direct product, non-abelian, soluble
Aliases: C32×GL2(𝔽3), Q8⋊(S3×C32), C6.21(C3×S4), (C3×C6).26S4, C2.3(C32×S4), SL2(𝔽3)⋊(C3×C6), (Q8×C32)⋊11S3, (C3×SL2(𝔽3))⋊7C6, (C32×SL2(𝔽3))⋊7C2, (C3×Q8)⋊2(C3×S3), SmallGroup(432,614)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C3×SL2(𝔽3) — C32×SL2(𝔽3) — C32×GL2(𝔽3) |
| SL2(𝔽3) — C32×GL2(𝔽3) |
Generators and relations for C32×GL2(𝔽3)
G = < a,b,c,d,e,f | a3=b3=c4=e3=f2=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=fdf=c-1, ece-1=cd, fcf=c2d, ede-1=c, fef=e-1 >
Subgroups: 474 in 120 conjugacy classes, 30 normal (10 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, Q8, C32, C32, C12, D6, C2×C6, SD16, C3×S3, C3×C6, C3×C6, C24, SL2(𝔽3), SL2(𝔽3), C3×D4, C3×Q8, C33, C3×C12, S3×C6, C62, C3×SD16, GL2(𝔽3), S3×C32, C32×C6, C3×C24, C3×SL2(𝔽3), C3×SL2(𝔽3), D4×C32, Q8×C32, S3×C3×C6, C32×SD16, C3×GL2(𝔽3), C32×SL2(𝔽3), C32×GL2(𝔽3)
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, S4, GL2(𝔽3), S3×C32, C3×S4, C3×GL2(𝔽3), C32×S4, C32×GL2(𝔽3)
►On 72 points
Generators in S72
(1 56 32)(2 53 29)(3 54 30)(4 55 31)(5 51 27)(6 52 28)(7 49 25)(8 50 26)(9 57 33)(10 58 34)(11 59 35)(12 60 36)(13 61 37)(14 62 38)(15 63 39)(16 64 40)(17 65 41)(18 66 42)(19 67 43)(20 68 44)(21 69 45)(22 70 46)(23 71 47)(24 72 48)
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 67 59)(6 68 60)(7 65 57)(8 66 58)(9 25 17)(10 26 18)(11 27 19)(12 28 20)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(33 49 41)(34 50 42)(35 51 43)(36 52 44)(53 69 61)(54 70 62)(55 71 63)(56 72 64)
(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)
(1 11 3 9)(2 10 4 12)(5 70 7 72)(6 69 8 71)(13 18 15 20)(14 17 16 19)(21 26 23 28)(22 25 24 27)(29 34 31 36)(30 33 32 35)(37 42 39 44)(38 41 40 43)(45 50 47 52)(46 49 48 51)(53 58 55 60)(54 57 56 59)(61 66 63 68)(62 65 64 67)
(2 11 10)(4 9 12)(5 8 69)(6 71 7)(13 19 18)(15 17 20)(21 27 26)(23 25 28)(29 35 34)(31 33 36)(37 43 42)(39 41 44)(45 51 50)(47 49 52)(53 59 58)(55 57 60)(61 67 66)(63 65 68)
(1 3)(2 11)(4 9)(5 69)(7 71)(13 19)(14 16)(15 17)(21 27)(22 24)(23 25)(29 35)(30 32)(31 33)(37 43)(38 40)(39 41)(45 51)(46 48)(47 49)(53 59)(54 56)(55 57)(61 67)(62 64)(63 65)(70 72)
G:=sub<Sym(72)| (1,56,32)(2,53,29)(3,54,30)(4,55,31)(5,51,27)(6,52,28)(7,49,25)(8,50,26)(9,57,33)(10,58,34)(11,59,35)(12,60,36)(13,61,37)(14,62,38)(15,63,39)(16,64,40)(17,65,41)(18,66,42)(19,67,43)(20,68,44)(21,69,45)(22,70,46)(23,71,47)(24,72,48), (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,67,59)(6,68,60)(7,65,57)(8,66,58)(9,25,17)(10,26,18)(11,27,19)(12,28,20)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(33,49,41)(34,50,42)(35,51,43)(36,52,44)(53,69,61)(54,70,62)(55,71,63)(56,72,64), (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), (1,11,3,9)(2,10,4,12)(5,70,7,72)(6,69,8,71)(13,18,15,20)(14,17,16,19)(21,26,23,28)(22,25,24,27)(29,34,31,36)(30,33,32,35)(37,42,39,44)(38,41,40,43)(45,50,47,52)(46,49,48,51)(53,58,55,60)(54,57,56,59)(61,66,63,68)(62,65,64,67), (2,11,10)(4,9,12)(5,8,69)(6,71,7)(13,19,18)(15,17,20)(21,27,26)(23,25,28)(29,35,34)(31,33,36)(37,43,42)(39,41,44)(45,51,50)(47,49,52)(53,59,58)(55,57,60)(61,67,66)(63,65,68), (1,3)(2,11)(4,9)(5,69)(7,71)(13,19)(14,16)(15,17)(21,27)(22,24)(23,25)(29,35)(30,32)(31,33)(37,43)(38,40)(39,41)(45,51)(46,48)(47,49)(53,59)(54,56)(55,57)(61,67)(62,64)(63,65)(70,72)>;
G:=Group( (1,56,32)(2,53,29)(3,54,30)(4,55,31)(5,51,27)(6,52,28)(7,49,25)(8,50,26)(9,57,33)(10,58,34)(11,59,35)(12,60,36)(13,61,37)(14,62,38)(15,63,39)(16,64,40)(17,65,41)(18,66,42)(19,67,43)(20,68,44)(21,69,45)(22,70,46)(23,71,47)(24,72,48), (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,67,59)(6,68,60)(7,65,57)(8,66,58)(9,25,17)(10,26,18)(11,27,19)(12,28,20)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(33,49,41)(34,50,42)(35,51,43)(36,52,44)(53,69,61)(54,70,62)(55,71,63)(56,72,64), (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), (1,11,3,9)(2,10,4,12)(5,70,7,72)(6,69,8,71)(13,18,15,20)(14,17,16,19)(21,26,23,28)(22,25,24,27)(29,34,31,36)(30,33,32,35)(37,42,39,44)(38,41,40,43)(45,50,47,52)(46,49,48,51)(53,58,55,60)(54,57,56,59)(61,66,63,68)(62,65,64,67), (2,11,10)(4,9,12)(5,8,69)(6,71,7)(13,19,18)(15,17,20)(21,27,26)(23,25,28)(29,35,34)(31,33,36)(37,43,42)(39,41,44)(45,51,50)(47,49,52)(53,59,58)(55,57,60)(61,67,66)(63,65,68), (1,3)(2,11)(4,9)(5,69)(7,71)(13,19)(14,16)(15,17)(21,27)(22,24)(23,25)(29,35)(30,32)(31,33)(37,43)(38,40)(39,41)(45,51)(46,48)(47,49)(53,59)(54,56)(55,57)(61,67)(62,64)(63,65)(70,72) );
G=PermutationGroup([[(1,56,32),(2,53,29),(3,54,30),(4,55,31),(5,51,27),(6,52,28),(7,49,25),(8,50,26),(9,57,33),(10,58,34),(11,59,35),(12,60,36),(13,61,37),(14,62,38),(15,63,39),(16,64,40),(17,65,41),(18,66,42),(19,67,43),(20,68,44),(21,69,45),(22,70,46),(23,71,47),(24,72,48)], [(1,24,16),(2,21,13),(3,22,14),(4,23,15),(5,67,59),(6,68,60),(7,65,57),(8,66,58),(9,25,17),(10,26,18),(11,27,19),(12,28,20),(29,45,37),(30,46,38),(31,47,39),(32,48,40),(33,49,41),(34,50,42),(35,51,43),(36,52,44),(53,69,61),(54,70,62),(55,71,63),(56,72,64)], [(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)], [(1,11,3,9),(2,10,4,12),(5,70,7,72),(6,69,8,71),(13,18,15,20),(14,17,16,19),(21,26,23,28),(22,25,24,27),(29,34,31,36),(30,33,32,35),(37,42,39,44),(38,41,40,43),(45,50,47,52),(46,49,48,51),(53,58,55,60),(54,57,56,59),(61,66,63,68),(62,65,64,67)], [(2,11,10),(4,9,12),(5,8,69),(6,71,7),(13,19,18),(15,17,20),(21,27,26),(23,25,28),(29,35,34),(31,33,36),(37,43,42),(39,41,44),(45,51,50),(47,49,52),(53,59,58),(55,57,60),(61,67,66),(63,65,68)], [(1,3),(2,11),(4,9),(5,69),(7,71),(13,19),(14,16),(15,17),(21,27),(22,24),(23,25),(29,35),(30,32),(31,33),(37,43),(38,40),(39,41),(45,51),(46,48),(47,49),(53,59),(54,56),(55,57),(61,67),(62,64),(63,65),(70,72)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 3A | ··· | 3H | 3I | ··· | 3Q | 4 | 6A | ··· | 6H | 6I | ··· | 6Q | 6R | ··· | 6Y | 8A | 8B | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 12 | 1 | ··· | 1 | 8 | ··· | 8 | 6 | 1 | ··· | 1 | 8 | ··· | 8 | 12 | ··· | 12 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C3 | C6 | S3 | C3×S3 | GL2(𝔽3) | C3×GL2(𝔽3) | S4 | C3×S4 | GL2(𝔽3) | C3×GL2(𝔽3) |
| kernel | C32×GL2(𝔽3) | C32×SL2(𝔽3) | C3×GL2(𝔽3) | C3×SL2(𝔽3) | Q8×C32 | C3×Q8 | C32 | C3 | C3×C6 | C6 | C32 | C3 |
| # reps | 1 | 1 | 8 | 8 | 1 | 8 | 2 | 16 | 2 | 16 | 1 | 8 |
Matrix representation of C32×GL2(𝔽3) ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 32 | 21 |
| 0 | 0 | 52 | 41 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 20 | 33 |
| 0 | 0 | 52 | 53 |
| 72 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 72 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,32,52,0,0,21,41],[1,0,0,0,0,1,0,0,0,0,20,52,0,0,33,53],[72,1,0,0,72,0,0,0,0,0,72,72,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,72,72] >;
C32×GL2(𝔽3) in GAP, Magma, Sage, TeX
C_3^2\times {\rm GL}_2({\mathbb F}_3) % in TeX
G:=Group("C3^2xGL(2,3)"); // GroupNames label
G:=SmallGroup(432,614);
// by ID
G=gap.SmallGroup(432,614);
# by ID
G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1011,3784,655,172,2273,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^4=e^3=f^2=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d^-1=f*d*f=c^-1,e*c*e^-1=c*d,f*c*f=c^2*d,e*d*e^-1=c,f*e*f=e^-1>;
// generators/relations