direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C4.C42, C24.18Q8, (C23×C8).5C2, (C2×C4).67C42, (C22×C8).28C4, C4.19(C2×C42), (C22×C4).88Q8, C23.75(C2×Q8), C23.62(C4⋊C4), C4○(C4.C42), (C22×C4).754D4, (C2×M4(2)).25C4, M4(2).27(C2×C4), (C23×C4).669C22, (C22×C8).467C22, C22.12(C8.C4), C4.19(C2.C42), (C22×C4).1303C23, (C22×M4(2)).13C2, (C2×M4(2)).297C22, C22.30(C2.C42), (C2×C8).201(C2×C4), C2.4(C2×C8.C4), C22.10(C2×C4⋊C4), (C2×C4).124(C4⋊C4), (C2×C4).1294(C2×D4), C4.104(C2×C22⋊C4), (C2×C4)○(C4.C42), (C2×C4).347(C22×C4), (C22×C4).403(C2×C4), (C2×C4).394(C22⋊C4), C2.14(C2×C2.C42), SmallGroup(128,469)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C4.C42
G = < a,b,c,d | a2=b4=1, c4=d4=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c >
Subgroups: 276 in 196 conjugacy classes, 116 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C23, C23, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C24, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C4.C42, C23×C8, C22×M4(2), C2×C4.C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2.C42, C8.C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4.C42, C2×C2.C42, C2×C8.C4, C2×C4.C42
►On 64 points
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 41)(16 42)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 49)(24 50)(33 64)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)
(1 27 5 31)(2 32 6 28)(3 29 7 25)(4 26 8 30)(9 41 13 45)(10 46 14 42)(11 43 15 47)(12 48 16 44)(17 49 21 53)(18 54 22 50)(19 51 23 55)(20 56 24 52)(33 58 37 62)(34 63 38 59)(35 60 39 64)(36 57 40 61)
(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 59 9 23 5 63 13 19)(2 35 14 52 6 39 10 56)(3 61 11 17 7 57 15 21)(4 37 16 54 8 33 12 50)(18 26 62 44 22 30 58 48)(20 28 64 46 24 32 60 42)(25 40 47 53 29 36 43 49)(27 34 41 55 31 38 45 51)
G:=sub<Sym(64)| (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,41)(16,42)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(33,64)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63), (1,27,5,31)(2,32,6,28)(3,29,7,25)(4,26,8,30)(9,41,13,45)(10,46,14,42)(11,43,15,47)(12,48,16,44)(17,49,21,53)(18,54,22,50)(19,51,23,55)(20,56,24,52)(33,58,37,62)(34,63,38,59)(35,60,39,64)(36,57,40,61), (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), (1,59,9,23,5,63,13,19)(2,35,14,52,6,39,10,56)(3,61,11,17,7,57,15,21)(4,37,16,54,8,33,12,50)(18,26,62,44,22,30,58,48)(20,28,64,46,24,32,60,42)(25,40,47,53,29,36,43,49)(27,34,41,55,31,38,45,51)>;
G:=Group( (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,41)(16,42)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(33,64)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63), (1,27,5,31)(2,32,6,28)(3,29,7,25)(4,26,8,30)(9,41,13,45)(10,46,14,42)(11,43,15,47)(12,48,16,44)(17,49,21,53)(18,54,22,50)(19,51,23,55)(20,56,24,52)(33,58,37,62)(34,63,38,59)(35,60,39,64)(36,57,40,61), (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), (1,59,9,23,5,63,13,19)(2,35,14,52,6,39,10,56)(3,61,11,17,7,57,15,21)(4,37,16,54,8,33,12,50)(18,26,62,44,22,30,58,48)(20,28,64,46,24,32,60,42)(25,40,47,53,29,36,43,49)(27,34,41,55,31,38,45,51) );
G=PermutationGroup([[(1,29),(2,30),(3,31),(4,32),(5,25),(6,26),(7,27),(8,28),(9,43),(10,44),(11,45),(12,46),(13,47),(14,48),(15,41),(16,42),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,49),(24,50),(33,64),(34,57),(35,58),(36,59),(37,60),(38,61),(39,62),(40,63)], [(1,27,5,31),(2,32,6,28),(3,29,7,25),(4,26,8,30),(9,41,13,45),(10,46,14,42),(11,43,15,47),(12,48,16,44),(17,49,21,53),(18,54,22,50),(19,51,23,55),(20,56,24,52),(33,58,37,62),(34,63,38,59),(35,60,39,64),(36,57,40,61)], [(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)], [(1,59,9,23,5,63,13,19),(2,35,14,52,6,39,10,56),(3,61,11,17,7,57,15,21),(4,37,16,54,8,33,12,50),(18,26,62,44,22,30,58,48),(20,28,64,46,24,32,60,42),(25,40,47,53,29,36,43,49),(27,34,41,55,31,38,45,51)]])
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 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | Q8 | C8.C4 |
| kernel | C2×C4.C42 | C4.C42 | C23×C8 | C22×M4(2) | C22×C8 | C2×M4(2) | C22×C4 | C22×C4 | C24 | C22 |
| # reps | 1 | 4 | 1 | 2 | 8 | 16 | 6 | 1 | 1 | 16 |
Matrix representation of C2×C4.C42 ►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 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 13 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 8 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,13],[1,0,0,0,0,1,0,0,0,0,0,13,0,0,16,0],[4,0,0,0,0,16,0,0,0,0,2,0,0,0,0,8] >;
C2×C4.C42 in GAP, Magma, Sage, TeX
C_2\times C_4.C_4^2
% in TeX
G:=Group("C2xC4.C4^2"); // GroupNames label
G:=SmallGroup(128,469);
// by ID
G=gap.SmallGroup(128,469);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,2019,248,172,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=1,c^4=d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c>;
// generators/relations