Aliases: Dic7.2A4, SL2(𝔽3)⋊2D7, Q8.(C3×D7), C7⋊3(C4.A4), C2.2(A4×D7), C14.7(C2×A4), Q8⋊2D7⋊1C3, (C7×Q8).1C6, (C7×SL2(𝔽3))⋊2C2, SmallGroup(336,131)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — C14 — C7×Q8 — C7×SL2(𝔽3) — Dic7.2A4 |
| C7×Q8 — Dic7.2A4 |
Generators and relations for Dic7.2A4
G = < a,b,c,d,e | a14=e3=1, b2=c2=d2=a7, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a7c, ece-1=a7cd, ede-1=c >
Smallest permutation representation of Dic7.2A4
►On 112 points
Generators in S112
(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)
(1 66 8 59)(2 65 9 58)(3 64 10 57)(4 63 11 70)(5 62 12 69)(6 61 13 68)(7 60 14 67)(15 74 22 81)(16 73 23 80)(17 72 24 79)(18 71 25 78)(19 84 26 77)(20 83 27 76)(21 82 28 75)(29 96 36 89)(30 95 37 88)(31 94 38 87)(32 93 39 86)(33 92 40 85)(34 91 41 98)(35 90 42 97)(43 102 50 109)(44 101 51 108)(45 100 52 107)(46 99 53 106)(47 112 54 105)(48 111 55 104)(49 110 56 103)
(1 26 8 19)(2 27 9 20)(3 28 10 21)(4 15 11 22)(5 16 12 23)(6 17 13 24)(7 18 14 25)(29 55 36 48)(30 56 37 49)(31 43 38 50)(32 44 39 51)(33 45 40 52)(34 46 41 53)(35 47 42 54)(57 82 64 75)(58 83 65 76)(59 84 66 77)(60 71 67 78)(61 72 68 79)(62 73 69 80)(63 74 70 81)(85 107 92 100)(86 108 93 101)(87 109 94 102)(88 110 95 103)(89 111 96 104)(90 112 97 105)(91 99 98 106)
(1 36 8 29)(2 37 9 30)(3 38 10 31)(4 39 11 32)(5 40 12 33)(6 41 13 34)(7 42 14 35)(15 44 22 51)(16 45 23 52)(17 46 24 53)(18 47 25 54)(19 48 26 55)(20 49 27 56)(21 50 28 43)(57 94 64 87)(58 95 65 88)(59 96 66 89)(60 97 67 90)(61 98 68 91)(62 85 69 92)(63 86 70 93)(71 112 78 105)(72 99 79 106)(73 100 80 107)(74 101 81 108)(75 102 82 109)(76 103 83 110)(77 104 84 111)
(15 39 51)(16 40 52)(17 41 53)(18 42 54)(19 29 55)(20 30 56)(21 31 43)(22 32 44)(23 33 45)(24 34 46)(25 35 47)(26 36 48)(27 37 49)(28 38 50)(71 97 105)(72 98 106)(73 85 107)(74 86 108)(75 87 109)(76 88 110)(77 89 111)(78 90 112)(79 91 99)(80 92 100)(81 93 101)(82 94 102)(83 95 103)(84 96 104)
G:=sub<Sym(112)| (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), (1,66,8,59)(2,65,9,58)(3,64,10,57)(4,63,11,70)(5,62,12,69)(6,61,13,68)(7,60,14,67)(15,74,22,81)(16,73,23,80)(17,72,24,79)(18,71,25,78)(19,84,26,77)(20,83,27,76)(21,82,28,75)(29,96,36,89)(30,95,37,88)(31,94,38,87)(32,93,39,86)(33,92,40,85)(34,91,41,98)(35,90,42,97)(43,102,50,109)(44,101,51,108)(45,100,52,107)(46,99,53,106)(47,112,54,105)(48,111,55,104)(49,110,56,103), (1,26,8,19)(2,27,9,20)(3,28,10,21)(4,15,11,22)(5,16,12,23)(6,17,13,24)(7,18,14,25)(29,55,36,48)(30,56,37,49)(31,43,38,50)(32,44,39,51)(33,45,40,52)(34,46,41,53)(35,47,42,54)(57,82,64,75)(58,83,65,76)(59,84,66,77)(60,71,67,78)(61,72,68,79)(62,73,69,80)(63,74,70,81)(85,107,92,100)(86,108,93,101)(87,109,94,102)(88,110,95,103)(89,111,96,104)(90,112,97,105)(91,99,98,106), (1,36,8,29)(2,37,9,30)(3,38,10,31)(4,39,11,32)(5,40,12,33)(6,41,13,34)(7,42,14,35)(15,44,22,51)(16,45,23,52)(17,46,24,53)(18,47,25,54)(19,48,26,55)(20,49,27,56)(21,50,28,43)(57,94,64,87)(58,95,65,88)(59,96,66,89)(60,97,67,90)(61,98,68,91)(62,85,69,92)(63,86,70,93)(71,112,78,105)(72,99,79,106)(73,100,80,107)(74,101,81,108)(75,102,82,109)(76,103,83,110)(77,104,84,111), (15,39,51)(16,40,52)(17,41,53)(18,42,54)(19,29,55)(20,30,56)(21,31,43)(22,32,44)(23,33,45)(24,34,46)(25,35,47)(26,36,48)(27,37,49)(28,38,50)(71,97,105)(72,98,106)(73,85,107)(74,86,108)(75,87,109)(76,88,110)(77,89,111)(78,90,112)(79,91,99)(80,92,100)(81,93,101)(82,94,102)(83,95,103)(84,96,104)>;
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), (1,66,8,59)(2,65,9,58)(3,64,10,57)(4,63,11,70)(5,62,12,69)(6,61,13,68)(7,60,14,67)(15,74,22,81)(16,73,23,80)(17,72,24,79)(18,71,25,78)(19,84,26,77)(20,83,27,76)(21,82,28,75)(29,96,36,89)(30,95,37,88)(31,94,38,87)(32,93,39,86)(33,92,40,85)(34,91,41,98)(35,90,42,97)(43,102,50,109)(44,101,51,108)(45,100,52,107)(46,99,53,106)(47,112,54,105)(48,111,55,104)(49,110,56,103), (1,26,8,19)(2,27,9,20)(3,28,10,21)(4,15,11,22)(5,16,12,23)(6,17,13,24)(7,18,14,25)(29,55,36,48)(30,56,37,49)(31,43,38,50)(32,44,39,51)(33,45,40,52)(34,46,41,53)(35,47,42,54)(57,82,64,75)(58,83,65,76)(59,84,66,77)(60,71,67,78)(61,72,68,79)(62,73,69,80)(63,74,70,81)(85,107,92,100)(86,108,93,101)(87,109,94,102)(88,110,95,103)(89,111,96,104)(90,112,97,105)(91,99,98,106), (1,36,8,29)(2,37,9,30)(3,38,10,31)(4,39,11,32)(5,40,12,33)(6,41,13,34)(7,42,14,35)(15,44,22,51)(16,45,23,52)(17,46,24,53)(18,47,25,54)(19,48,26,55)(20,49,27,56)(21,50,28,43)(57,94,64,87)(58,95,65,88)(59,96,66,89)(60,97,67,90)(61,98,68,91)(62,85,69,92)(63,86,70,93)(71,112,78,105)(72,99,79,106)(73,100,80,107)(74,101,81,108)(75,102,82,109)(76,103,83,110)(77,104,84,111), (15,39,51)(16,40,52)(17,41,53)(18,42,54)(19,29,55)(20,30,56)(21,31,43)(22,32,44)(23,33,45)(24,34,46)(25,35,47)(26,36,48)(27,37,49)(28,38,50)(71,97,105)(72,98,106)(73,85,107)(74,86,108)(75,87,109)(76,88,110)(77,89,111)(78,90,112)(79,91,99)(80,92,100)(81,93,101)(82,94,102)(83,95,103)(84,96,104) );
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)], [(1,66,8,59),(2,65,9,58),(3,64,10,57),(4,63,11,70),(5,62,12,69),(6,61,13,68),(7,60,14,67),(15,74,22,81),(16,73,23,80),(17,72,24,79),(18,71,25,78),(19,84,26,77),(20,83,27,76),(21,82,28,75),(29,96,36,89),(30,95,37,88),(31,94,38,87),(32,93,39,86),(33,92,40,85),(34,91,41,98),(35,90,42,97),(43,102,50,109),(44,101,51,108),(45,100,52,107),(46,99,53,106),(47,112,54,105),(48,111,55,104),(49,110,56,103)], [(1,26,8,19),(2,27,9,20),(3,28,10,21),(4,15,11,22),(5,16,12,23),(6,17,13,24),(7,18,14,25),(29,55,36,48),(30,56,37,49),(31,43,38,50),(32,44,39,51),(33,45,40,52),(34,46,41,53),(35,47,42,54),(57,82,64,75),(58,83,65,76),(59,84,66,77),(60,71,67,78),(61,72,68,79),(62,73,69,80),(63,74,70,81),(85,107,92,100),(86,108,93,101),(87,109,94,102),(88,110,95,103),(89,111,96,104),(90,112,97,105),(91,99,98,106)], [(1,36,8,29),(2,37,9,30),(3,38,10,31),(4,39,11,32),(5,40,12,33),(6,41,13,34),(7,42,14,35),(15,44,22,51),(16,45,23,52),(17,46,24,53),(18,47,25,54),(19,48,26,55),(20,49,27,56),(21,50,28,43),(57,94,64,87),(58,95,65,88),(59,96,66,89),(60,97,67,90),(61,98,68,91),(62,85,69,92),(63,86,70,93),(71,112,78,105),(72,99,79,106),(73,100,80,107),(74,101,81,108),(75,102,82,109),(76,103,83,110),(77,104,84,111)], [(15,39,51),(16,40,52),(17,41,53),(18,42,54),(19,29,55),(20,30,56),(21,31,43),(22,32,44),(23,33,45),(24,34,46),(25,35,47),(26,36,48),(27,37,49),(28,38,50),(71,97,105),(72,98,106),(73,85,107),(74,86,108),(75,87,109),(76,88,110),(77,89,111),(78,90,112),(79,91,99),(80,92,100),(81,93,101),(82,94,102),(83,95,103),(84,96,104)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 7A | 7B | 7C | 12A | 12B | 12C | 12D | 14A | 14B | 14C | 21A | ··· | 21F | 28A | 28B | 28C | 42A | ··· | 42F |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 7 | 7 | 7 | 12 | 12 | 12 | 12 | 14 | 14 | 14 | 21 | ··· | 21 | 28 | 28 | 28 | 42 | ··· | 42 |
| size | 1 | 1 | 42 | 4 | 4 | 6 | 7 | 7 | 4 | 4 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 8 | ··· | 8 | 12 | 12 | 12 | 8 | ··· | 8 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 6 |
| type | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C3 | C6 | D7 | C3×D7 | C4.A4 | A4 | C2×A4 | Dic7.2A4 | Dic7.2A4 | A4×D7 |
| kernel | Dic7.2A4 | C7×SL2(𝔽3) | Q8⋊2D7 | C7×Q8 | SL2(𝔽3) | Q8 | C7 | Dic7 | C14 | C1 | C1 | C2 |
| # reps | 1 | 1 | 2 | 2 | 3 | 6 | 6 | 1 | 1 | 3 | 6 | 3 |
Matrix representation of Dic7.2A4 ►in GL4(𝔽337) generated by
| 336 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 239 | 142 |
| 189 | 0 | 0 | 0 |
| 0 | 189 | 0 | 0 |
| 0 | 0 | 188 | 136 |
| 0 | 0 | 35 | 149 |
| 0 | 1 | 0 | 0 |
| 336 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 209 | 208 | 0 | 0 |
| 208 | 128 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 208 | 128 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(337))| [336,0,0,0,0,336,0,0,0,0,1,239,0,0,2,142],[189,0,0,0,0,189,0,0,0,0,188,35,0,0,136,149],[0,336,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[209,208,0,0,208,128,0,0,0,0,1,0,0,0,0,1],[1,208,0,0,0,128,0,0,0,0,1,0,0,0,0,1] >;
Dic7.2A4 in GAP, Magma, Sage, TeX
{\rm Dic}_7._2A_4 % in TeX
G:=Group("Dic7.2A4"); // GroupNames label
G:=SmallGroup(336,131);
// by ID
G=gap.SmallGroup(336,131);
# by ID
G:=PCGroup([6,-2,-3,-2,2,-7,-2,1008,170,518,81,735,357,4324]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=e^3=1,b^2=c^2=d^2=a^7,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^7*c,e*c*e^-1=a^7*c*d,e*d*e^-1=c>;
// generators/relations
Export