Copied to
clipboard

## G = C7×CSU2(𝔽3)  order 336 = 24·3·7

### Direct product of C7 and CSU2(𝔽3)

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

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C7×CSU2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C7×SL2(𝔽3) — C7×CSU2(𝔽3)
 Lower central SL2(𝔽3) — C7×CSU2(𝔽3)
 Upper central C1 — C14

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 78 59)(2 51 79 60)(3 52 80 61)(4 53 81 62)(5 54 82 63)(6 55 83 57)(7 56 84 58)(8 42 25 35)(9 36 26 29)(10 37 27 30)(11 38 28 31)(12 39 22 32)(13 40 23 33)(14 41 24 34)(15 100 109 44)(16 101 110 45)(17 102 111 46)(18 103 112 47)(19 104 106 48)(20 105 107 49)(21 99 108 43)(64 90 71 98)(65 91 72 92)(66 85 73 93)(67 86 74 94)(68 87 75 95)(69 88 76 96)(70 89 77 97)
(1 68 78 75)(2 69 79 76)(3 70 80 77)(4 64 81 71)(5 65 82 72)(6 66 83 73)(7 67 84 74)(8 44 25 100)(9 45 26 101)(10 46 27 102)(11 47 28 103)(12 48 22 104)(13 49 23 105)(14 43 24 99)(15 42 109 35)(16 36 110 29)(17 37 111 30)(18 38 112 31)(19 39 106 32)(20 40 107 33)(21 41 108 34)(50 95 59 87)(51 96 60 88)(52 97 61 89)(53 98 62 90)(54 92 63 91)(55 93 57 85)(56 94 58 86)
(8 15 44)(9 16 45)(10 17 46)(11 18 47)(12 19 48)(13 20 49)(14 21 43)(22 106 104)(23 107 105)(24 108 99)(25 109 100)(26 110 101)(27 111 102)(28 112 103)(50 68 95)(51 69 96)(52 70 97)(53 64 98)(54 65 92)(55 66 93)(56 67 94)(57 73 85)(58 74 86)(59 75 87)(60 76 88)(61 77 89)(62 71 90)(63 72 91)
(1 39 78 32)(2 40 79 33)(3 41 80 34)(4 42 81 35)(5 36 82 29)(6 37 83 30)(7 38 84 31)(8 71 25 64)(9 72 26 65)(10 73 27 66)(11 74 28 67)(12 75 22 68)(13 76 23 69)(14 77 24 70)(15 62 109 53)(16 63 110 54)(17 57 111 55)(18 58 112 56)(19 59 106 50)(20 60 107 51)(21 61 108 52)(43 89 99 97)(44 90 100 98)(45 91 101 92)(46 85 102 93)(47 86 103 94)(48 87 104 95)(49 88 105 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), (1,50,78,59)(2,51,79,60)(3,52,80,61)(4,53,81,62)(5,54,82,63)(6,55,83,57)(7,56,84,58)(8,42,25,35)(9,36,26,29)(10,37,27,30)(11,38,28,31)(12,39,22,32)(13,40,23,33)(14,41,24,34)(15,100,109,44)(16,101,110,45)(17,102,111,46)(18,103,112,47)(19,104,106,48)(20,105,107,49)(21,99,108,43)(64,90,71,98)(65,91,72,92)(66,85,73,93)(67,86,74,94)(68,87,75,95)(69,88,76,96)(70,89,77,97), (1,68,78,75)(2,69,79,76)(3,70,80,77)(4,64,81,71)(5,65,82,72)(6,66,83,73)(7,67,84,74)(8,44,25,100)(9,45,26,101)(10,46,27,102)(11,47,28,103)(12,48,22,104)(13,49,23,105)(14,43,24,99)(15,42,109,35)(16,36,110,29)(17,37,111,30)(18,38,112,31)(19,39,106,32)(20,40,107,33)(21,41,108,34)(50,95,59,87)(51,96,60,88)(52,97,61,89)(53,98,62,90)(54,92,63,91)(55,93,57,85)(56,94,58,86), (8,15,44)(9,16,45)(10,17,46)(11,18,47)(12,19,48)(13,20,49)(14,21,43)(22,106,104)(23,107,105)(24,108,99)(25,109,100)(26,110,101)(27,111,102)(28,112,103)(50,68,95)(51,69,96)(52,70,97)(53,64,98)(54,65,92)(55,66,93)(56,67,94)(57,73,85)(58,74,86)(59,75,87)(60,76,88)(61,77,89)(62,71,90)(63,72,91), (1,39,78,32)(2,40,79,33)(3,41,80,34)(4,42,81,35)(5,36,82,29)(6,37,83,30)(7,38,84,31)(8,71,25,64)(9,72,26,65)(10,73,27,66)(11,74,28,67)(12,75,22,68)(13,76,23,69)(14,77,24,70)(15,62,109,53)(16,63,110,54)(17,57,111,55)(18,58,112,56)(19,59,106,50)(20,60,107,51)(21,61,108,52)(43,89,99,97)(44,90,100,98)(45,91,101,92)(46,85,102,93)(47,86,103,94)(48,87,104,95)(49,88,105,96)>;

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), (1,50,78,59)(2,51,79,60)(3,52,80,61)(4,53,81,62)(5,54,82,63)(6,55,83,57)(7,56,84,58)(8,42,25,35)(9,36,26,29)(10,37,27,30)(11,38,28,31)(12,39,22,32)(13,40,23,33)(14,41,24,34)(15,100,109,44)(16,101,110,45)(17,102,111,46)(18,103,112,47)(19,104,106,48)(20,105,107,49)(21,99,108,43)(64,90,71,98)(65,91,72,92)(66,85,73,93)(67,86,74,94)(68,87,75,95)(69,88,76,96)(70,89,77,97), (1,68,78,75)(2,69,79,76)(3,70,80,77)(4,64,81,71)(5,65,82,72)(6,66,83,73)(7,67,84,74)(8,44,25,100)(9,45,26,101)(10,46,27,102)(11,47,28,103)(12,48,22,104)(13,49,23,105)(14,43,24,99)(15,42,109,35)(16,36,110,29)(17,37,111,30)(18,38,112,31)(19,39,106,32)(20,40,107,33)(21,41,108,34)(50,95,59,87)(51,96,60,88)(52,97,61,89)(53,98,62,90)(54,92,63,91)(55,93,57,85)(56,94,58,86), (8,15,44)(9,16,45)(10,17,46)(11,18,47)(12,19,48)(13,20,49)(14,21,43)(22,106,104)(23,107,105)(24,108,99)(25,109,100)(26,110,101)(27,111,102)(28,112,103)(50,68,95)(51,69,96)(52,70,97)(53,64,98)(54,65,92)(55,66,93)(56,67,94)(57,73,85)(58,74,86)(59,75,87)(60,76,88)(61,77,89)(62,71,90)(63,72,91), (1,39,78,32)(2,40,79,33)(3,41,80,34)(4,42,81,35)(5,36,82,29)(6,37,83,30)(7,38,84,31)(8,71,25,64)(9,72,26,65)(10,73,27,66)(11,74,28,67)(12,75,22,68)(13,76,23,69)(14,77,24,70)(15,62,109,53)(16,63,110,54)(17,57,111,55)(18,58,112,56)(19,59,106,50)(20,60,107,51)(21,61,108,52)(43,89,99,97)(44,90,100,98)(45,91,101,92)(46,85,102,93)(47,86,103,94)(48,87,104,95)(49,88,105,96) );

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)], [(1,50,78,59),(2,51,79,60),(3,52,80,61),(4,53,81,62),(5,54,82,63),(6,55,83,57),(7,56,84,58),(8,42,25,35),(9,36,26,29),(10,37,27,30),(11,38,28,31),(12,39,22,32),(13,40,23,33),(14,41,24,34),(15,100,109,44),(16,101,110,45),(17,102,111,46),(18,103,112,47),(19,104,106,48),(20,105,107,49),(21,99,108,43),(64,90,71,98),(65,91,72,92),(66,85,73,93),(67,86,74,94),(68,87,75,95),(69,88,76,96),(70,89,77,97)], [(1,68,78,75),(2,69,79,76),(3,70,80,77),(4,64,81,71),(5,65,82,72),(6,66,83,73),(7,67,84,74),(8,44,25,100),(9,45,26,101),(10,46,27,102),(11,47,28,103),(12,48,22,104),(13,49,23,105),(14,43,24,99),(15,42,109,35),(16,36,110,29),(17,37,111,30),(18,38,112,31),(19,39,106,32),(20,40,107,33),(21,41,108,34),(50,95,59,87),(51,96,60,88),(52,97,61,89),(53,98,62,90),(54,92,63,91),(55,93,57,85),(56,94,58,86)], [(8,15,44),(9,16,45),(10,17,46),(11,18,47),(12,19,48),(13,20,49),(14,21,43),(22,106,104),(23,107,105),(24,108,99),(25,109,100),(26,110,101),(27,111,102),(28,112,103),(50,68,95),(51,69,96),(52,70,97),(53,64,98),(54,65,92),(55,66,93),(56,67,94),(57,73,85),(58,74,86),(59,75,87),(60,76,88),(61,77,89),(62,71,90),(63,72,91)], [(1,39,78,32),(2,40,79,33),(3,41,80,34),(4,42,81,35),(5,36,82,29),(6,37,83,30),(7,38,84,31),(8,71,25,64),(9,72,26,65),(10,73,27,66),(11,74,28,67),(12,75,22,68),(13,76,23,69),(14,77,24,70),(15,62,109,53),(16,63,110,54),(17,57,111,55),(18,58,112,56),(19,59,106,50),(20,60,107,51),(21,61,108,52),(43,89,99,97),(44,90,100,98),(45,91,101,92),(46,85,102,93),(47,86,103,94),(48,87,104,95),(49,88,105,96)])

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

Export

׿
×
𝔽