Aliases: Q8:He3, C32:2SL2(F3), (C3xC6).2A4, C6.11(C3xA4), C2.(C32:A4), (Q8xC32):1C3, (C3xSL2(F3)):C3, (C3xQ8).5C32, C3.5(C3xSL2(F3)), SmallGroup(216,42)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C3xQ8 — C3xSL2(F3) — Q8:He3 |
Generators and relations for Q8:He3
G = < a,b,c,d,e | a4=c3=d3=e3=1, b2=a2, bab-1=a-1, ac=ca, ad=da, eae-1=b, bc=cb, bd=db, ebe-1=ab, cd=dc, ece-1=cd-1, de=ed >
Quotients: C1, C3, C32, A4, SL2(F3), He3, C3xA4, C3xSL2(F3), C32:A4, Q8:He3
Smallest permutation representation of Q8:He3
►On 72 points
Generators in S72
(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 7 3 5)(2 6 4 8)(9 15 11 13)(10 14 12 16)(17 23 19 21)(18 22 20 24)(25 31 27 29)(26 30 28 32)(33 39 35 37)(34 38 36 40)(41 47 43 45)(42 46 44 48)(49 55 51 53)(50 54 52 56)(57 63 59 61)(58 62 60 64)(65 71 67 69)(66 70 68 72)
(1 49 25)(2 50 26)(3 51 27)(4 52 28)(5 53 29)(6 54 30)(7 55 31)(8 56 32)(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 17 9)(2 18 10)(3 19 11)(4 20 12)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(49 65 57)(50 66 58)(51 67 59)(52 68 60)(53 69 61)(54 70 62)(55 71 63)(56 72 64)
(1 25 57)(2 30 63)(3 27 59)(4 32 61)(5 28 64)(6 31 58)(7 26 62)(8 29 60)(9 33 65)(10 38 71)(11 35 67)(12 40 69)(13 36 72)(14 39 66)(15 34 70)(16 37 68)(17 41 49)(18 46 55)(19 43 51)(20 48 53)(21 44 56)(22 47 50)(23 42 54)(24 45 52)
G:=sub<Sym(72)| (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), (1,7,3,5)(2,6,4,8)(9,15,11,13)(10,14,12,16)(17,23,19,21)(18,22,20,24)(25,31,27,29)(26,30,28,32)(33,39,35,37)(34,38,36,40)(41,47,43,45)(42,46,44,48)(49,55,51,53)(50,54,52,56)(57,63,59,61)(58,62,60,64)(65,71,67,69)(66,70,68,72), (1,49,25)(2,50,26)(3,51,27)(4,52,28)(5,53,29)(6,54,30)(7,55,31)(8,56,32)(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), (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(49,65,57)(50,66,58)(51,67,59)(52,68,60)(53,69,61)(54,70,62)(55,71,63)(56,72,64), (1,25,57)(2,30,63)(3,27,59)(4,32,61)(5,28,64)(6,31,58)(7,26,62)(8,29,60)(9,33,65)(10,38,71)(11,35,67)(12,40,69)(13,36,72)(14,39,66)(15,34,70)(16,37,68)(17,41,49)(18,46,55)(19,43,51)(20,48,53)(21,44,56)(22,47,50)(23,42,54)(24,45,52)>;
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), (1,7,3,5)(2,6,4,8)(9,15,11,13)(10,14,12,16)(17,23,19,21)(18,22,20,24)(25,31,27,29)(26,30,28,32)(33,39,35,37)(34,38,36,40)(41,47,43,45)(42,46,44,48)(49,55,51,53)(50,54,52,56)(57,63,59,61)(58,62,60,64)(65,71,67,69)(66,70,68,72), (1,49,25)(2,50,26)(3,51,27)(4,52,28)(5,53,29)(6,54,30)(7,55,31)(8,56,32)(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), (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(49,65,57)(50,66,58)(51,67,59)(52,68,60)(53,69,61)(54,70,62)(55,71,63)(56,72,64), (1,25,57)(2,30,63)(3,27,59)(4,32,61)(5,28,64)(6,31,58)(7,26,62)(8,29,60)(9,33,65)(10,38,71)(11,35,67)(12,40,69)(13,36,72)(14,39,66)(15,34,70)(16,37,68)(17,41,49)(18,46,55)(19,43,51)(20,48,53)(21,44,56)(22,47,50)(23,42,54)(24,45,52) );
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)], [(1,7,3,5),(2,6,4,8),(9,15,11,13),(10,14,12,16),(17,23,19,21),(18,22,20,24),(25,31,27,29),(26,30,28,32),(33,39,35,37),(34,38,36,40),(41,47,43,45),(42,46,44,48),(49,55,51,53),(50,54,52,56),(57,63,59,61),(58,62,60,64),(65,71,67,69),(66,70,68,72)], [(1,49,25),(2,50,26),(3,51,27),(4,52,28),(5,53,29),(6,54,30),(7,55,31),(8,56,32),(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)], [(1,17,9),(2,18,10),(3,19,11),(4,20,12),(5,21,13),(6,22,14),(7,23,15),(8,24,16),(25,41,33),(26,42,34),(27,43,35),(28,44,36),(29,45,37),(30,46,38),(31,47,39),(32,48,40),(49,65,57),(50,66,58),(51,67,59),(52,68,60),(53,69,61),(54,70,62),(55,71,63),(56,72,64)], [(1,25,57),(2,30,63),(3,27,59),(4,32,61),(5,28,64),(6,31,58),(7,26,62),(8,29,60),(9,33,65),(10,38,71),(11,35,67),(12,40,69),(13,36,72),(14,39,66),(15,34,70),(16,37,68),(17,41,49),(18,46,55),(19,43,51),(20,48,53),(21,44,56),(22,47,50),(23,42,54),(24,45,52)]])
Q8:He3 is a maximal subgroup of
C32:CSU2(F3) C32:2GL2(F3) C32:2CSU2(F3) C32:3GL2(F3) C6.(S3xA4) Q8:He3:C2 C4oD4:He3
31 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | ··· | 3J | 4 | 6A | 6B | 6C | 6D | 6E | ··· | 6J | 12A | ··· | 12H |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 12 | ··· | 12 | 6 | 1 | 1 | 3 | 3 | 12 | ··· | 12 | 6 | ··· | 6 |
31 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 |
| type | + | - | + | ||||||||
| image | C1 | C3 | C3 | SL2(F3) | SL2(F3) | C3xSL2(F3) | A4 | He3 | C3xA4 | C32:A4 | Q8:He3 |
| kernel | Q8:He3 | C3xSL2(F3) | Q8xC32 | C32 | C32 | C3 | C3xC6 | Q8 | C6 | C2 | C1 |
| # reps | 1 | 6 | 2 | 1 | 2 | 6 | 1 | 2 | 2 | 6 | 2 |
Matrix representation of Q8:He3 ►in GL5(F13)
| 0 | 1 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 3 | 4 | 0 | 0 | 0 |
| 4 | 10 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 8 | 0 |
| 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 0 | 1 | 0 | 0 | 0 |
| 4 | 10 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 7 |
| 0 | 0 | 12 | 0 | 4 |
| 0 | 0 | 9 | 3 | 4 |
G:=sub<GL(5,GF(13))| [0,12,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[3,4,0,0,0,4,10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,8,12,12,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[0,4,0,0,0,1,10,0,0,0,0,0,9,12,9,0,0,0,0,3,0,0,7,4,4] >;
Q8:He3 in GAP, Magma, Sage, TeX
Q_8\rtimes {\rm He}_3 % in TeX
G:=Group("Q8:He3"); // GroupNames label
G:=SmallGroup(216,42);
// by ID
G=gap.SmallGroup(216,42);
# by ID
G:=PCGroup([6,-3,-3,-3,-2,2,-2,145,1299,117,2434,202,88]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^3=d^3=e^3=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=b,b*c=c*b,b*d=d*b,e*b*e^-1=a*b,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;
// generators/relations
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