Aliases: C2.SU3(𝔽2), C6.3PSU3(𝔽2), He3⋊C4⋊2C4, (C2×He3).Q8, He3⋊2(C4⋊C4), He3⋊C2.2D4, C3.(C2.PSU3(𝔽2)), (C2×He3⋊C4).4C2, He3⋊C2.2(C2×C4), (C2×He3⋊C2).3C22, SmallGroup(432,239)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2.SU3(𝔽2)
G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=1, f2=ae2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd-1=bc-1, ebe-1=dc=cd, fbf-1=bd, ce=ec, cf=fc, ede-1=b-1, fdf-1=bc-1d-1, fef-1=ae-1 >
9C2
9C2
12C3
9C4
9C4
9C22
18C4
18C4
9C6
9C6
12S3
12S3
12C6
4C32
9C12
9C12
12D6
18C12
18C12
12C3×S3
12C3×S3
12S3×C6
Smallest permutation representation of C2.SU3(𝔽2)
►On 72 points
Generators in S72
(1 3)(2 4)(5 7)(6 8)(9 25)(10 26)(11 27)(12 28)(13 38)(14 39)(15 40)(16 37)(17 19)(18 20)(21 23)(22 24)(29 34)(30 35)(31 36)(32 33)(41 61)(42 62)(43 63)(44 64)(45 47)(46 48)(49 51)(50 52)(53 65)(54 66)(55 67)(56 68)(57 71)(58 72)(59 69)(60 70)
(1 40 30)(2 21 5)(3 15 35)(4 23 7)(6 42 38)(8 62 13)(9 18 68)(10 48 65)(11 71 49)(12 72 17)(14 41 31)(19 28 58)(20 56 25)(22 32 64)(24 33 44)(26 46 53)(27 57 51)(36 39 61)(45 59 66)(47 69 54)(50 60 67)(52 70 55)
(1 8 24)(2 5 21)(3 6 22)(4 7 23)(9 54 57)(10 55 58)(11 56 59)(12 53 60)(13 44 30)(14 41 31)(15 42 32)(16 43 29)(17 46 50)(18 47 51)(19 48 52)(20 45 49)(25 66 71)(26 67 72)(27 68 69)(28 65 70)(33 40 62)(34 37 63)(35 38 64)(36 39 61)
(1 8 24)(2 31 37)(3 6 22)(4 36 16)(5 14 63)(7 39 43)(9 18 69)(10 65 19)(11 66 45)(12 50 72)(15 32 42)(17 26 53)(20 59 25)(21 41 34)(23 61 29)(27 54 47)(28 52 58)(33 62 40)(46 67 60)(48 55 70)(49 56 71)(51 68 57)
(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 18)(2 19)(3 20)(4 17)(5 48)(6 45)(7 46)(8 47)(9 62 27 44)(10 41 28 63)(11 64 25 42)(12 43 26 61)(13 57 40 69)(14 70 37 58)(15 59 38 71)(16 72 39 60)(21 52)(22 49)(23 50)(24 51)(29 67 36 53)(30 54 33 68)(31 65 34 55)(32 56 35 66)
G:=sub<Sym(72)| (1,3)(2,4)(5,7)(6,8)(9,25)(10,26)(11,27)(12,28)(13,38)(14,39)(15,40)(16,37)(17,19)(18,20)(21,23)(22,24)(29,34)(30,35)(31,36)(32,33)(41,61)(42,62)(43,63)(44,64)(45,47)(46,48)(49,51)(50,52)(53,65)(54,66)(55,67)(56,68)(57,71)(58,72)(59,69)(60,70), (1,40,30)(2,21,5)(3,15,35)(4,23,7)(6,42,38)(8,62,13)(9,18,68)(10,48,65)(11,71,49)(12,72,17)(14,41,31)(19,28,58)(20,56,25)(22,32,64)(24,33,44)(26,46,53)(27,57,51)(36,39,61)(45,59,66)(47,69,54)(50,60,67)(52,70,55), (1,8,24)(2,5,21)(3,6,22)(4,7,23)(9,54,57)(10,55,58)(11,56,59)(12,53,60)(13,44,30)(14,41,31)(15,42,32)(16,43,29)(17,46,50)(18,47,51)(19,48,52)(20,45,49)(25,66,71)(26,67,72)(27,68,69)(28,65,70)(33,40,62)(34,37,63)(35,38,64)(36,39,61), (1,8,24)(2,31,37)(3,6,22)(4,36,16)(5,14,63)(7,39,43)(9,18,69)(10,65,19)(11,66,45)(12,50,72)(15,32,42)(17,26,53)(20,59,25)(21,41,34)(23,61,29)(27,54,47)(28,52,58)(33,62,40)(46,67,60)(48,55,70)(49,56,71)(51,68,57), (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), (1,18)(2,19)(3,20)(4,17)(5,48)(6,45)(7,46)(8,47)(9,62,27,44)(10,41,28,63)(11,64,25,42)(12,43,26,61)(13,57,40,69)(14,70,37,58)(15,59,38,71)(16,72,39,60)(21,52)(22,49)(23,50)(24,51)(29,67,36,53)(30,54,33,68)(31,65,34,55)(32,56,35,66)>;
G:=Group( (1,3)(2,4)(5,7)(6,8)(9,25)(10,26)(11,27)(12,28)(13,38)(14,39)(15,40)(16,37)(17,19)(18,20)(21,23)(22,24)(29,34)(30,35)(31,36)(32,33)(41,61)(42,62)(43,63)(44,64)(45,47)(46,48)(49,51)(50,52)(53,65)(54,66)(55,67)(56,68)(57,71)(58,72)(59,69)(60,70), (1,40,30)(2,21,5)(3,15,35)(4,23,7)(6,42,38)(8,62,13)(9,18,68)(10,48,65)(11,71,49)(12,72,17)(14,41,31)(19,28,58)(20,56,25)(22,32,64)(24,33,44)(26,46,53)(27,57,51)(36,39,61)(45,59,66)(47,69,54)(50,60,67)(52,70,55), (1,8,24)(2,5,21)(3,6,22)(4,7,23)(9,54,57)(10,55,58)(11,56,59)(12,53,60)(13,44,30)(14,41,31)(15,42,32)(16,43,29)(17,46,50)(18,47,51)(19,48,52)(20,45,49)(25,66,71)(26,67,72)(27,68,69)(28,65,70)(33,40,62)(34,37,63)(35,38,64)(36,39,61), (1,8,24)(2,31,37)(3,6,22)(4,36,16)(5,14,63)(7,39,43)(9,18,69)(10,65,19)(11,66,45)(12,50,72)(15,32,42)(17,26,53)(20,59,25)(21,41,34)(23,61,29)(27,54,47)(28,52,58)(33,62,40)(46,67,60)(48,55,70)(49,56,71)(51,68,57), (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), (1,18)(2,19)(3,20)(4,17)(5,48)(6,45)(7,46)(8,47)(9,62,27,44)(10,41,28,63)(11,64,25,42)(12,43,26,61)(13,57,40,69)(14,70,37,58)(15,59,38,71)(16,72,39,60)(21,52)(22,49)(23,50)(24,51)(29,67,36,53)(30,54,33,68)(31,65,34,55)(32,56,35,66) );
G=PermutationGroup([[(1,3),(2,4),(5,7),(6,8),(9,25),(10,26),(11,27),(12,28),(13,38),(14,39),(15,40),(16,37),(17,19),(18,20),(21,23),(22,24),(29,34),(30,35),(31,36),(32,33),(41,61),(42,62),(43,63),(44,64),(45,47),(46,48),(49,51),(50,52),(53,65),(54,66),(55,67),(56,68),(57,71),(58,72),(59,69),(60,70)], [(1,40,30),(2,21,5),(3,15,35),(4,23,7),(6,42,38),(8,62,13),(9,18,68),(10,48,65),(11,71,49),(12,72,17),(14,41,31),(19,28,58),(20,56,25),(22,32,64),(24,33,44),(26,46,53),(27,57,51),(36,39,61),(45,59,66),(47,69,54),(50,60,67),(52,70,55)], [(1,8,24),(2,5,21),(3,6,22),(4,7,23),(9,54,57),(10,55,58),(11,56,59),(12,53,60),(13,44,30),(14,41,31),(15,42,32),(16,43,29),(17,46,50),(18,47,51),(19,48,52),(20,45,49),(25,66,71),(26,67,72),(27,68,69),(28,65,70),(33,40,62),(34,37,63),(35,38,64),(36,39,61)], [(1,8,24),(2,31,37),(3,6,22),(4,36,16),(5,14,63),(7,39,43),(9,18,69),(10,65,19),(11,66,45),(12,50,72),(15,32,42),(17,26,53),(20,59,25),(21,41,34),(23,61,29),(27,54,47),(28,52,58),(33,62,40),(46,67,60),(48,55,70),(49,56,71),(51,68,57)], [(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)], [(1,18),(2,19),(3,20),(4,17),(5,48),(6,45),(7,46),(8,47),(9,62,27,44),(10,41,28,63),(11,64,25,42),(12,43,26,61),(13,57,40,69),(14,70,37,58),(15,59,38,71),(16,72,39,60),(21,52),(22,49),(23,50),(24,51),(29,67,36,53),(30,54,33,68),(31,65,34,55),(32,56,35,66)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | ··· | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 9 | 9 | 1 | 1 | 24 | 18 | ··· | 18 | 1 | 1 | 9 | 9 | 9 | 9 | 24 | 18 | ··· | 18 |
32 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 6 | 6 | 8 | 8 |
| type | + | + | + | - | + | + | |||||
| image | C1 | C2 | C4 | D4 | Q8 | SU3(𝔽2) | C2.SU3(𝔽2) | SU3(𝔽2) | C2.SU3(𝔽2) | PSU3(𝔽2) | C2.PSU3(𝔽2) |
| kernel | C2.SU3(𝔽2) | C2×He3⋊C4 | He3⋊C4 | He3⋊C2 | C2×He3 | C2 | C1 | C2 | C1 | C6 | C3 |
| # reps | 1 | 3 | 4 | 1 | 1 | 8 | 8 | 2 | 2 | 1 | 1 |
Matrix representation of C2.SU3(𝔽2) ►in GL3(𝔽13) generated by
| 12 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 0 | 0 | 9 |
| 9 | 0 | 0 |
| 0 | 9 | 0 |
| 9 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 9 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 1 |
| 3 | 1 | 9 |
| 9 | 9 | 9 |
| 1 | 3 | 9 |
| 11 | 11 | 8 |
| 11 | 8 | 11 |
| 7 | 8 | 8 |
G:=sub<GL(3,GF(13))| [12,0,0,0,12,0,0,0,12],[0,9,0,0,0,9,9,0,0],[9,0,0,0,9,0,0,0,9],[9,0,0,0,3,0,0,0,1],[3,9,1,1,9,3,9,9,9],[11,11,7,11,8,8,8,11,8] >;
C2.SU3(𝔽2) in GAP, Magma, Sage, TeX
C_2.{\rm SU}_3({\mathbb F}_2) % in TeX
G:=Group("C2.SU(3,2)"); // GroupNames label
G:=SmallGroup(432,239);
// by ID
G=gap.SmallGroup(432,239);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,85,92,3924,851,1558,17477,3708,2539,537]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^4=1,f^2=a*e^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,d*b*d^-1=b*c^-1,e*b*e^-1=d*c=c*d,f*b*f^-1=b*d,c*e=e*c,c*f=f*c,e*d*e^-1=b^-1,f*d*f^-1=b*c^-1*d^-1,f*e*f^-1=a*e^-1>;
// generators/relations
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