direct product, metacyclic, nilpotent (class 2), monomial
Aliases: Q8xC32, C12.5C6, C2.2C62, C4.(C3xC6), C6.9(C2xC6), (C3xC12).5C2, (C3xC6).17C22, SmallGroup(72,38)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8xC32
G = < a,b,c,d | a3=b3=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Quotients: C1, C2, C3, C22, C6, Q8, C32, C2xC6, C3xC6, C3xQ8, C62, Q8xC32
Smallest permutation representation of Q8xC32
►Regular action on 72 points
Generators in S72
(1 40 31)(2 37 32)(3 38 29)(4 39 30)(5 35 27)(6 36 28)(7 33 25)(8 34 26)(9 24 14)(10 21 15)(11 22 16)(12 23 13)(17 54 50)(18 55 51)(19 56 52)(20 53 49)(41 71 45)(42 72 46)(43 69 47)(44 70 48)(57 68 61)(58 65 62)(59 66 63)(60 67 64)
(1 13 5)(2 14 6)(3 15 7)(4 16 8)(9 36 37)(10 33 38)(11 34 39)(12 35 40)(17 62 69)(18 63 70)(19 64 71)(20 61 72)(21 25 29)(22 26 30)(23 27 31)(24 28 32)(41 52 67)(42 49 68)(43 50 65)(44 51 66)(45 56 60)(46 53 57)(47 54 58)(48 55 59)
(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 52 3 50)(2 51 4 49)(5 41 7 43)(6 44 8 42)(9 63 11 61)(10 62 12 64)(13 67 15 65)(14 66 16 68)(17 40 19 38)(18 39 20 37)(21 58 23 60)(22 57 24 59)(25 47 27 45)(26 46 28 48)(29 54 31 56)(30 53 32 55)(33 69 35 71)(34 72 36 70)
G:=sub<Sym(72)| (1,40,31)(2,37,32)(3,38,29)(4,39,30)(5,35,27)(6,36,28)(7,33,25)(8,34,26)(9,24,14)(10,21,15)(11,22,16)(12,23,13)(17,54,50)(18,55,51)(19,56,52)(20,53,49)(41,71,45)(42,72,46)(43,69,47)(44,70,48)(57,68,61)(58,65,62)(59,66,63)(60,67,64), (1,13,5)(2,14,6)(3,15,7)(4,16,8)(9,36,37)(10,33,38)(11,34,39)(12,35,40)(17,62,69)(18,63,70)(19,64,71)(20,61,72)(21,25,29)(22,26,30)(23,27,31)(24,28,32)(41,52,67)(42,49,68)(43,50,65)(44,51,66)(45,56,60)(46,53,57)(47,54,58)(48,55,59), (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), (1,52,3,50)(2,51,4,49)(5,41,7,43)(6,44,8,42)(9,63,11,61)(10,62,12,64)(13,67,15,65)(14,66,16,68)(17,40,19,38)(18,39,20,37)(21,58,23,60)(22,57,24,59)(25,47,27,45)(26,46,28,48)(29,54,31,56)(30,53,32,55)(33,69,35,71)(34,72,36,70)>;
G:=Group( (1,40,31)(2,37,32)(3,38,29)(4,39,30)(5,35,27)(6,36,28)(7,33,25)(8,34,26)(9,24,14)(10,21,15)(11,22,16)(12,23,13)(17,54,50)(18,55,51)(19,56,52)(20,53,49)(41,71,45)(42,72,46)(43,69,47)(44,70,48)(57,68,61)(58,65,62)(59,66,63)(60,67,64), (1,13,5)(2,14,6)(3,15,7)(4,16,8)(9,36,37)(10,33,38)(11,34,39)(12,35,40)(17,62,69)(18,63,70)(19,64,71)(20,61,72)(21,25,29)(22,26,30)(23,27,31)(24,28,32)(41,52,67)(42,49,68)(43,50,65)(44,51,66)(45,56,60)(46,53,57)(47,54,58)(48,55,59), (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), (1,52,3,50)(2,51,4,49)(5,41,7,43)(6,44,8,42)(9,63,11,61)(10,62,12,64)(13,67,15,65)(14,66,16,68)(17,40,19,38)(18,39,20,37)(21,58,23,60)(22,57,24,59)(25,47,27,45)(26,46,28,48)(29,54,31,56)(30,53,32,55)(33,69,35,71)(34,72,36,70) );
G=PermutationGroup([[(1,40,31),(2,37,32),(3,38,29),(4,39,30),(5,35,27),(6,36,28),(7,33,25),(8,34,26),(9,24,14),(10,21,15),(11,22,16),(12,23,13),(17,54,50),(18,55,51),(19,56,52),(20,53,49),(41,71,45),(42,72,46),(43,69,47),(44,70,48),(57,68,61),(58,65,62),(59,66,63),(60,67,64)], [(1,13,5),(2,14,6),(3,15,7),(4,16,8),(9,36,37),(10,33,38),(11,34,39),(12,35,40),(17,62,69),(18,63,70),(19,64,71),(20,61,72),(21,25,29),(22,26,30),(23,27,31),(24,28,32),(41,52,67),(42,49,68),(43,50,65),(44,51,66),(45,56,60),(46,53,57),(47,54,58),(48,55,59)], [(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)], [(1,52,3,50),(2,51,4,49),(5,41,7,43),(6,44,8,42),(9,63,11,61),(10,62,12,64),(13,67,15,65),(14,66,16,68),(17,40,19,38),(18,39,20,37),(21,58,23,60),(22,57,24,59),(25,47,27,45),(26,46,28,48),(29,54,31,56),(30,53,32,55),(33,69,35,71),(34,72,36,70)]])
Q8xC32 is a maximal subgroup of
C32:11SD16 C32:7Q16 C12.26D6 Q8:3- 1+2 Q8:He3
45 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 4A | 4B | 4C | 6A | ··· | 6H | 12A | ··· | 12X |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | - | |||
| image | C1 | C2 | C3 | C6 | Q8 | C3xQ8 |
| kernel | Q8xC32 | C3xC12 | C3xQ8 | C12 | C32 | C3 |
| # reps | 1 | 3 | 8 | 24 | 1 | 8 |
Matrix representation of Q8xC32 ►in GL3(F13) generated by
| 1 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 9 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 12 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 12 | 0 |
| 1 | 0 | 0 |
| 0 | 9 | 10 |
| 0 | 10 | 4 |
G:=sub<GL(3,GF(13))| [1,0,0,0,3,0,0,0,3],[9,0,0,0,3,0,0,0,3],[12,0,0,0,0,12,0,1,0],[1,0,0,0,9,10,0,10,4] >;
Q8xC32 in GAP, Magma, Sage, TeX
Q_8\times C_3^2
% in TeX
G:=Group("Q8xC3^2"); // GroupNames label
G:=SmallGroup(72,38);
// by ID
G=gap.SmallGroup(72,38);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-2,180,381,186]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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