metabelian, supersoluble, monomial
Aliases: He3⋊3Q8, C32⋊3Dic6, C6.9(S3×C6), (C3×C6).6D6, C3⋊Dic3.C6, C32⋊4Q8⋊C3, C12.3(C3×S3), (C3×C12).1C6, (C3×C12).1S3, C4.(C32⋊C6), C32⋊2(C3×Q8), (C4×He3).1C2, C3.2(C3×Dic6), C32⋊C12.2C2, (C2×He3).6C22, (C3×C6).1(C2×C6), C2.3(C2×C32⋊C6), SmallGroup(216,49)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for He3⋊3Q8
G = < a,b,c,d,e | a3=b3=c3=d4=1, e2=d2, ab=ba, cac-1=ab-1, ad=da, eae-1=a-1, bc=cb, bd=db, ebe-1=b-1, cd=dc, ce=ec, ede-1=d-1 >
3C3
3C3
6C3
9C4
9C4
3C6
3C6
6C6
2C32
9Q8
3C12
3C12
3Dic3
3Dic3
6C12
9C12
9Dic3
9C12
9Dic3
3Dic6
9Dic6
Smallest permutation representation of He3⋊3Q8
►On 72 points
Generators in S72
(1 40 7)(2 37 8)(3 38 5)(4 39 6)(9 66 46)(10 67 47)(11 68 48)(12 65 45)(13 51 31)(14 52 32)(15 49 29)(16 50 30)(17 42 62)(18 43 63)(19 44 64)(20 41 61)(21 58 33)(22 59 34)(23 60 35)(24 57 36)(25 56 71)(26 53 72)(27 54 69)(28 55 70)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 35 51)(6 36 52)(7 33 49)(8 34 50)(9 69 43)(10 70 44)(11 71 41)(12 72 42)(13 38 60)(14 39 57)(15 40 58)(16 37 59)(17 45 53)(18 46 54)(19 47 55)(20 48 56)(25 61 68)(26 62 65)(27 63 66)(28 64 67)
(1 7 15)(2 8 16)(3 5 13)(4 6 14)(9 27 46)(10 28 47)(11 25 48)(12 26 45)(17 42 65)(18 43 66)(19 44 67)(20 41 68)(21 33 40)(22 34 37)(23 35 38)(24 36 39)(29 49 58)(30 50 59)(31 51 60)(32 52 57)(53 72 62)(54 69 63)(55 70 64)(56 71 61)
(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 3 20)(2 17 4 19)(5 41 7 43)(6 44 8 42)(9 51 11 49)(10 50 12 52)(13 68 15 66)(14 67 16 65)(21 54 23 56)(22 53 24 55)(25 58 27 60)(26 57 28 59)(29 46 31 48)(30 45 32 47)(33 69 35 71)(34 72 36 70)(37 62 39 64)(38 61 40 63)
G:=sub<Sym(72)| (1,40,7)(2,37,8)(3,38,5)(4,39,6)(9,66,46)(10,67,47)(11,68,48)(12,65,45)(13,51,31)(14,52,32)(15,49,29)(16,50,30)(17,42,62)(18,43,63)(19,44,64)(20,41,61)(21,58,33)(22,59,34)(23,60,35)(24,57,36)(25,56,71)(26,53,72)(27,54,69)(28,55,70), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,51)(6,36,52)(7,33,49)(8,34,50)(9,69,43)(10,70,44)(11,71,41)(12,72,42)(13,38,60)(14,39,57)(15,40,58)(16,37,59)(17,45,53)(18,46,54)(19,47,55)(20,48,56)(25,61,68)(26,62,65)(27,63,66)(28,64,67), (1,7,15)(2,8,16)(3,5,13)(4,6,14)(9,27,46)(10,28,47)(11,25,48)(12,26,45)(17,42,65)(18,43,66)(19,44,67)(20,41,68)(21,33,40)(22,34,37)(23,35,38)(24,36,39)(29,49,58)(30,50,59)(31,51,60)(32,52,57)(53,72,62)(54,69,63)(55,70,64)(56,71,61), (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), (1,18,3,20)(2,17,4,19)(5,41,7,43)(6,44,8,42)(9,51,11,49)(10,50,12,52)(13,68,15,66)(14,67,16,65)(21,54,23,56)(22,53,24,55)(25,58,27,60)(26,57,28,59)(29,46,31,48)(30,45,32,47)(33,69,35,71)(34,72,36,70)(37,62,39,64)(38,61,40,63)>;
G:=Group( (1,40,7)(2,37,8)(3,38,5)(4,39,6)(9,66,46)(10,67,47)(11,68,48)(12,65,45)(13,51,31)(14,52,32)(15,49,29)(16,50,30)(17,42,62)(18,43,63)(19,44,64)(20,41,61)(21,58,33)(22,59,34)(23,60,35)(24,57,36)(25,56,71)(26,53,72)(27,54,69)(28,55,70), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,51)(6,36,52)(7,33,49)(8,34,50)(9,69,43)(10,70,44)(11,71,41)(12,72,42)(13,38,60)(14,39,57)(15,40,58)(16,37,59)(17,45,53)(18,46,54)(19,47,55)(20,48,56)(25,61,68)(26,62,65)(27,63,66)(28,64,67), (1,7,15)(2,8,16)(3,5,13)(4,6,14)(9,27,46)(10,28,47)(11,25,48)(12,26,45)(17,42,65)(18,43,66)(19,44,67)(20,41,68)(21,33,40)(22,34,37)(23,35,38)(24,36,39)(29,49,58)(30,50,59)(31,51,60)(32,52,57)(53,72,62)(54,69,63)(55,70,64)(56,71,61), (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), (1,18,3,20)(2,17,4,19)(5,41,7,43)(6,44,8,42)(9,51,11,49)(10,50,12,52)(13,68,15,66)(14,67,16,65)(21,54,23,56)(22,53,24,55)(25,58,27,60)(26,57,28,59)(29,46,31,48)(30,45,32,47)(33,69,35,71)(34,72,36,70)(37,62,39,64)(38,61,40,63) );
G=PermutationGroup([[(1,40,7),(2,37,8),(3,38,5),(4,39,6),(9,66,46),(10,67,47),(11,68,48),(12,65,45),(13,51,31),(14,52,32),(15,49,29),(16,50,30),(17,42,62),(18,43,63),(19,44,64),(20,41,61),(21,58,33),(22,59,34),(23,60,35),(24,57,36),(25,56,71),(26,53,72),(27,54,69),(28,55,70)], [(1,21,29),(2,22,30),(3,23,31),(4,24,32),(5,35,51),(6,36,52),(7,33,49),(8,34,50),(9,69,43),(10,70,44),(11,71,41),(12,72,42),(13,38,60),(14,39,57),(15,40,58),(16,37,59),(17,45,53),(18,46,54),(19,47,55),(20,48,56),(25,61,68),(26,62,65),(27,63,66),(28,64,67)], [(1,7,15),(2,8,16),(3,5,13),(4,6,14),(9,27,46),(10,28,47),(11,25,48),(12,26,45),(17,42,65),(18,43,66),(19,44,67),(20,41,68),(21,33,40),(22,34,37),(23,35,38),(24,36,39),(29,49,58),(30,50,59),(31,51,60),(32,52,57),(53,72,62),(54,69,63),(55,70,64),(56,71,61)], [(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)], [(1,18,3,20),(2,17,4,19),(5,41,7,43),(6,44,8,42),(9,51,11,49),(10,50,12,52),(13,68,15,66),(14,67,16,65),(21,54,23,56),(22,53,24,55),(25,58,27,60),(26,57,28,59),(29,46,31,48),(30,45,32,47),(33,69,35,71),(34,72,36,70),(37,62,39,64),(38,61,40,63)]])
He3⋊3Q8 is a maximal subgroup of
He3⋊3SD16 He3⋊2Q16 He3⋊4SD16 He3⋊3Q16 He3⋊4Q16 He3⋊6SD16 He3⋊8SD16 He3⋊6Q16 C3⋊S3⋊Dic6 C12.84S32 C12.85S32 C12.S32 C62.36D6 C62.13D6 Q8×C32⋊C6
He3⋊3Q8 is a maximal quotient of
C62.19D6 C62.20D6
31 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | ··· | 12J | 12K | 12L | 12M | 12N |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 3 | 3 | 6 | 6 | 6 | 2 | 18 | 18 | 2 | 3 | 3 | 6 | 6 | 6 | 2 | 2 | 6 | ··· | 6 | 18 | 18 | 18 | 18 |
31 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 |
| type | + | + | + | + | - | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | Q8 | D6 | C3×S3 | Dic6 | C3×Q8 | S3×C6 | C3×Dic6 | C32⋊C6 | C2×C32⋊C6 | He3⋊3Q8 |
| kernel | He3⋊3Q8 | C32⋊C12 | C4×He3 | C32⋊4Q8 | C3⋊Dic3 | C3×C12 | C3×C12 | He3 | C3×C6 | C12 | C32 | C32 | C6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of He3⋊3Q8 ►in GL6(𝔽13)
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 3 | 7 | 0 | 0 | 0 | 0 |
| 6 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 7 | 0 | 0 |
| 0 | 0 | 6 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 7 |
| 0 | 0 | 0 | 0 | 6 | 10 |
| 0 | 11 | 11 | 2 | 0 | 11 |
| 11 | 0 | 0 | 2 | 11 | 0 |
| 11 | 2 | 0 | 11 | 0 | 11 |
| 0 | 2 | 11 | 0 | 11 | 0 |
| 0 | 11 | 0 | 11 | 11 | 2 |
| 11 | 0 | 11 | 0 | 0 | 2 |
G:=sub<GL(6,GF(13))| [0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,12,12,0,0,0,0,1,0,0,0,0,0],[3,6,0,0,0,0,7,10,0,0,0,0,0,0,3,6,0,0,0,0,7,10,0,0,0,0,0,0,3,6,0,0,0,0,7,10],[0,11,11,0,0,11,11,0,2,2,11,0,11,0,0,11,0,11,2,2,11,0,11,0,0,11,0,11,11,0,11,0,11,0,2,2] >;
He3⋊3Q8 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_3Q_8 % in TeX
G:=Group("He3:3Q8"); // GroupNames label
G:=SmallGroup(216,49);
// by ID
G=gap.SmallGroup(216,49);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,169,79,1444,736,5189]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,e*a*e^-1=a^-1,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
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