direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C42⋊6C4, C24.151D4, (C2×C42)⋊17C4, C42⋊42(C2×C4), C4○(C42⋊6C4), C22.40C4≀C2, (C2×C4).64C42, C4.14(C2×C42), C42⋊C2⋊13C4, (C22×C4).85Q8, C23.59(C4⋊C4), M4(2)⋊18(C2×C4), (C2×M4(2))⋊12C4, C23.536(C2×D4), (C22×C4).751D4, (C22×C42).15C2, (C23×C4).666C22, C23.225(C22⋊C4), C4.15(C2.C42), (C22×C4).1298C23, (C2×C42).1044C22, (C22×M4(2)).10C2, C42⋊C2.256C22, (C2×M4(2)).295C22, C22.28(C2.C42), (C2×C4⋊C4)⋊20C4, C4⋊C4⋊33(C2×C4), C2.4(C2×C4≀C2), C4.72(C2×C4⋊C4), C22.8(C2×C4⋊C4), (C2×C4).177(C2×Q8), (C2×C4).121(C4⋊C4), (C2×C4)○(C42⋊6C4), (C2×C4).1491(C2×D4), C4.102(C2×C22⋊C4), (C2×C42⋊C2).5C2, (C22×C4).400(C2×C4), (C2×C4).344(C22×C4), C2.9(C2×C2.C42), (C2×C4).392(C22⋊C4), C22.103(C2×C22⋊C4), SmallGroup(128,464)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C42⋊6C4
G = < a,b,c,d | a2=b4=c4=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=c-1 >
Subgroups: 388 in 244 conjugacy classes, 116 normal (32 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C22×C4, C24, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C23×C4, C42⋊6C4, C22×C42, C2×C42⋊C2, C22×M4(2), C2×C42⋊6C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2.C42, C4≀C2, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊6C4, C2×C2.C42, C2×C4≀C2, C2×C42⋊6C4
►On 32 points
(1 10)(2 9)(3 6)(4 5)(7 11)(8 12)(13 15)(14 16)(17 26)(18 27)(19 28)(20 25)(21 31)(22 32)(23 29)(24 30)
(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)
(1 7 13 4)(2 8 14 3)(5 10 11 15)(6 9 12 16)(17 32 19 30)(18 29 20 31)(21 27 23 25)(22 28 24 26)
(1 23 2 26)(3 22 4 25)(5 20 6 32)(7 27 8 24)(9 17 10 29)(11 18 12 30)(13 21 14 28)(15 31 16 19)
G:=sub<Sym(32)| (1,10)(2,9)(3,6)(4,5)(7,11)(8,12)(13,15)(14,16)(17,26)(18,27)(19,28)(20,25)(21,31)(22,32)(23,29)(24,30), (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), (1,7,13,4)(2,8,14,3)(5,10,11,15)(6,9,12,16)(17,32,19,30)(18,29,20,31)(21,27,23,25)(22,28,24,26), (1,23,2,26)(3,22,4,25)(5,20,6,32)(7,27,8,24)(9,17,10,29)(11,18,12,30)(13,21,14,28)(15,31,16,19)>;
G:=Group( (1,10)(2,9)(3,6)(4,5)(7,11)(8,12)(13,15)(14,16)(17,26)(18,27)(19,28)(20,25)(21,31)(22,32)(23,29)(24,30), (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), (1,7,13,4)(2,8,14,3)(5,10,11,15)(6,9,12,16)(17,32,19,30)(18,29,20,31)(21,27,23,25)(22,28,24,26), (1,23,2,26)(3,22,4,25)(5,20,6,32)(7,27,8,24)(9,17,10,29)(11,18,12,30)(13,21,14,28)(15,31,16,19) );
G=PermutationGroup([[(1,10),(2,9),(3,6),(4,5),(7,11),(8,12),(13,15),(14,16),(17,26),(18,27),(19,28),(20,25),(21,31),(22,32),(23,29),(24,30)], [(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)], [(1,7,13,4),(2,8,14,3),(5,10,11,15),(6,9,12,16),(17,32,19,30),(18,29,20,31),(21,27,23,25),(22,28,24,26)], [(1,23,2,26),(3,22,4,25),(5,20,6,32),(7,27,8,24),(9,17,10,29),(11,18,12,30),(13,21,14,28),(15,31,16,19)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4AB | 4AC | ··· | 4AJ | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | Q8 | D4 | C4≀C2 |
| kernel | C2×C42⋊6C4 | C42⋊6C4 | C22×C42 | C2×C42⋊C2 | C22×M4(2) | C2×C42 | C2×C4⋊C4 | C42⋊C2 | C2×M4(2) | C22×C4 | C22×C4 | C24 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 4 | 4 | 8 | 5 | 2 | 1 | 16 |
Matrix representation of C2×C42⋊6C4 ►in GL4(𝔽17) generated by
| 16 | 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 | 1 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 13 |
| 4 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 16 | 0 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,13],[4,0,0,0,0,16,0,0,0,0,0,16,0,0,16,0] >;
C2×C42⋊6C4 in GAP, Magma, Sage, TeX
C_2\times C_4^2\rtimes_6C_4
% in TeX
G:=Group("C2xC4^2:6C4"); // GroupNames label
G:=SmallGroup(128,464);
// by ID
G=gap.SmallGroup(128,464);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,2019,248,4037]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=c^-1>;
// generators/relations