direct product, p-group, metabelian, nilpotent (class 4), monomial
Aliases: C4×SD32, C42.330D4, C16⋊8(C2×C4), (C4×C16)⋊12C2, Q16⋊2(C2×C4), (C4×Q16)⋊1C2, (C4×D8).4C2, D8.2(C2×C4), C2.14(C4×D8), C4.26(C4×D4), C4○2(C16⋊4C4), C16⋊4C4⋊14C2, (C2×C8).232D4, (C2×C4).173D8, C4○3(C2.D16), C2.4(C2×SD32), C2.D16.8C2, C2.4(C4○D16), C8.39(C4○D4), C4.12(C4○D8), C8.36(C22×C4), (C2×SD32).5C2, C22.62(C2×D8), C4○2(C2.Q32), C2.Q32⋊21C2, (C2×C16).70C22, (C2×C8).503C23, (C4×C8).397C22, (C2×D8).104C22, C2.D8.151C22, (C2×Q16).102C22, (C2×C4).769(C2×D4), SmallGroup(128,905)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4×SD32
G = < a,b,c | a4=b16=c2=1, ab=ba, ac=ca, cbc=b7 >
Subgroups: 196 in 81 conjugacy classes, 42 normal (30 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C16, C16, C42, C42, C22⋊C4, C4⋊C4, C2×C8, D8, D8, Q16, Q16, C22×C4, C2×D4, C2×Q8, C4×C8, D4⋊C4, Q8⋊C4, C2.D8, C2×C16, SD32, C4×D4, C4×Q8, C2×D8, C2×Q16, C4×C16, C2.D16, C2.Q32, C16⋊4C4, C4×D8, C4×Q16, C2×SD32, C4×SD32
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D8, C22×C4, C2×D4, C4○D4, SD32, C4×D4, C2×D8, C4○D8, C4×D8, C2×SD32, C4○D16, C4×SD32
►On 64 points
(1 47 55 26)(2 48 56 27)(3 33 57 28)(4 34 58 29)(5 35 59 30)(6 36 60 31)(7 37 61 32)(8 38 62 17)(9 39 63 18)(10 40 64 19)(11 41 49 20)(12 42 50 21)(13 43 51 22)(14 44 52 23)(15 45 53 24)(16 46 54 25)
(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)
(1 55)(2 62)(3 53)(4 60)(5 51)(6 58)(7 49)(8 56)(9 63)(10 54)(11 61)(12 52)(13 59)(14 50)(15 57)(16 64)(17 48)(18 39)(19 46)(20 37)(21 44)(22 35)(23 42)(24 33)(25 40)(26 47)(27 38)(28 45)(29 36)(30 43)(31 34)(32 41)
G:=sub<Sym(64)| (1,47,55,26)(2,48,56,27)(3,33,57,28)(4,34,58,29)(5,35,59,30)(6,36,60,31)(7,37,61,32)(8,38,62,17)(9,39,63,18)(10,40,64,19)(11,41,49,20)(12,42,50,21)(13,43,51,22)(14,44,52,23)(15,45,53,24)(16,46,54,25), (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), (1,55)(2,62)(3,53)(4,60)(5,51)(6,58)(7,49)(8,56)(9,63)(10,54)(11,61)(12,52)(13,59)(14,50)(15,57)(16,64)(17,48)(18,39)(19,46)(20,37)(21,44)(22,35)(23,42)(24,33)(25,40)(26,47)(27,38)(28,45)(29,36)(30,43)(31,34)(32,41)>;
G:=Group( (1,47,55,26)(2,48,56,27)(3,33,57,28)(4,34,58,29)(5,35,59,30)(6,36,60,31)(7,37,61,32)(8,38,62,17)(9,39,63,18)(10,40,64,19)(11,41,49,20)(12,42,50,21)(13,43,51,22)(14,44,52,23)(15,45,53,24)(16,46,54,25), (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), (1,55)(2,62)(3,53)(4,60)(5,51)(6,58)(7,49)(8,56)(9,63)(10,54)(11,61)(12,52)(13,59)(14,50)(15,57)(16,64)(17,48)(18,39)(19,46)(20,37)(21,44)(22,35)(23,42)(24,33)(25,40)(26,47)(27,38)(28,45)(29,36)(30,43)(31,34)(32,41) );
G=PermutationGroup([[(1,47,55,26),(2,48,56,27),(3,33,57,28),(4,34,58,29),(5,35,59,30),(6,36,60,31),(7,37,61,32),(8,38,62,17),(9,39,63,18),(10,40,64,19),(11,41,49,20),(12,42,50,21),(13,43,51,22),(14,44,52,23),(15,45,53,24),(16,46,54,25)], [(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)], [(1,55),(2,62),(3,53),(4,60),(5,51),(6,58),(7,49),(8,56),(9,63),(10,54),(11,61),(12,52),(13,59),(14,50),(15,57),(16,64),(17,48),(18,39),(19,46),(20,37),(21,44),(22,35),(23,42),(24,33),(25,40),(26,47),(27,38),(28,45),(29,36),(30,43),(31,34),(32,41)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | C4○D4 | D8 | SD32 | C4○D8 | C4○D16 |
| kernel | C4×SD32 | C4×C16 | C2.D16 | C2.Q32 | C16⋊4C4 | C4×D8 | C4×Q16 | C2×SD32 | SD32 | C42 | C2×C8 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 2 | 4 | 8 | 4 | 8 |
Matrix representation of C4×SD32 ►in GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 14 | 14 | 0 | 0 |
| 3 | 14 | 0 | 0 |
| 0 | 0 | 1 | 7 |
| 0 | 0 | 10 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,4,0,0,0,0,4],[14,3,0,0,14,14,0,0,0,0,1,10,0,0,7,1],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1] >;
C4×SD32 in GAP, Magma, Sage, TeX
C_4\times {\rm SD}_{32} % in TeX
G:=Group("C4xSD32"); // GroupNames label
G:=SmallGroup(128,905);
// by ID
G=gap.SmallGroup(128,905);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,100,1123,570,360,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c|a^4=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^7>;
// generators/relations