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 47)(2 48 69)(3 70 41)(4 42 71)(5 72 43)(6 44 65)(7 66 45)(8 46 67)(9 17 31)(10 32 18)(11 19 25)(12 26 20)(13 21 27)(14 28 22)(15 23 29)(16 30 24)(33 54 57)(34 58 55)(35 56 59)(36 60 49)(37 50 61)(38 62 51)(39 52 63)(40 64 53)
(1 52 31)(2 32 53)(3 54 25)(4 26 55)(5 56 27)(6 28 49)(7 50 29)(8 30 51)(9 68 63)(10 64 69)(11 70 57)(12 58 71)(13 72 59)(14 60 65)(15 66 61)(16 62 67)(17 47 39)(18 40 48)(19 41 33)(20 34 42)(21 43 35)(22 36 44)(23 45 37)(24 38 46)
(1 47 9)(2 48 10)(3 41 11)(4 42 12)(5 43 13)(6 44 14)(7 45 15)(8 46 16)(17 63 31)(18 64 32)(19 57 25)(20 58 26)(21 59 27)(22 60 28)(23 61 29)(24 62 30)(33 70 54)(34 71 55)(35 72 56)(36 65 49)(37 66 50)(38 67 51)(39 68 52)(40 69 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)| (1,68,47)(2,48,69)(3,70,41)(4,42,71)(5,72,43)(6,44,65)(7,66,45)(8,46,67)(9,17,31)(10,32,18)(11,19,25)(12,26,20)(13,21,27)(14,28,22)(15,23,29)(16,30,24)(33,54,57)(34,58,55)(35,56,59)(36,60,49)(37,50,61)(38,62,51)(39,52,63)(40,64,53), (1,52,31)(2,32,53)(3,54,25)(4,26,55)(5,56,27)(6,28,49)(7,50,29)(8,30,51)(9,68,63)(10,64,69)(11,70,57)(12,58,71)(13,72,59)(14,60,65)(15,66,61)(16,62,67)(17,47,39)(18,40,48)(19,41,33)(20,34,42)(21,43,35)(22,36,44)(23,45,37)(24,38,46), (1,47,9)(2,48,10)(3,41,11)(4,42,12)(5,43,13)(6,44,14)(7,45,15)(8,46,16)(17,63,31)(18,64,32)(19,57,25)(20,58,26)(21,59,27)(22,60,28)(23,61,29)(24,62,30)(33,70,54)(34,71,55)(35,72,56)(36,65,49)(37,66,50)(38,67,51)(39,68,52)(40,69,53), (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( (1,68,47)(2,48,69)(3,70,41)(4,42,71)(5,72,43)(6,44,65)(7,66,45)(8,46,67)(9,17,31)(10,32,18)(11,19,25)(12,26,20)(13,21,27)(14,28,22)(15,23,29)(16,30,24)(33,54,57)(34,58,55)(35,56,59)(36,60,49)(37,50,61)(38,62,51)(39,52,63)(40,64,53), (1,52,31)(2,32,53)(3,54,25)(4,26,55)(5,56,27)(6,28,49)(7,50,29)(8,30,51)(9,68,63)(10,64,69)(11,70,57)(12,58,71)(13,72,59)(14,60,65)(15,66,61)(16,62,67)(17,47,39)(18,40,48)(19,41,33)(20,34,42)(21,43,35)(22,36,44)(23,45,37)(24,38,46), (1,47,9)(2,48,10)(3,41,11)(4,42,12)(5,43,13)(6,44,14)(7,45,15)(8,46,16)(17,63,31)(18,64,32)(19,57,25)(20,58,26)(21,59,27)(22,60,28)(23,61,29)(24,62,30)(33,70,54)(34,71,55)(35,72,56)(36,65,49)(37,66,50)(38,67,51)(39,68,52)(40,69,53), (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([[(1,68,47),(2,48,69),(3,70,41),(4,42,71),(5,72,43),(6,44,65),(7,66,45),(8,46,67),(9,17,31),(10,32,18),(11,19,25),(12,26,20),(13,21,27),(14,28,22),(15,23,29),(16,30,24),(33,54,57),(34,58,55),(35,56,59),(36,60,49),(37,50,61),(38,62,51),(39,52,63),(40,64,53)], [(1,52,31),(2,32,53),(3,54,25),(4,26,55),(5,56,27),(6,28,49),(7,50,29),(8,30,51),(9,68,63),(10,64,69),(11,70,57),(12,58,71),(13,72,59),(14,60,65),(15,66,61),(16,62,67),(17,47,39),(18,40,48),(19,41,33),(20,34,42),(21,43,35),(22,36,44),(23,45,37),(24,38,46)], [(1,47,9),(2,48,10),(3,41,11),(4,42,12),(5,43,13),(6,44,14),(7,45,15),(8,46,16),(17,63,31),(18,64,32),(19,57,25),(20,58,26),(21,59,27),(22,60,28),(23,61,29),(24,62,30),(33,70,54),(34,71,55),(35,72,56),(36,65,49),(37,66,50),(38,67,51),(39,68,52),(40,69,53)], [(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⋊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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