direct product, non-abelian, soluble
Aliases: C7×CSU2(𝔽3), C14.4S4, SL2(𝔽3).C14, Q8.(S3×C7), C2.2(C7×S4), (C7×Q8).2S3, (C7×SL2(𝔽3)).2C2, SmallGroup(336,115)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C7×SL2(𝔽3) — C7×CSU2(𝔽3) |
| SL2(𝔽3) — C7×CSU2(𝔽3) |
Generators and relations for C7×CSU2(𝔽3)
G = < a,b,c,d,e | a7=b4=d3=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece-1=b-1, dbd-1=bc, ebe-1=b2c, dcd-1=b, ede-1=d-1 >
Smallest permutation representation of C7×CSU2(𝔽3)
►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 50 77 59)(2 51 71 60)(3 52 72 61)(4 53 73 62)(5 54 74 63)(6 55 75 57)(7 56 76 58)(8 29 39 21)(9 30 40 15)(10 31 41 16)(11 32 42 17)(12 33 36 18)(13 34 37 19)(14 35 38 20)(22 99 108 43)(23 100 109 44)(24 101 110 45)(25 102 111 46)(26 103 112 47)(27 104 106 48)(28 105 107 49)(64 84 96 86)(65 78 97 87)(66 79 98 88)(67 80 92 89)(68 81 93 90)(69 82 94 91)(70 83 95 85)
(1 85 77 83)(2 86 71 84)(3 87 72 78)(4 88 73 79)(5 89 74 80)(6 90 75 81)(7 91 76 82)(8 27 39 106)(9 28 40 107)(10 22 41 108)(11 23 42 109)(12 24 36 110)(13 25 37 111)(14 26 38 112)(15 105 30 49)(16 99 31 43)(17 100 32 44)(18 101 33 45)(19 102 34 46)(20 103 35 47)(21 104 29 48)(50 95 59 70)(51 96 60 64)(52 97 61 65)(53 98 62 66)(54 92 63 67)(55 93 57 68)(56 94 58 69)
(15 107 105)(16 108 99)(17 109 100)(18 110 101)(19 111 102)(20 112 103)(21 106 104)(22 43 31)(23 44 32)(24 45 33)(25 46 34)(26 47 35)(27 48 29)(28 49 30)(50 85 95)(51 86 96)(52 87 97)(53 88 98)(54 89 92)(55 90 93)(56 91 94)(57 81 68)(58 82 69)(59 83 70)(60 84 64)(61 78 65)(62 79 66)(63 80 67)
(1 39 77 8)(2 40 71 9)(3 41 72 10)(4 42 73 11)(5 36 74 12)(6 37 75 13)(7 38 76 14)(15 86 30 84)(16 87 31 78)(17 88 32 79)(18 89 33 80)(19 90 34 81)(20 91 35 82)(21 85 29 83)(22 61 108 52)(23 62 109 53)(24 63 110 54)(25 57 111 55)(26 58 112 56)(27 59 106 50)(28 60 107 51)(43 65 99 97)(44 66 100 98)(45 67 101 92)(46 68 102 93)(47 69 103 94)(48 70 104 95)(49 64 105 96)
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), (1,50,77,59)(2,51,71,60)(3,52,72,61)(4,53,73,62)(5,54,74,63)(6,55,75,57)(7,56,76,58)(8,29,39,21)(9,30,40,15)(10,31,41,16)(11,32,42,17)(12,33,36,18)(13,34,37,19)(14,35,38,20)(22,99,108,43)(23,100,109,44)(24,101,110,45)(25,102,111,46)(26,103,112,47)(27,104,106,48)(28,105,107,49)(64,84,96,86)(65,78,97,87)(66,79,98,88)(67,80,92,89)(68,81,93,90)(69,82,94,91)(70,83,95,85), (1,85,77,83)(2,86,71,84)(3,87,72,78)(4,88,73,79)(5,89,74,80)(6,90,75,81)(7,91,76,82)(8,27,39,106)(9,28,40,107)(10,22,41,108)(11,23,42,109)(12,24,36,110)(13,25,37,111)(14,26,38,112)(15,105,30,49)(16,99,31,43)(17,100,32,44)(18,101,33,45)(19,102,34,46)(20,103,35,47)(21,104,29,48)(50,95,59,70)(51,96,60,64)(52,97,61,65)(53,98,62,66)(54,92,63,67)(55,93,57,68)(56,94,58,69), (15,107,105)(16,108,99)(17,109,100)(18,110,101)(19,111,102)(20,112,103)(21,106,104)(22,43,31)(23,44,32)(24,45,33)(25,46,34)(26,47,35)(27,48,29)(28,49,30)(50,85,95)(51,86,96)(52,87,97)(53,88,98)(54,89,92)(55,90,93)(56,91,94)(57,81,68)(58,82,69)(59,83,70)(60,84,64)(61,78,65)(62,79,66)(63,80,67), (1,39,77,8)(2,40,71,9)(3,41,72,10)(4,42,73,11)(5,36,74,12)(6,37,75,13)(7,38,76,14)(15,86,30,84)(16,87,31,78)(17,88,32,79)(18,89,33,80)(19,90,34,81)(20,91,35,82)(21,85,29,83)(22,61,108,52)(23,62,109,53)(24,63,110,54)(25,57,111,55)(26,58,112,56)(27,59,106,50)(28,60,107,51)(43,65,99,97)(44,66,100,98)(45,67,101,92)(46,68,102,93)(47,69,103,94)(48,70,104,95)(49,64,105,96)>;
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), (1,50,77,59)(2,51,71,60)(3,52,72,61)(4,53,73,62)(5,54,74,63)(6,55,75,57)(7,56,76,58)(8,29,39,21)(9,30,40,15)(10,31,41,16)(11,32,42,17)(12,33,36,18)(13,34,37,19)(14,35,38,20)(22,99,108,43)(23,100,109,44)(24,101,110,45)(25,102,111,46)(26,103,112,47)(27,104,106,48)(28,105,107,49)(64,84,96,86)(65,78,97,87)(66,79,98,88)(67,80,92,89)(68,81,93,90)(69,82,94,91)(70,83,95,85), (1,85,77,83)(2,86,71,84)(3,87,72,78)(4,88,73,79)(5,89,74,80)(6,90,75,81)(7,91,76,82)(8,27,39,106)(9,28,40,107)(10,22,41,108)(11,23,42,109)(12,24,36,110)(13,25,37,111)(14,26,38,112)(15,105,30,49)(16,99,31,43)(17,100,32,44)(18,101,33,45)(19,102,34,46)(20,103,35,47)(21,104,29,48)(50,95,59,70)(51,96,60,64)(52,97,61,65)(53,98,62,66)(54,92,63,67)(55,93,57,68)(56,94,58,69), (15,107,105)(16,108,99)(17,109,100)(18,110,101)(19,111,102)(20,112,103)(21,106,104)(22,43,31)(23,44,32)(24,45,33)(25,46,34)(26,47,35)(27,48,29)(28,49,30)(50,85,95)(51,86,96)(52,87,97)(53,88,98)(54,89,92)(55,90,93)(56,91,94)(57,81,68)(58,82,69)(59,83,70)(60,84,64)(61,78,65)(62,79,66)(63,80,67), (1,39,77,8)(2,40,71,9)(3,41,72,10)(4,42,73,11)(5,36,74,12)(6,37,75,13)(7,38,76,14)(15,86,30,84)(16,87,31,78)(17,88,32,79)(18,89,33,80)(19,90,34,81)(20,91,35,82)(21,85,29,83)(22,61,108,52)(23,62,109,53)(24,63,110,54)(25,57,111,55)(26,58,112,56)(27,59,106,50)(28,60,107,51)(43,65,99,97)(44,66,100,98)(45,67,101,92)(46,68,102,93)(47,69,103,94)(48,70,104,95)(49,64,105,96) );
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)], [(1,50,77,59),(2,51,71,60),(3,52,72,61),(4,53,73,62),(5,54,74,63),(6,55,75,57),(7,56,76,58),(8,29,39,21),(9,30,40,15),(10,31,41,16),(11,32,42,17),(12,33,36,18),(13,34,37,19),(14,35,38,20),(22,99,108,43),(23,100,109,44),(24,101,110,45),(25,102,111,46),(26,103,112,47),(27,104,106,48),(28,105,107,49),(64,84,96,86),(65,78,97,87),(66,79,98,88),(67,80,92,89),(68,81,93,90),(69,82,94,91),(70,83,95,85)], [(1,85,77,83),(2,86,71,84),(3,87,72,78),(4,88,73,79),(5,89,74,80),(6,90,75,81),(7,91,76,82),(8,27,39,106),(9,28,40,107),(10,22,41,108),(11,23,42,109),(12,24,36,110),(13,25,37,111),(14,26,38,112),(15,105,30,49),(16,99,31,43),(17,100,32,44),(18,101,33,45),(19,102,34,46),(20,103,35,47),(21,104,29,48),(50,95,59,70),(51,96,60,64),(52,97,61,65),(53,98,62,66),(54,92,63,67),(55,93,57,68),(56,94,58,69)], [(15,107,105),(16,108,99),(17,109,100),(18,110,101),(19,111,102),(20,112,103),(21,106,104),(22,43,31),(23,44,32),(24,45,33),(25,46,34),(26,47,35),(27,48,29),(28,49,30),(50,85,95),(51,86,96),(52,87,97),(53,88,98),(54,89,92),(55,90,93),(56,91,94),(57,81,68),(58,82,69),(59,83,70),(60,84,64),(61,78,65),(62,79,66),(63,80,67)], [(1,39,77,8),(2,40,71,9),(3,41,72,10),(4,42,73,11),(5,36,74,12),(6,37,75,13),(7,38,76,14),(15,86,30,84),(16,87,31,78),(17,88,32,79),(18,89,33,80),(19,90,34,81),(20,91,35,82),(21,85,29,83),(22,61,108,52),(23,62,109,53),(24,63,110,54),(25,57,111,55),(26,58,112,56),(27,59,106,50),(28,60,107,51),(43,65,99,97),(44,66,100,98),(45,67,101,92),(46,68,102,93),(47,69,103,94),(48,70,104,95),(49,64,105,96)]])
56 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 7A | ··· | 7F | 8A | 8B | 14A | ··· | 14F | 21A | ··· | 21F | 28A | ··· | 28F | 28G | ··· | 28L | 42A | ··· | 42F | 56A | ··· | 56L |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 7 | ··· | 7 | 8 | 8 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 28 | ··· | 28 | 42 | ··· | 42 | 56 | ··· | 56 |
| size | 1 | 1 | 8 | 6 | 12 | 8 | 1 | ··· | 1 | 6 | 6 | 1 | ··· | 1 | 8 | ··· | 8 | 6 | ··· | 6 | 12 | ··· | 12 | 8 | ··· | 8 | 6 | ··· | 6 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 |
| type | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C7 | C14 | S3 | S3×C7 | CSU2(𝔽3) | C7×CSU2(𝔽3) | S4 | C7×S4 | CSU2(𝔽3) | C7×CSU2(𝔽3) |
| kernel | C7×CSU2(𝔽3) | C7×SL2(𝔽3) | CSU2(𝔽3) | SL2(𝔽3) | C7×Q8 | Q8 | C7 | C1 | C14 | C2 | C7 | C1 |
| # reps | 1 | 1 | 6 | 6 | 1 | 6 | 2 | 12 | 2 | 12 | 1 | 6 |
Matrix representation of C7×CSU2(𝔽3) ►in GL2(𝔽337) generated by
| 175 | 0 |
| 0 | 175 |
| 49 | 38 |
| 327 | 288 |
| 326 | 287 |
| 299 | 11 |
| 336 | 336 |
| 1 | 0 |
| 106 | 133 |
| 27 | 231 |
G:=sub<GL(2,GF(337))| [175,0,0,175],[49,327,38,288],[326,299,287,11],[336,1,336,0],[106,27,133,231] >;
C7×CSU2(𝔽3) in GAP, Magma, Sage, TeX
C_7\times {\rm CSU}_2({\mathbb F}_3) % in TeX
G:=Group("C7xCSU(2,3)"); // GroupNames label
G:=SmallGroup(336,115);
// by ID
G=gap.SmallGroup(336,115);
# by ID
G:=PCGroup([6,-2,-7,-3,-2,2,-2,1008,506,2019,447,117,1264,286,202,88]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=d^3=1,c^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*c*e^-1=b^-1,d*b*d^-1=b*c,e*b*e^-1=b^2*c,d*c*d^-1=b,e*d*e^-1=d^-1>;
// generators/relations
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