direct product, metacyclic, nilpotent (class 3), monomial
Aliases: C32×SD16, C24⋊6C6, C4.2C62, C8⋊2(C3×C6), D4.(C3×C6), Q8⋊2(C3×C6), (C3×Q8)⋊6C6, (C3×C24)⋊10C2, (C3×D4).4C6, (C3×C6).42D4, C6.21(C3×D4), C12.24(C2×C6), (Q8×C32)⋊7C2, C2.4(D4×C32), (D4×C32).3C2, (C3×C12).51C22, SmallGroup(144,107)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×SD16
G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >
Subgroups: 90 in 60 conjugacy classes, 42 normal (14 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, D4, Q8, C32, C12, C12, C2×C6, SD16, C3×C6, C3×C6, C24, C3×D4, C3×Q8, C3×C12, C3×C12, C62, C3×SD16, C3×C24, D4×C32, Q8×C32, C32×SD16
Quotients: C1, C2, C3, C22, C6, D4, C32, C2×C6, SD16, C3×C6, C3×D4, C62, C3×SD16, D4×C32, C32×SD16
►On 72 points
Generators in S72
(1 34 59)(2 35 60)(3 36 61)(4 37 62)(5 38 63)(6 39 64)(7 40 57)(8 33 58)(9 30 17)(10 31 18)(11 32 19)(12 25 20)(13 26 21)(14 27 22)(15 28 23)(16 29 24)(41 67 51)(42 68 52)(43 69 53)(44 70 54)(45 71 55)(46 72 56)(47 65 49)(48 66 50)
(1 30 41)(2 31 42)(3 32 43)(4 25 44)(5 26 45)(6 27 46)(7 28 47)(8 29 48)(9 51 59)(10 52 60)(11 53 61)(12 54 62)(13 55 63)(14 56 64)(15 49 57)(16 50 58)(17 67 34)(18 68 35)(19 69 36)(20 70 37)(21 71 38)(22 72 39)(23 65 40)(24 66 33)
(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)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(18 20)(19 23)(22 24)(25 31)(27 29)(28 32)(33 39)(35 37)(36 40)(42 44)(43 47)(46 48)(49 53)(50 56)(52 54)(57 61)(58 64)(60 62)(65 69)(66 72)(68 70)
G:=sub<Sym(72)| (1,34,59)(2,35,60)(3,36,61)(4,37,62)(5,38,63)(6,39,64)(7,40,57)(8,33,58)(9,30,17)(10,31,18)(11,32,19)(12,25,20)(13,26,21)(14,27,22)(15,28,23)(16,29,24)(41,67,51)(42,68,52)(43,69,53)(44,70,54)(45,71,55)(46,72,56)(47,65,49)(48,66,50), (1,30,41)(2,31,42)(3,32,43)(4,25,44)(5,26,45)(6,27,46)(7,28,47)(8,29,48)(9,51,59)(10,52,60)(11,53,61)(12,54,62)(13,55,63)(14,56,64)(15,49,57)(16,50,58)(17,67,34)(18,68,35)(19,69,36)(20,70,37)(21,71,38)(22,72,39)(23,65,40)(24,66,33), (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), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(18,20)(19,23)(22,24)(25,31)(27,29)(28,32)(33,39)(35,37)(36,40)(42,44)(43,47)(46,48)(49,53)(50,56)(52,54)(57,61)(58,64)(60,62)(65,69)(66,72)(68,70)>;
G:=Group( (1,34,59)(2,35,60)(3,36,61)(4,37,62)(5,38,63)(6,39,64)(7,40,57)(8,33,58)(9,30,17)(10,31,18)(11,32,19)(12,25,20)(13,26,21)(14,27,22)(15,28,23)(16,29,24)(41,67,51)(42,68,52)(43,69,53)(44,70,54)(45,71,55)(46,72,56)(47,65,49)(48,66,50), (1,30,41)(2,31,42)(3,32,43)(4,25,44)(5,26,45)(6,27,46)(7,28,47)(8,29,48)(9,51,59)(10,52,60)(11,53,61)(12,54,62)(13,55,63)(14,56,64)(15,49,57)(16,50,58)(17,67,34)(18,68,35)(19,69,36)(20,70,37)(21,71,38)(22,72,39)(23,65,40)(24,66,33), (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), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(18,20)(19,23)(22,24)(25,31)(27,29)(28,32)(33,39)(35,37)(36,40)(42,44)(43,47)(46,48)(49,53)(50,56)(52,54)(57,61)(58,64)(60,62)(65,69)(66,72)(68,70) );
G=PermutationGroup([[(1,34,59),(2,35,60),(3,36,61),(4,37,62),(5,38,63),(6,39,64),(7,40,57),(8,33,58),(9,30,17),(10,31,18),(11,32,19),(12,25,20),(13,26,21),(14,27,22),(15,28,23),(16,29,24),(41,67,51),(42,68,52),(43,69,53),(44,70,54),(45,71,55),(46,72,56),(47,65,49),(48,66,50)], [(1,30,41),(2,31,42),(3,32,43),(4,25,44),(5,26,45),(6,27,46),(7,28,47),(8,29,48),(9,51,59),(10,52,60),(11,53,61),(12,54,62),(13,55,63),(14,56,64),(15,49,57),(16,50,58),(17,67,34),(18,68,35),(19,69,36),(20,70,37),(21,71,38),(22,72,39),(23,65,40),(24,66,33)], [(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)], [(2,4),(3,7),(6,8),(10,12),(11,15),(14,16),(18,20),(19,23),(22,24),(25,31),(27,29),(28,32),(33,39),(35,37),(36,40),(42,44),(43,47),(46,48),(49,53),(50,56),(52,54),(57,61),(58,64),(60,62),(65,69),(66,72),(68,70)]])
C32×SD16 is a maximal subgroup of
C24⋊7D6 C24.32D6 C24.40D6
63 conjugacy classes
| class | 1 | 2A | 2B | 3A | ··· | 3H | 4A | 4B | 6A | ··· | 6H | 6I | ··· | 6P | 8A | 8B | 12A | ··· | 12H | 12I | ··· | 12P | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 4 | 1 | ··· | 1 | 2 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | SD16 | C3×D4 | C3×SD16 |
| kernel | C32×SD16 | C3×C24 | D4×C32 | Q8×C32 | C3×SD16 | C24 | C3×D4 | C3×Q8 | C3×C6 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 8 | 1 | 2 | 8 | 16 |
Matrix representation of C32×SD16 ►in GL3(𝔽73) generated by
| 64 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 64 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 1 | 0 | 0 |
| 0 | 6 | 67 |
| 0 | 6 | 6 |
| 72 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 72 |
G:=sub<GL(3,GF(73))| [64,0,0,0,1,0,0,0,1],[64,0,0,0,64,0,0,0,64],[1,0,0,0,6,6,0,67,6],[72,0,0,0,1,0,0,0,72] >;
C32×SD16 in GAP, Magma, Sage, TeX
C_3^2\times {\rm SD}_{16} % in TeX
G:=Group("C3^2xSD16"); // GroupNames label
G:=SmallGroup(144,107);
// by ID
G=gap.SmallGroup(144,107);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-2,-2,432,457,3244,1630,88]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations