direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C4.4D8, C42.352D4, C42.703C23, C4.19(C2×D8), (C2×C4).92D8, (C4×C8)⋊70C22, C4⋊Q8⋊61C22, C4⋊C4.81C23, (C2×C4).79SD16, C4.20(C2×SD16), C22.72(C2×D8), C2.10(C22×D8), (C2×C4).326C24, (C2×C8).488C23, (C2×D4).96C23, C23.870(C2×D4), (C22×C4).608D4, D4⋊C4⋊59C22, C4.18(C4.4D4), C22.86(C2×SD16), C2.16(C22×SD16), C4⋊1D4.143C22, (C22×C8).517C22, C22.586(C22×D4), (C22×C4).1548C23, (C2×C42).1121C22, C22.80(C4.4D4), (C22×D4).364C22, (C2×C4×C8)⋊20C2, (C2×C4⋊Q8)⋊34C2, C4.35(C2×C4○D4), (C2×C4).849(C2×D4), (C2×D4⋊C4)⋊18C2, (C2×C4⋊1D4).22C2, C2.37(C2×C4.4D4), (C2×C4).705(C4○D4), (C2×C4⋊C4).618C22, SmallGroup(128,1860)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 628 in 264 conjugacy classes, 116 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×12], C4 [×4], C22, C22 [×6], C22 [×20], C8 [×4], C2×C4 [×2], C2×C4 [×16], C2×C4 [×8], D4 [×28], Q8 [×8], C23, C23 [×16], C42 [×4], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×2], C2×D4 [×4], C2×D4 [×26], C2×Q8 [×8], C24 [×2], C4×C8 [×4], D4⋊C4 [×16], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C4⋊1D4 [×4], C4⋊1D4 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C22×D4 [×2], C22×D4 [×2], C22×Q8, C2×C4×C8, C2×D4⋊C4 [×4], C4.4D8 [×8], C2×C4⋊1D4, C2×C4⋊Q8, C2×C4.4D8
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], SD16 [×4], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C2×D8 [×6], C2×SD16 [×6], C22×D4, C2×C4○D4 [×2], C4.4D8 [×4], C2×C4.4D4, C22×D8, C22×SD16, C2×C4.4D8
Generators and relations
G = < a,b,c,d | a2=b4=c8=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >
►On 64 points
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 57)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 41)
(1 60 24 40)(2 61 17 33)(3 62 18 34)(4 63 19 35)(5 64 20 36)(6 57 21 37)(7 58 22 38)(8 59 23 39)(9 41 55 27)(10 42 56 28)(11 43 49 29)(12 44 50 30)(13 45 51 31)(14 46 52 32)(15 47 53 25)(16 48 54 26)
(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 23 24 8)(2 7 17 22)(3 21 18 6)(4 5 19 20)(9 54 55 16)(10 15 56 53)(11 52 49 14)(12 13 50 51)(25 28 47 42)(26 41 48 27)(29 32 43 46)(30 45 44 31)(33 58 61 38)(34 37 62 57)(35 64 63 36)(39 60 59 40)
G:=sub<Sym(64)| (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,57)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,41), (1,60,24,40)(2,61,17,33)(3,62,18,34)(4,63,19,35)(5,64,20,36)(6,57,21,37)(7,58,22,38)(8,59,23,39)(9,41,55,27)(10,42,56,28)(11,43,49,29)(12,44,50,30)(13,45,51,31)(14,46,52,32)(15,47,53,25)(16,48,54,26), (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), (1,23,24,8)(2,7,17,22)(3,21,18,6)(4,5,19,20)(9,54,55,16)(10,15,56,53)(11,52,49,14)(12,13,50,51)(25,28,47,42)(26,41,48,27)(29,32,43,46)(30,45,44,31)(33,58,61,38)(34,37,62,57)(35,64,63,36)(39,60,59,40)>;
G:=Group( (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,57)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,41), (1,60,24,40)(2,61,17,33)(3,62,18,34)(4,63,19,35)(5,64,20,36)(6,57,21,37)(7,58,22,38)(8,59,23,39)(9,41,55,27)(10,42,56,28)(11,43,49,29)(12,44,50,30)(13,45,51,31)(14,46,52,32)(15,47,53,25)(16,48,54,26), (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), (1,23,24,8)(2,7,17,22)(3,21,18,6)(4,5,19,20)(9,54,55,16)(10,15,56,53)(11,52,49,14)(12,13,50,51)(25,28,47,42)(26,41,48,27)(29,32,43,46)(30,45,44,31)(33,58,61,38)(34,37,62,57)(35,64,63,36)(39,60,59,40) );
G=PermutationGroup([(1,55),(2,56),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,24),(10,17),(11,18),(12,19),(13,20),(14,21),(15,22),(16,23),(25,58),(26,59),(27,60),(28,61),(29,62),(30,63),(31,64),(32,57),(33,42),(34,43),(35,44),(36,45),(37,46),(38,47),(39,48),(40,41)], [(1,60,24,40),(2,61,17,33),(3,62,18,34),(4,63,19,35),(5,64,20,36),(6,57,21,37),(7,58,22,38),(8,59,23,39),(9,41,55,27),(10,42,56,28),(11,43,49,29),(12,44,50,30),(13,45,51,31),(14,46,52,32),(15,47,53,25),(16,48,54,26)], [(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)], [(1,23,24,8),(2,7,17,22),(3,21,18,6),(4,5,19,20),(9,54,55,16),(10,15,56,53),(11,52,49,14),(12,13,50,51),(25,28,47,42),(26,41,48,27),(29,32,43,46),(30,45,44,31),(33,58,61,38),(34,37,62,57),(35,64,63,36),(39,60,59,40)])
Matrix representation ►G ⊆ GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 16 | 0 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 5 |
| 0 | 0 | 0 | 12 | 5 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 5 | 5 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,0,0,0,16,0,0,0,1,0],[16,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,5,12,0,0,0,5,5],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,12,5,0,0,0,5,5] >;
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4L | 4M | 4N | 4O | 4P | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D8 | SD16 | C4○D4 |
| kernel | C2×C4.4D8 | C2×C4×C8 | C2×D4⋊C4 | C4.4D8 | C2×C4⋊1D4 | C2×C4⋊Q8 | C42 | C22×C4 | C2×C4 | C2×C4 | C2×C4 |
| # reps | 1 | 1 | 4 | 8 | 1 | 1 | 2 | 2 | 8 | 8 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_4._4D_8
% in TeX
G:=Group("C2xC4.4D8"); // GroupNames label
G:=SmallGroup(128,1860);
// by ID
G=gap.SmallGroup(128,1860);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,232,758,100,2804,172,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations