101,90,73)(63,72,91,100)(64,99,92,71)(65,70,93,98)(66,97,94,69)(67,68,95,96)(79,112,107,84)(80,83,108,111)(81,110,109,82)(113,138,141,166)(114,165,142,137)(115,136,143,164)(116,163,144,135)(117,134,145,162)(118,161,146,133)(119,132,147,160)(120,159,148,131)(121,130,149,158)(122,157,150,129)(123,128,151,156)(124,155,152,127)(125,126,153,154)(139,168,167,140) );

G=PermutationGroup([(1,68,126,29,96,154),(2,155,97,30,127,69),(3,70,128,31,98,156),(4,157,99,32,129,71),(5,72,130,33,100,158),(6,159,101,34,131,73),(7,74,132,35,102,160),(8,161,103,36,133,75),(9,76,134,37,104,162),(10,163,105,38,135,77),(11,78,136,39,106,164),(12,165,107,40,137,79),(13,80,138,41,108,166),(14,167,109,42,139,81),(15,82,140,43,110,168),(16,113,111,44,141,83),(17,84,142,45,112,114),(18,115,57,46,143,85),(19,86,144,47,58,116),(20,117,59,48,145,87),(21,88,146,49,60,118),(22,119,61,50,147,89),(23,90,148,51,62,120),(24,121,63,52,149,91),(25,92,150,53,64,122),(26,123,65,54,151,93),(27,94,152,55,66,124),(28,125,67,56,153,95)], [(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),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)], [(1,28,29,56),(2,55,30,27),(3,26,31,54),(4,53,32,25),(5,24,33,52),(6,51,34,23),(7,22,35,50),(8,49,36,21),(9,20,37,48),(10,47,38,19),(11,18,39,46),(12,45,40,17),(13,16,41,44),(14,43,42,15),(57,78,85,106),(58,105,86,77),(59,76,87,104),(60,103,88,75),(61,74,89,102),(62,101,90,73),(63,72,91,100),(64,99,92,71),(65,70,93,98),(66,97,94,69),(67,68,95,96),(79,112,107,84),(80,83,108,111),(81,110,109,82),(113,138,141,166),(114,165,142,137),(115,136,143,164),(116,163,144,135),(117,134,145,162),(118,161,146,133),(119,132,147,160),(120,159,148,131),(121,130,149,158),(122,157,150,129),(123,128,151,156),(124,155,152,127),(125,126,153,154),(139,168,167,140)])

48 conjugacy classes

class 1 2A2B 3 4A4B6A6B6C7A7B7C8A8B 12 14A14B14C21A21B21C28A···28F42A42B42C56A···56L84A···84F
order122344666777881214141421212128···2842424256···5684···84
size11282284228282226642224442···24446···64···4

48 irreducible representations

dim11112222222224444
type++++++++++-++-
imageC1C2C2C2S3D4D6D7SD16C3⋊D4D14D28C56⋊C2D4.S3S3×D7C3⋊D28C6.D28
kernelC6.D28C7×C3⋊C8C3×D28Dic42D28C42C28C3⋊C8C21C14C12C6C3C7C4C2C1
# reps111111132236121336

Matrix representation of C6.D28 in GL6(𝔽337)

100000
010000
00336000
00033600
00002080
0000178128
,
1091430000
19410000
00029000
004319600
0000165331
0000269172
,
1091430000
2282280000
00029000
00294000
0000165331
000044172

G:=sub<GL(6,GF(337))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,336,0,0,0,0,0,0,336,0,0,0,0,0,0,208,178,0,0,0,0,0,128],[109,194,0,0,0,0,143,1,0,0,0,0,0,0,0,43,0,0,0,0,290,196,0,0,0,0,0,0,165,269,0,0,0,0,331,172],[109,228,0,0,0,0,143,228,0,0,0,0,0,0,0,294,0,0,0,0,290,0,0,0,0,0,0,0,165,44,0,0,0,0,331,172] >;

C6.D28 in GAP, Magma, Sage, TeX

C_6.D_{28}
% in TeX

G:=Group("C6.D28");
// GroupNames label

G:=SmallGroup(336,34);
// by ID

G=gap.SmallGroup(336,34);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,73,31,218,50,490,10373]);
// Polycyclic

G:=Group<a,b,c|a^6=1,b^28=c^2=a^3,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=b^27>;
// generators/relations

Export

Subgroup lattice of C6.D28 in TeX

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