direct product, p-group, metabelian, nilpotent (class 4), monomial
Aliases: C2×D8.C4, C23.27SD16, C4○D8.2C4, (C2×D8).9C4, D8.9(C2×C4), C4.88(C2×D8), (C22×C16)⋊6C2, C8.114(C2×D4), (C2×C4).169D8, (C2×C8).269D4, (C2×Q16).9C4, Q16.9(C2×C4), (C2×C16)⋊17C22, C4○(D8.C4), C8.33(C22×C4), (C2×C4).78SD16, C8.26(C22⋊C4), C8.C4⋊7C22, (C2×C8).576C23, C4○D8.12C22, (C22×C4).585D4, C4.25(D4⋊C4), C22.1(C2×SD16), C22.8(D4⋊C4), (C22×C8).552C22, (C2×C4○D8).4C2, (C2×C8).180(C2×C4), (C2×C4).760(C2×D4), C4.54(C2×C22⋊C4), (C2×C8.C4)⋊17C2, C2.32(C2×D4⋊C4), (C2×C4).272(C22⋊C4), SmallGroup(128,874)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×D8.C4
G = < a,b,c,d | a2=b8=c2=1, d4=b4, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=b5c >
Subgroups: 244 in 112 conjugacy classes, 52 normal (34 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C16, C2×C8, C2×C8, M4(2), D8, D8, SD16, Q16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C8.C4, C8.C4, C2×C16, C2×C16, C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C4○D8, C4○D8, C2×C4○D4, D8.C4, C2×C8.C4, C22×C16, C2×C4○D8, C2×D8.C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, D8.C4, C2×D4⋊C4, C2×D8.C4
►On 64 points
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 56)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 33)(24 34)(41 61)(42 62)(43 63)(44 64)(45 57)(46 58)(47 59)(48 60)
(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 8)(2 7)(3 6)(4 5)(9 13)(10 12)(14 16)(17 20)(18 19)(21 24)(22 23)(25 30)(26 29)(27 28)(31 32)(33 40)(34 39)(35 38)(36 37)(42 48)(43 47)(44 46)(49 51)(52 56)(53 55)(58 64)(59 63)(60 62)
(1 12 23 58 5 16 19 62)(2 11 24 57 6 15 20 61)(3 10 17 64 7 14 21 60)(4 9 18 63 8 13 22 59)(25 54 38 41 29 50 34 45)(26 53 39 48 30 49 35 44)(27 52 40 47 31 56 36 43)(28 51 33 46 32 55 37 42)
G:=sub<Sym(64)| (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,56)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,33)(24,34)(41,61)(42,62)(43,63)(44,64)(45,57)(46,58)(47,59)(48,60), (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), (1,8)(2,7)(3,6)(4,5)(9,13)(10,12)(14,16)(17,20)(18,19)(21,24)(22,23)(25,30)(26,29)(27,28)(31,32)(33,40)(34,39)(35,38)(36,37)(42,48)(43,47)(44,46)(49,51)(52,56)(53,55)(58,64)(59,63)(60,62), (1,12,23,58,5,16,19,62)(2,11,24,57,6,15,20,61)(3,10,17,64,7,14,21,60)(4,9,18,63,8,13,22,59)(25,54,38,41,29,50,34,45)(26,53,39,48,30,49,35,44)(27,52,40,47,31,56,36,43)(28,51,33,46,32,55,37,42)>;
G:=Group( (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,56)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,33)(24,34)(41,61)(42,62)(43,63)(44,64)(45,57)(46,58)(47,59)(48,60), (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), (1,8)(2,7)(3,6)(4,5)(9,13)(10,12)(14,16)(17,20)(18,19)(21,24)(22,23)(25,30)(26,29)(27,28)(31,32)(33,40)(34,39)(35,38)(36,37)(42,48)(43,47)(44,46)(49,51)(52,56)(53,55)(58,64)(59,63)(60,62), (1,12,23,58,5,16,19,62)(2,11,24,57,6,15,20,61)(3,10,17,64,7,14,21,60)(4,9,18,63,8,13,22,59)(25,54,38,41,29,50,34,45)(26,53,39,48,30,49,35,44)(27,52,40,47,31,56,36,43)(28,51,33,46,32,55,37,42) );
G=PermutationGroup([[(1,28),(2,29),(3,30),(4,31),(5,32),(6,25),(7,26),(8,27),(9,56),(10,49),(11,50),(12,51),(13,52),(14,53),(15,54),(16,55),(17,35),(18,36),(19,37),(20,38),(21,39),(22,40),(23,33),(24,34),(41,61),(42,62),(43,63),(44,64),(45,57),(46,58),(47,59),(48,60)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,13),(10,12),(14,16),(17,20),(18,19),(21,24),(22,23),(25,30),(26,29),(27,28),(31,32),(33,40),(34,39),(35,38),(36,37),(42,48),(43,47),(44,46),(49,51),(52,56),(53,55),(58,64),(59,63),(60,62)], [(1,12,23,58,5,16,19,62),(2,11,24,57,6,15,20,61),(3,10,17,64,7,14,21,60),(4,9,18,63,8,13,22,59),(25,54,38,41,29,50,34,45),(26,53,39,48,30,49,35,44),(27,52,40,47,31,56,36,43),(28,51,33,46,32,55,37,42)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D8 | SD16 | SD16 | D8.C4 |
| kernel | C2×D8.C4 | D8.C4 | C2×C8.C4 | C22×C16 | C2×C4○D8 | C2×D8 | C2×Q16 | C4○D8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 3 | 1 | 4 | 2 | 2 | 16 |
Matrix representation of C2×D8.C4 ►in GL3(𝔽17) generated by
| 16 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 0 | 6 |
| 0 | 14 | 6 |
| 16 | 0 | 0 |
| 0 | 0 | 6 |
| 0 | 3 | 0 |
| 13 | 0 | 0 |
| 0 | 16 | 13 |
| 0 | 14 | 1 |
G:=sub<GL(3,GF(17))| [16,0,0,0,1,0,0,0,1],[1,0,0,0,0,14,0,6,6],[16,0,0,0,0,3,0,6,0],[13,0,0,0,16,14,0,13,1] >;
C2×D8.C4 in GAP, Magma, Sage, TeX
C_2\times D_8.C_4
% in TeX
G:=Group("C2xD8.C4"); // GroupNames label
G:=SmallGroup(128,874);
// by ID
G=gap.SmallGroup(128,874);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1123,570,360,172,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^2=1,d^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b^5*c>;
// generators/relations