metabelian, supersoluble, monomial
Aliases: He3⋊3C8, C32⋊2C24, (C3×C6).C12, C32⋊4C8⋊C3, C12.9(C3×S3), C32⋊2(C3⋊C8), (C3×C12).2C6, (C3×C12).5S3, C2.(C32⋊C12), (C2×He3).2C4, (C4×He3).3C2, (C3×C6).1Dic3, C6.2(C3×Dic3), C4.2(C32⋊C6), C3.2(C3×C3⋊C8), SmallGroup(216,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — He3⋊3C8 |
Generators and relations for He3⋊3C8
G = < a,b,c,d | a3=b3=c3=d8=1, ab=ba, cac-1=ab-1, dad-1=a-1, bc=cb, dbd-1=b-1, cd=dc >
Smallest permutation representation of He3⋊3C8
►On 72 points
Generators in S72
(1 68 11)(2 12 69)(3 70 13)(4 14 71)(5 72 15)(6 16 65)(7 66 9)(8 10 67)(17 30 43)(18 44 31)(19 32 45)(20 46 25)(21 26 47)(22 48 27)(23 28 41)(24 42 29)(33 56 62)(34 63 49)(35 50 64)(36 57 51)(37 52 58)(38 59 53)(39 54 60)(40 61 55)
(1 34 31)(2 32 35)(3 36 25)(4 26 37)(5 38 27)(6 28 39)(7 40 29)(8 30 33)(9 55 42)(10 43 56)(11 49 44)(12 45 50)(13 51 46)(14 47 52)(15 53 48)(16 41 54)(17 62 67)(18 68 63)(19 64 69)(20 70 57)(21 58 71)(22 72 59)(23 60 65)(24 66 61)
(1 11 18)(2 12 19)(3 13 20)(4 14 21)(5 15 22)(6 16 23)(7 9 24)(8 10 17)(25 46 57)(26 47 58)(27 48 59)(28 41 60)(29 42 61)(30 43 62)(31 44 63)(32 45 64)(33 56 67)(34 49 68)(35 50 69)(36 51 70)(37 52 71)(38 53 72)(39 54 65)(40 55 66)
(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)| (1,68,11)(2,12,69)(3,70,13)(4,14,71)(5,72,15)(6,16,65)(7,66,9)(8,10,67)(17,30,43)(18,44,31)(19,32,45)(20,46,25)(21,26,47)(22,48,27)(23,28,41)(24,42,29)(33,56,62)(34,63,49)(35,50,64)(36,57,51)(37,52,58)(38,59,53)(39,54,60)(40,61,55), (1,34,31)(2,32,35)(3,36,25)(4,26,37)(5,38,27)(6,28,39)(7,40,29)(8,30,33)(9,55,42)(10,43,56)(11,49,44)(12,45,50)(13,51,46)(14,47,52)(15,53,48)(16,41,54)(17,62,67)(18,68,63)(19,64,69)(20,70,57)(21,58,71)(22,72,59)(23,60,65)(24,66,61), (1,11,18)(2,12,19)(3,13,20)(4,14,21)(5,15,22)(6,16,23)(7,9,24)(8,10,17)(25,46,57)(26,47,58)(27,48,59)(28,41,60)(29,42,61)(30,43,62)(31,44,63)(32,45,64)(33,56,67)(34,49,68)(35,50,69)(36,51,70)(37,52,71)(38,53,72)(39,54,65)(40,55,66), (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( (1,68,11)(2,12,69)(3,70,13)(4,14,71)(5,72,15)(6,16,65)(7,66,9)(8,10,67)(17,30,43)(18,44,31)(19,32,45)(20,46,25)(21,26,47)(22,48,27)(23,28,41)(24,42,29)(33,56,62)(34,63,49)(35,50,64)(36,57,51)(37,52,58)(38,59,53)(39,54,60)(40,61,55), (1,34,31)(2,32,35)(3,36,25)(4,26,37)(5,38,27)(6,28,39)(7,40,29)(8,30,33)(9,55,42)(10,43,56)(11,49,44)(12,45,50)(13,51,46)(14,47,52)(15,53,48)(16,41,54)(17,62,67)(18,68,63)(19,64,69)(20,70,57)(21,58,71)(22,72,59)(23,60,65)(24,66,61), (1,11,18)(2,12,19)(3,13,20)(4,14,21)(5,15,22)(6,16,23)(7,9,24)(8,10,17)(25,46,57)(26,47,58)(27,48,59)(28,41,60)(29,42,61)(30,43,62)(31,44,63)(32,45,64)(33,56,67)(34,49,68)(35,50,69)(36,51,70)(37,52,71)(38,53,72)(39,54,65)(40,55,66), (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([(1,68,11),(2,12,69),(3,70,13),(4,14,71),(5,72,15),(6,16,65),(7,66,9),(8,10,67),(17,30,43),(18,44,31),(19,32,45),(20,46,25),(21,26,47),(22,48,27),(23,28,41),(24,42,29),(33,56,62),(34,63,49),(35,50,64),(36,57,51),(37,52,58),(38,59,53),(39,54,60),(40,61,55)], [(1,34,31),(2,32,35),(3,36,25),(4,26,37),(5,38,27),(6,28,39),(7,40,29),(8,30,33),(9,55,42),(10,43,56),(11,49,44),(12,45,50),(13,51,46),(14,47,52),(15,53,48),(16,41,54),(17,62,67),(18,68,63),(19,64,69),(20,70,57),(21,58,71),(22,72,59),(23,60,65),(24,66,61)], [(1,11,18),(2,12,19),(3,13,20),(4,14,21),(5,15,22),(6,16,23),(7,9,24),(8,10,17),(25,46,57),(26,47,58),(27,48,59),(28,41,60),(29,42,61),(30,43,62),(31,44,63),(32,45,64),(33,56,67),(34,49,68),(35,50,69),(36,51,70),(37,52,71),(38,53,72),(39,54,65),(40,55,66)], [(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⋊3C8 is a maximal subgroup of
C32⋊C6⋊C8 He3⋊M4(2) C12.89S32 He3⋊3M4(2) He3⋊3D8 He3⋊4SD16 He3⋊5SD16 He3⋊3Q16 C8×C32⋊C6 He3⋊5M4(2) He3⋊7M4(2) He3⋊8SD16 He3⋊6D8 He3⋊6Q16 He3⋊10SD16
He3⋊3C8 is a maximal quotient of
He3⋊3C16
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 | 12F | 12G | ··· | 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 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 3 | 3 | 6 | 6 | 6 | 1 | 1 | 2 | 3 | 3 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 9 | ··· | 9 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 |
| type | + | + | + | - | + | - | |||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 | S3 | Dic3 | C3×S3 | C3⋊C8 | C3×Dic3 | C3×C3⋊C8 | C32⋊C6 | C32⋊C12 | He3⋊3C8 |
| kernel | He3⋊3C8 | C4×He3 | C32⋊4C8 | C2×He3 | C3×C12 | He3 | C3×C6 | C32 | C3×C12 | C3×C6 | C12 | C32 | C6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of He3⋊3C8 ►in GL6(𝔽73)
| 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 9 | 20 | 11 | 64 | 9 | 20 |
| 11 | 64 | 53 | 62 | 11 | 64 |
| 11 | 64 | 9 | 20 | 9 | 20 |
| 53 | 62 | 11 | 64 | 11 | 64 |
| 9 | 20 | 9 | 20 | 11 | 64 |
| 11 | 64 | 11 | 64 | 53 | 62 |
G:=sub<GL(6,GF(73))| [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],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,72,0,0,0,0,1,72,0,0,0,0],[9,11,11,53,9,11,20,64,64,62,20,64,11,53,9,11,9,11,64,62,20,64,20,64,9,11,9,11,11,53,20,64,20,64,64,62] >;
He3⋊3C8 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_3C_8 % in TeX
G:=Group("He3:3C8"); // GroupNames label
G:=SmallGroup(216,14);
// by ID
G=gap.SmallGroup(216,14);
# by ID
G:=PCGroup([6,-2,-3,-2,-2,-3,-3,36,50,1444,1450,5189]);
// 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,d*b*d^-1=b^-1,c*d=d*c>;
// generators/relations
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