direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C4.4D8, C42.352D4, C42.703C23, (C2×C4).92D8, C4.19(C2×D8), (C4×C8)⋊70C22, C4⋊Q8⋊61C22, C4⋊C4.81C23, C4.20(C2×SD16), (C2×C4).79SD16, C22.72(C2×D8), C2.10(C22×D8), (C2×C8).488C23, (C2×C4).326C24, (C2×D4).96C23, (C22×C4).608D4, C23.870(C2×D4), D4⋊C4⋊59C22, C4.18(C4.4D4), C2.16(C22×SD16), C22.86(C2×SD16), C4⋊1D4.143C22, (C22×C8).517C22, C22.586(C22×D4), (C2×C42).1121C22, (C22×C4).1548C23, 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
Generators and relations for C2×C4.4D8
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 >
Subgroups: 628 in 264 conjugacy classes, 116 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C4×C8, D4⋊C4, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4⋊1D4, C4⋊1D4, C4⋊Q8, C4⋊Q8, C22×C8, C22×D4, C22×D4, C22×Q8, C2×C4×C8, C2×D4⋊C4, C4.4D8, C2×C4⋊1D4, C2×C4⋊Q8, C2×C4.4D8
Quotients: C1, C2, C22, D4, C23, D8, SD16, C2×D4, C4○D4, C24, C4.4D4, C2×D8, C2×SD16, C22×D4, C2×C4○D4, C4.4D8, C2×C4.4D4, C22×D8, C22×SD16, C2×C4.4D8
►On 64 points
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 41)(32 42)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)
(1 60 24 31)(2 61 17 32)(3 62 18 25)(4 63 19 26)(5 64 20 27)(6 57 21 28)(7 58 22 29)(8 59 23 30)(9 45 51 40)(10 46 52 33)(11 47 53 34)(12 48 54 35)(13 41 55 36)(14 42 56 37)(15 43 49 38)(16 44 50 39)
(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 50 51 16)(10 15 52 49)(11 56 53 14)(12 13 54 55)(25 28 62 57)(26 64 63 27)(29 32 58 61)(30 60 59 31)(33 43 46 38)(34 37 47 42)(35 41 48 36)(39 45 44 40)
G:=sub<Sym(64)| (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,41)(32,42)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64), (1,60,24,31)(2,61,17,32)(3,62,18,25)(4,63,19,26)(5,64,20,27)(6,57,21,28)(7,58,22,29)(8,59,23,30)(9,45,51,40)(10,46,52,33)(11,47,53,34)(12,48,54,35)(13,41,55,36)(14,42,56,37)(15,43,49,38)(16,44,50,39), (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)(9,50,51,16)(10,15,52,49)(11,56,53,14)(12,13,54,55)(25,28,62,57)(26,64,63,27)(29,32,58,61)(30,60,59,31)(33,43,46,38)(34,37,47,42)(35,41,48,36)(39,45,44,40)>;
G:=Group( (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,41)(32,42)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64), (1,60,24,31)(2,61,17,32)(3,62,18,25)(4,63,19,26)(5,64,20,27)(6,57,21,28)(7,58,22,29)(8,59,23,30)(9,45,51,40)(10,46,52,33)(11,47,53,34)(12,48,54,35)(13,41,55,36)(14,42,56,37)(15,43,49,38)(16,44,50,39), (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)(9,50,51,16)(10,15,52,49)(11,56,53,14)(12,13,54,55)(25,28,62,57)(26,64,63,27)(29,32,58,61)(30,60,59,31)(33,43,46,38)(34,37,47,42)(35,41,48,36)(39,45,44,40) );
G=PermutationGroup([[(1,55),(2,56),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,20),(10,21),(11,22),(12,23),(13,24),(14,17),(15,18),(16,19),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,41),(32,42),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64)], [(1,60,24,31),(2,61,17,32),(3,62,18,25),(4,63,19,26),(5,64,20,27),(6,57,21,28),(7,58,22,29),(8,59,23,30),(9,45,51,40),(10,46,52,33),(11,47,53,34),(12,48,54,35),(13,41,55,36),(14,42,56,37),(15,43,49,38),(16,44,50,39)], [(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),(9,50,51,16),(10,15,52,49),(11,56,53,14),(12,13,54,55),(25,28,62,57),(26,64,63,27),(29,32,58,61),(30,60,59,31),(33,43,46,38),(34,37,47,42),(35,41,48,36),(39,45,44,40)]])
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 |
Matrix representation of C2×C4.4D8 ►in 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] >;
C2×C4.4D8 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