direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C22×C8.C4, C24.22Q8, M4(2).27C23, C8.60(C22×C4), (C23×C8).19C2, C4.52(C23×C4), (C22×C8).38C4, C23.77(C4⋊C4), (C2×C8).582C23, (C2×C4).190C24, C4.188(C22×D4), (C22×C4).823D4, (C22×C4).104Q8, C23.107(C2×Q8), C22.1(C22×Q8), (C22×C8).555C22, (C23×C4).697C22, (C22×C4).1507C23, (C22×M4(2)).30C2, (C2×M4(2)).339C22, C4.67(C2×C4⋊C4), C4○(C2×C8.C4), (C2×C4)○(C8.C4), (C2×C8).228(C2×C4), C2.29(C22×C4⋊C4), C22.38(C2×C4⋊C4), (C2×C4).240(C2×Q8), (C2×C4).152(C4⋊C4), (C2×C4).1569(C2×D4), (C2×C4).575(C22×C4), (C22×C4).497(C2×C4), (C2×C4)○(C2×C8.C4), SmallGroup(128,1646)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22×C8.C4
G = < a,b,c,d | a2=b2=c8=1, d4=c4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 300 in 240 conjugacy classes, 180 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C23, C23, C23, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C24, C8.C4, C22×C8, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C2×C8.C4, C23×C8, C22×M4(2), C22×C8.C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, C8.C4, C2×C4⋊C4, C23×C4, C22×D4, C22×Q8, C2×C8.C4, C22×C4⋊C4, C22×C8.C4
►On 64 points
(1 47)(2 48)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 64)(26 57)(27 58)(28 59)(29 60)(30 61)(31 62)(32 63)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 49)(40 50)
(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 38 45 57 5 34 41 61)(2 37 46 64 6 33 42 60)(3 36 47 63 7 40 43 59)(4 35 48 62 8 39 44 58)(9 50 19 28 13 54 23 32)(10 49 20 27 14 53 24 31)(11 56 21 26 15 52 17 30)(12 55 22 25 16 51 18 29)
G:=sub<Sym(64)| (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,57), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,64)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(32,63)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,49)(40,50), (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), (1,38,45,57,5,34,41,61)(2,37,46,64,6,33,42,60)(3,36,47,63,7,40,43,59)(4,35,48,62,8,39,44,58)(9,50,19,28,13,54,23,32)(10,49,20,27,14,53,24,31)(11,56,21,26,15,52,17,30)(12,55,22,25,16,51,18,29)>;
G:=Group( (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,57), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,64)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(32,63)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,49)(40,50), (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), (1,38,45,57,5,34,41,61)(2,37,46,64,6,33,42,60)(3,36,47,63,7,40,43,59)(4,35,48,62,8,39,44,58)(9,50,19,28,13,54,23,32)(10,49,20,27,14,53,24,31)(11,56,21,26,15,52,17,30)(12,55,22,25,16,51,18,29) );
G=PermutationGroup([[(1,47),(2,48),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,21),(10,22),(11,23),(12,24),(13,17),(14,18),(15,19),(16,20),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,58),(34,59),(35,60),(36,61),(37,62),(38,63),(39,64),(40,57)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,64),(26,57),(27,58),(28,59),(29,60),(30,61),(31,62),(32,63),(33,51),(34,52),(35,53),(36,54),(37,55),(38,56),(39,49),(40,50)], [(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)], [(1,38,45,57,5,34,41,61),(2,37,46,64,6,33,42,60),(3,36,47,63,7,40,43,59),(4,35,48,62,8,39,44,58),(9,50,19,28,13,54,23,32),(10,49,20,27,14,53,24,31),(11,56,21,26,15,52,17,30),(12,55,22,25,16,51,18,29)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 8A | ··· | 8P | 8Q | ··· | 8AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 | Q8 | C8.C4 |
| kernel | C22×C8.C4 | C2×C8.C4 | C23×C8 | C22×M4(2) | C22×C8 | C22×C4 | C22×C4 | C24 | C22 |
| # reps | 1 | 12 | 1 | 2 | 16 | 4 | 3 | 1 | 16 |
Matrix representation of C22×C8.C4 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 15 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 13 | 0 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,8,0,0,0,0,15],[1,0,0,0,0,1,0,0,0,0,0,13,0,0,1,0] >;
C22×C8.C4 in GAP, Magma, Sage, TeX
C_2^2\times C_8.C_4
% in TeX
G:=Group("C2^2xC8.C4"); // GroupNames label
G:=SmallGroup(128,1646);
// by ID
G=gap.SmallGroup(128,1646);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,2804,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^8=1,d^4=c^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations