direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C4⋊SD16, C42.208D4, C42.317C23, Q8⋊1(C2×D4), (C2×Q8)⋊27D4, C4⋊4(C2×SD16), C4⋊C8⋊73C22, (C2×C4)⋊16SD16, (C4×Q8)⋊77C22, C4.63(C22×D4), C4.66(C4⋊D4), C4⋊C4.373C23, (C2×C8).305C23, (C2×C4).236C24, (C2×D4).46C23, C23.855(C2×D4), (C22×C4).791D4, D4⋊C4⋊74C22, C2.8(C22×SD16), (C22×SD16)⋊22C2, (C2×SD16)⋊73C22, (C2×Q8).357C23, C22.83(C2×SD16), C4⋊1D4.135C22, (C2×C42).805C22, (C22×C8).338C22, C22.496(C22×D4), C22.168(C4⋊D4), C22.114(C8⋊C22), (C22×C4).1526C23, (C22×D4).336C22, (C22×Q8).469C22, (C2×C4⋊C8)⋊36C2, (C2×C4×Q8)⋊34C2, C4.146(C2×C4○D4), C2.54(C2×C4⋊D4), C2.14(C2×C8⋊C22), (C2×D4⋊C4)⋊38C2, (C2×C4).1416(C2×D4), (C2×C4⋊1D4).21C2, (C2×C4).903(C4○D4), (C2×C4⋊C4).917C22, SmallGroup(128,1764)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C4⋊SD16
G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c3 >
Subgroups: 652 in 292 conjugacy classes, 116 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, D4⋊C4, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C4×Q8, C4⋊1D4, C4⋊1D4, C22×C8, C2×SD16, C2×SD16, C22×D4, C22×D4, C22×Q8, C2×D4⋊C4, C2×C4⋊C8, C4⋊SD16, C2×C4×Q8, C2×C4⋊1D4, C22×SD16, C2×C4⋊SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C24, C4⋊D4, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C4⋊SD16, C2×C4⋊D4, C22×SD16, C2×C8⋊C22, C2×C4⋊SD16
►On 64 points
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(25 39)(26 40)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 57)
(1 37 42 63)(2 64 43 38)(3 39 44 57)(4 58 45 40)(5 33 46 59)(6 60 47 34)(7 35 48 61)(8 62 41 36)(9 54 19 31)(10 32 20 55)(11 56 21 25)(12 26 22 49)(13 50 23 27)(14 28 24 51)(15 52 17 29)(16 30 18 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)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(17 21)(18 24)(20 22)(25 52)(26 55)(27 50)(28 53)(29 56)(30 51)(31 54)(32 49)(33 59)(34 62)(35 57)(36 60)(37 63)(38 58)(39 61)(40 64)(41 47)(43 45)(44 48)
G:=sub<Sym(64)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,39)(26,40)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,57), (1,37,42,63)(2,64,43,38)(3,39,44,57)(4,58,45,40)(5,33,46,59)(6,60,47,34)(7,35,48,61)(8,62,41,36)(9,54,19,31)(10,32,20,55)(11,56,21,25)(12,26,22,49)(13,50,23,27)(14,28,24,51)(15,52,17,29)(16,30,18,53), (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), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(17,21)(18,24)(20,22)(25,52)(26,55)(27,50)(28,53)(29,56)(30,51)(31,54)(32,49)(33,59)(34,62)(35,57)(36,60)(37,63)(38,58)(39,61)(40,64)(41,47)(43,45)(44,48)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,39)(26,40)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,57), (1,37,42,63)(2,64,43,38)(3,39,44,57)(4,58,45,40)(5,33,46,59)(6,60,47,34)(7,35,48,61)(8,62,41,36)(9,54,19,31)(10,32,20,55)(11,56,21,25)(12,26,22,49)(13,50,23,27)(14,28,24,51)(15,52,17,29)(16,30,18,53), (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), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(17,21)(18,24)(20,22)(25,52)(26,55)(27,50)(28,53)(29,56)(30,51)(31,54)(32,49)(33,59)(34,62)(35,57)(36,60)(37,63)(38,58)(39,61)(40,64)(41,47)(43,45)(44,48) );
G=PermutationGroup([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(25,39),(26,40),(27,33),(28,34),(29,35),(30,36),(31,37),(32,38),(49,58),(50,59),(51,60),(52,61),(53,62),(54,63),(55,64),(56,57)], [(1,37,42,63),(2,64,43,38),(3,39,44,57),(4,58,45,40),(5,33,46,59),(6,60,47,34),(7,35,48,61),(8,62,41,36),(9,54,19,31),(10,32,20,55),(11,56,21,25),(12,26,22,49),(13,50,23,27),(14,28,24,51),(15,52,17,29),(16,30,18,53)], [(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)], [(2,4),(3,7),(6,8),(10,12),(11,15),(14,16),(17,21),(18,24),(20,22),(25,52),(26,55),(27,50),(28,53),(29,56),(30,51),(31,54),(32,49),(33,59),(34,62),(35,57),(36,60),(37,63),(38,58),(39,61),(40,64),(41,47),(43,45),(44,48)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4R | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | SD16 | C4○D4 | C8⋊C22 |
| kernel | C2×C4⋊SD16 | C2×D4⋊C4 | C2×C4⋊C8 | C4⋊SD16 | C2×C4×Q8 | C2×C4⋊1D4 | C22×SD16 | C42 | C22×C4 | C2×Q8 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 4 | 2 |
Matrix representation of C2×C4⋊SD16 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 2 | 0 | 0 |
| 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 10 | 7 |
| 0 | 0 | 0 | 5 | 0 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 16 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,16,0,0,0,2,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,1,0,0,0,0,1,0,0,0,0,0,10,5,0,0,0,7,0],[16,0,0,0,0,0,1,16,0,0,0,0,16,0,0,0,0,0,1,1,0,0,0,0,16] >;
C2×C4⋊SD16 in GAP, Magma, Sage, TeX
C_2\times C_4\rtimes {\rm SD}_{16} % in TeX
G:=Group("C2xC4:SD16"); // GroupNames label
G:=SmallGroup(128,1764);
// by ID
G=gap.SmallGroup(128,1764);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,352,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^3>;
// generators/relations