p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4.22(C4×D4), C8⋊3(C22⋊C4), (C2×SD16)⋊5C4, (C2×C8).208D4, C2.7(C8⋊D4), C2.5(C8⋊3D4), C4.86(C4⋊D4), C2.5(C8.2D4), C4.6(C4.4D4), C23.806(C2×D4), C22.186(C4×D4), (C22×C4).136D4, (C22×SD16).1C2, C22.42(C4⋊1D4), C22.97(C8⋊C22), (C22×C8).408C22, (C2×C42).324C22, (C22×D4).51C22, C2.19(SD16⋊C4), (C22×Q8).41C22, C22.148(C4⋊D4), (C22×C4).1413C23, C23.67C23⋊5C2, C22.86(C8.C22), C24.3C22.11C2, C2.25(C24.3C22), (C2×C8⋊C4)⋊2C2, (C2×C8).73(C2×C4), (C2×C2.D8)⋊34C2, C4.41(C2×C22⋊C4), (C2×Q8).97(C2×C4), (C2×Q8⋊C4)⋊48C2, (C2×D4).112(C2×C4), (C2×C4).1358(C2×D4), (C2×C4⋊C4).94C22, (C2×D4⋊C4).36C2, (C2×C4).604(C4○D4), (C2×C4).427(C22×C4), SmallGroup(128,705)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊(C22⋊C4)
G = < a,b,c,d | a8=b2=c2=d4=1, bab=a3, ac=ca, dad-1=a5, dbd-1=bc=cb, cd=dc >
Subgroups: 388 in 170 conjugacy classes, 64 normal (34 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C2.C42, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×C8, C2×SD16, C2×SD16, C22×D4, C22×Q8, C24.3C22, C23.67C23, C2×C8⋊C4, C2×D4⋊C4, C2×Q8⋊C4, C2×C2.D8, C22×SD16, C8⋊(C22⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C8⋊C22, C8.C22, C24.3C22, SD16⋊C4, C8⋊D4, C8⋊3D4, C8.2D4, C8⋊(C22⋊C4)
►On 64 points
(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 3)(2 6)(5 7)(9 11)(10 14)(13 15)(17 19)(18 22)(21 23)(25 63)(26 58)(27 61)(28 64)(29 59)(30 62)(31 57)(32 60)(33 54)(34 49)(35 52)(36 55)(37 50)(38 53)(39 56)(40 51)(41 43)(42 46)(45 47)
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 63)(26 64)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 49)
(1 40 13 30)(2 37 14 27)(3 34 15 32)(4 39 16 29)(5 36 9 26)(6 33 10 31)(7 38 11 28)(8 35 12 25)(17 53 41 64)(18 50 42 61)(19 55 43 58)(20 52 44 63)(21 49 45 60)(22 54 46 57)(23 51 47 62)(24 56 48 59)
G:=sub<Sym(64)| (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), (1,3)(2,6)(5,7)(9,11)(10,14)(13,15)(17,19)(18,22)(21,23)(25,63)(26,58)(27,61)(28,64)(29,59)(30,62)(31,57)(32,60)(33,54)(34,49)(35,52)(36,55)(37,50)(38,53)(39,56)(40,51)(41,43)(42,46)(45,47), (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,63)(26,64)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,49), (1,40,13,30)(2,37,14,27)(3,34,15,32)(4,39,16,29)(5,36,9,26)(6,33,10,31)(7,38,11,28)(8,35,12,25)(17,53,41,64)(18,50,42,61)(19,55,43,58)(20,52,44,63)(21,49,45,60)(22,54,46,57)(23,51,47,62)(24,56,48,59)>;
G:=Group( (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), (1,3)(2,6)(5,7)(9,11)(10,14)(13,15)(17,19)(18,22)(21,23)(25,63)(26,58)(27,61)(28,64)(29,59)(30,62)(31,57)(32,60)(33,54)(34,49)(35,52)(36,55)(37,50)(38,53)(39,56)(40,51)(41,43)(42,46)(45,47), (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,63)(26,64)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,49), (1,40,13,30)(2,37,14,27)(3,34,15,32)(4,39,16,29)(5,36,9,26)(6,33,10,31)(7,38,11,28)(8,35,12,25)(17,53,41,64)(18,50,42,61)(19,55,43,58)(20,52,44,63)(21,49,45,60)(22,54,46,57)(23,51,47,62)(24,56,48,59) );
G=PermutationGroup([[(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)], [(1,3),(2,6),(5,7),(9,11),(10,14),(13,15),(17,19),(18,22),(21,23),(25,63),(26,58),(27,61),(28,64),(29,59),(30,62),(31,57),(32,60),(33,54),(34,49),(35,52),(36,55),(37,50),(38,53),(39,56),(40,51),(41,43),(42,46),(45,47)], [(1,21),(2,22),(3,23),(4,24),(5,17),(6,18),(7,19),(8,20),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(25,63),(26,64),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62),(33,50),(34,51),(35,52),(36,53),(37,54),(38,55),(39,56),(40,49)], [(1,40,13,30),(2,37,14,27),(3,34,15,32),(4,39,16,29),(5,36,9,26),(6,33,10,31),(7,38,11,28),(8,35,12,25),(17,53,41,64),(18,50,42,61),(19,55,43,58),(20,52,44,63),(21,49,45,60),(22,54,46,57),(23,51,47,62),(24,56,48,59)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | C8⋊(C22⋊C4) | C24.3C22 | C23.67C23 | C2×C8⋊C4 | C2×D4⋊C4 | C2×Q8⋊C4 | C2×C2.D8 | C22×SD16 | C2×SD16 | C2×C8 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 6 | 2 | 4 | 2 | 2 |
Matrix representation of C8⋊(C22⋊C4) ►in GL8(𝔽17)
| 1 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 2 | 0 | 15 |
| 0 | 0 | 0 | 0 | 16 | 2 | 16 | 16 |
| 0 | 0 | 0 | 0 | 16 | 1 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 | 2 | 16 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 16 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 1 | 13 | 13 |
| 0 | 0 | 0 | 0 | 10 | 9 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 14 | 15 | 7 |
| 0 | 0 | 0 | 0 | 14 | 3 | 10 | 2 |
G:=sub<GL(8,GF(17))| [1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,16,16,16,0,0,0,0,2,2,1,2,0,0,0,0,0,16,16,16,0,0,0,0,15,16,0,0],[16,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,1,0,0,0,0,2,1,1,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,8,10,0,14,0,0,0,0,1,9,14,3,0,0,0,0,13,13,15,10,0,0,0,0,13,0,7,2] >;
C8⋊(C22⋊C4) in GAP, Magma, Sage, TeX
C_8\rtimes (C_2^2\rtimes C_4)
% in TeX
G:=Group("C8:(C2^2:C4)"); // GroupNames label
G:=SmallGroup(128,705);
// by ID
G=gap.SmallGroup(128,705);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,436,2019,248,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^2=d^4=1,b*a*b=a^3,a*c=c*a,d*a*d^-1=a^5,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations