non-abelian, supersoluble, monomial
Aliases: He3⋊4C8, C32⋊3(C3⋊C8), (C3×C12).7S3, C2.(He3⋊3C4), (C4×He3).4C2, (C2×He3).3C4, C12.10(C3⋊S3), (C3×C6).3Dic3, C6.4(C3⋊Dic3), C4.2(He3⋊C2), C3.2(C32⋊4C8), SmallGroup(216,17)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — He3⋊4C8 |
Generators and relations for He3⋊4C8
G = < a,b,c,d | a3=b3=c3=d8=1, ab=ba, cac-1=ab-1, dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of He3⋊4C8
►On 72 points
Generators in S72
(1 68 56)(2 49 69)(3 70 50)(4 51 71)(5 72 52)(6 53 65)(7 66 54)(8 55 67)(9 57 33)(10 34 58)(11 59 35)(12 36 60)(13 61 37)(14 38 62)(15 63 39)(16 40 64)(17 41 27)(18 28 42)(19 43 29)(20 30 44)(21 45 31)(22 32 46)(23 47 25)(24 26 48)
(1 15 41)(2 16 42)(3 9 43)(4 10 44)(5 11 45)(6 12 46)(7 13 47)(8 14 48)(17 56 39)(18 49 40)(19 50 33)(20 51 34)(21 52 35)(22 53 36)(23 54 37)(24 55 38)(25 66 61)(26 67 62)(27 68 63)(28 69 64)(29 70 57)(30 71 58)(31 72 59)(32 65 60)
(1 68 39)(2 40 69)(3 70 33)(4 34 71)(5 72 35)(6 36 65)(7 66 37)(8 38 67)(9 57 19)(10 20 58)(11 59 21)(12 22 60)(13 61 23)(14 24 62)(15 63 17)(16 18 64)(25 54 47)(26 48 55)(27 56 41)(28 42 49)(29 50 43)(30 44 51)(31 52 45)(32 46 53)
(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)
G:=sub<Sym(72)| (1,68,56)(2,49,69)(3,70,50)(4,51,71)(5,72,52)(6,53,65)(7,66,54)(8,55,67)(9,57,33)(10,34,58)(11,59,35)(12,36,60)(13,61,37)(14,38,62)(15,63,39)(16,40,64)(17,41,27)(18,28,42)(19,43,29)(20,30,44)(21,45,31)(22,32,46)(23,47,25)(24,26,48), (1,15,41)(2,16,42)(3,9,43)(4,10,44)(5,11,45)(6,12,46)(7,13,47)(8,14,48)(17,56,39)(18,49,40)(19,50,33)(20,51,34)(21,52,35)(22,53,36)(23,54,37)(24,55,38)(25,66,61)(26,67,62)(27,68,63)(28,69,64)(29,70,57)(30,71,58)(31,72,59)(32,65,60), (1,68,39)(2,40,69)(3,70,33)(4,34,71)(5,72,35)(6,36,65)(7,66,37)(8,38,67)(9,57,19)(10,20,58)(11,59,21)(12,22,60)(13,61,23)(14,24,62)(15,63,17)(16,18,64)(25,54,47)(26,48,55)(27,56,41)(28,42,49)(29,50,43)(30,44,51)(31,52,45)(32,46,53), (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)>;
G:=Group( (1,68,56)(2,49,69)(3,70,50)(4,51,71)(5,72,52)(6,53,65)(7,66,54)(8,55,67)(9,57,33)(10,34,58)(11,59,35)(12,36,60)(13,61,37)(14,38,62)(15,63,39)(16,40,64)(17,41,27)(18,28,42)(19,43,29)(20,30,44)(21,45,31)(22,32,46)(23,47,25)(24,26,48), (1,15,41)(2,16,42)(3,9,43)(4,10,44)(5,11,45)(6,12,46)(7,13,47)(8,14,48)(17,56,39)(18,49,40)(19,50,33)(20,51,34)(21,52,35)(22,53,36)(23,54,37)(24,55,38)(25,66,61)(26,67,62)(27,68,63)(28,69,64)(29,70,57)(30,71,58)(31,72,59)(32,65,60), (1,68,39)(2,40,69)(3,70,33)(4,34,71)(5,72,35)(6,36,65)(7,66,37)(8,38,67)(9,57,19)(10,20,58)(11,59,21)(12,22,60)(13,61,23)(14,24,62)(15,63,17)(16,18,64)(25,54,47)(26,48,55)(27,56,41)(28,42,49)(29,50,43)(30,44,51)(31,52,45)(32,46,53), (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) );
G=PermutationGroup([[(1,68,56),(2,49,69),(3,70,50),(4,51,71),(5,72,52),(6,53,65),(7,66,54),(8,55,67),(9,57,33),(10,34,58),(11,59,35),(12,36,60),(13,61,37),(14,38,62),(15,63,39),(16,40,64),(17,41,27),(18,28,42),(19,43,29),(20,30,44),(21,45,31),(22,32,46),(23,47,25),(24,26,48)], [(1,15,41),(2,16,42),(3,9,43),(4,10,44),(5,11,45),(6,12,46),(7,13,47),(8,14,48),(17,56,39),(18,49,40),(19,50,33),(20,51,34),(21,52,35),(22,53,36),(23,54,37),(24,55,38),(25,66,61),(26,67,62),(27,68,63),(28,69,64),(29,70,57),(30,71,58),(31,72,59),(32,65,60)], [(1,68,39),(2,40,69),(3,70,33),(4,34,71),(5,72,35),(6,36,65),(7,66,37),(8,38,67),(9,57,19),(10,20,58),(11,59,21),(12,22,60),(13,61,23),(14,24,62),(15,63,17),(16,18,64),(25,54,47),(26,48,55),(27,56,41),(28,42,49),(29,50,43),(30,44,51),(31,52,45),(32,46,53)], [(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)]])
He3⋊4C8 is a maximal subgroup of
He3⋊2C16 C32⋊C6⋊C8 He3⋊M4(2) He3⋊3SD16 He3⋊2D8 He3⋊2Q16 C8×He3⋊C2 He3⋊6M4(2) He3⋊8M4(2) He3⋊7D8 He3⋊9SD16 He3⋊11SD16 He3⋊7Q16
He3⋊4C8 is a maximal quotient of
He3⋊4C16
40 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 24A | ··· | 24H |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 1 | 1 | 1 | 1 | 6 | ··· | 6 | 9 | ··· | 9 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C8 | S3 | Dic3 | C3⋊C8 | He3⋊C2 | He3⋊3C4 | He3⋊4C8 |
| kernel | He3⋊4C8 | C4×He3 | C2×He3 | He3 | C3×C12 | C3×C6 | C32 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 4 | 4 | 8 |
Matrix representation of He3⋊4C8 ►in GL3(𝔽73) generated by
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 8 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 0 | 64 | 0 |
| 0 | 0 | 8 |
| 1 | 0 | 0 |
| 0 | 0 | 63 |
| 0 | 63 | 0 |
| 63 | 0 | 0 |
G:=sub<GL(3,GF(73))| [0,0,1,1,0,0,0,1,0],[8,0,0,0,8,0,0,0,8],[0,0,1,64,0,0,0,8,0],[0,0,63,0,63,0,63,0,0] >;
He3⋊4C8 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_4C_8 % in TeX
G:=Group("He3:4C8"); // GroupNames label
G:=SmallGroup(216,17);
// by ID
G=gap.SmallGroup(216,17);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,12,31,387,1444,382]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^8=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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