non-abelian, supersoluble, monomial
Aliases: He3⋊4Q8, C32⋊4Dic6, (C3×C12).4S3, (C3×C6).16D6, C12.7(C3⋊S3), C4.(He3⋊C2), (C4×He3).2C2, He3⋊3C4.3C2, C3.2(C32⋊4Q8), (C2×He3).11C22, C6.27(C2×C3⋊S3), C2.3(C2×He3⋊C2), SmallGroup(216,66)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for He3⋊4Q8
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, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 186 in 66 conjugacy classes, 24 normal (10 characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, Q8, C32, Dic3, C12, C12, C3×C6, Dic6, C3×Q8, He3, C3×Dic3, C3×C12, C2×He3, C3×Dic6, He3⋊3C4, C4×He3, He3⋊4Q8
Quotients: C1, C2, C22, S3, Q8, D6, C3⋊S3, Dic6, C2×C3⋊S3, He3⋊C2, C32⋊4Q8, C2×He3⋊C2, He3⋊4Q8
►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 43 69)(10 44 70)(11 41 71)(12 42 72)(13 38 60)(14 39 57)(15 40 58)(16 37 59)(17 53 45)(18 54 46)(19 55 47)(20 56 48)(25 68 61)(26 65 62)(27 66 63)(28 67 64)
(1 40 33)(2 37 34)(3 38 35)(4 39 36)(5 31 13)(6 32 14)(7 29 15)(8 30 16)(9 27 54)(10 28 55)(11 25 56)(12 26 53)(17 72 62)(18 69 63)(19 70 64)(20 71 61)(21 58 49)(22 59 50)(23 60 51)(24 57 52)(41 68 48)(42 65 45)(43 66 46)(44 67 47)
(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)(9,43,69)(10,44,70)(11,41,71)(12,42,72)(13,38,60)(14,39,57)(15,40,58)(16,37,59)(17,53,45)(18,54,46)(19,55,47)(20,56,48)(25,68,61)(26,65,62)(27,66,63)(28,67,64), (1,40,33)(2,37,34)(3,38,35)(4,39,36)(5,31,13)(6,32,14)(7,29,15)(8,30,16)(9,27,54)(10,28,55)(11,25,56)(12,26,53)(17,72,62)(18,69,63)(19,70,64)(20,71,61)(21,58,49)(22,59,50)(23,60,51)(24,57,52)(41,68,48)(42,65,45)(43,66,46)(44,67,47), (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)(9,43,69)(10,44,70)(11,41,71)(12,42,72)(13,38,60)(14,39,57)(15,40,58)(16,37,59)(17,53,45)(18,54,46)(19,55,47)(20,56,48)(25,68,61)(26,65,62)(27,66,63)(28,67,64), (1,40,33)(2,37,34)(3,38,35)(4,39,36)(5,31,13)(6,32,14)(7,29,15)(8,30,16)(9,27,54)(10,28,55)(11,25,56)(12,26,53)(17,72,62)(18,69,63)(19,70,64)(20,71,61)(21,58,49)(22,59,50)(23,60,51)(24,57,52)(41,68,48)(42,65,45)(43,66,46)(44,67,47), (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),(9,43,69),(10,44,70),(11,41,71),(12,42,72),(13,38,60),(14,39,57),(15,40,58),(16,37,59),(17,53,45),(18,54,46),(19,55,47),(20,56,48),(25,68,61),(26,65,62),(27,66,63),(28,67,64)], [(1,40,33),(2,37,34),(3,38,35),(4,39,36),(5,31,13),(6,32,14),(7,29,15),(8,30,16),(9,27,54),(10,28,55),(11,25,56),(12,26,53),(17,72,62),(18,69,63),(19,70,64),(20,71,61),(21,58,49),(22,59,50),(23,60,51),(24,57,52),(41,68,48),(42,65,45),(43,66,46),(44,67,47)], [(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⋊4Q8 is a maximal subgroup of
He3⋊5SD16 He3⋊3Q16 He3⋊7SD16 He3⋊5Q16 He3⋊9SD16 He3⋊7Q16 C3⋊S3⋊Dic6 C12⋊S3⋊S3 C62.47D6 C62.16D6 Q8×He3⋊C2
He3⋊4Q8 is a maximal quotient of
C62.29D6 C62.30D6
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 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 18 | 18 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 6 | ··· | 6 | 18 | 18 | 18 | 18 |
31 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 6 |
| type | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | S3 | Q8 | D6 | Dic6 | He3⋊C2 | C2×He3⋊C2 | He3⋊4Q8 |
| kernel | He3⋊4Q8 | He3⋊3C4 | C4×He3 | C3×C12 | He3 | C3×C6 | C32 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 4 | 1 | 4 | 8 | 4 | 4 | 2 |
Matrix representation of He3⋊4Q8 ►in GL5(𝔽13)
| 0 | 12 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 12 | 2 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 12 | 1 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 10 | 10 | 6 |
| 0 | 0 | 12 | 3 | 3 |
| 3 | 7 | 0 | 0 | 0 |
| 6 | 10 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 11 | 4 | 0 | 0 | 0 |
| 2 | 2 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 2 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(13))| [0,1,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,1,12,0,0,0,0,2,1],[1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[12,12,0,0,0,1,0,0,0,0,0,0,0,10,12,0,0,9,10,3,0,0,0,6,3],[3,6,0,0,0,7,10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[11,2,0,0,0,4,2,0,0,0,0,0,12,0,0,0,0,12,1,0,0,0,2,0,1] >;
He3⋊4Q8 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_4Q_8 % in TeX
G:=Group("He3:4Q8"); // GroupNames label
G:=SmallGroup(216,66);
// by ID
G=gap.SmallGroup(216,66);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,73,31,387,1444,382]);
// 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,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations