p-group, metabelian, nilpotent (class 4), monomial
Aliases: C2.5C2≀C4, (C2×C4).3D8, (C2×C4).3SD16, C22.15C4≀C2, (C22×D4).3C4, (C22×C4).31D4, C2.C42.3C4, C22.C42⋊11C2, C22.56(C23⋊C4), C2.4(C42.C4), C23.10D4.1C2, C22.19(D4⋊C4), C23.156(C22⋊C4), C2.10(C22.SD16), C22.M4(2)⋊2C2, (C2×C4⋊C4).5C22, (C22×C4).3(C2×C4), SmallGroup(128,77)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
C1 — C22 — C23 — C2×C4⋊C4 — C2.C2≀C4 |
C1 — C22 — C23 — C2×C4⋊C4 — C2.C2≀C4 |
Generators and relations for C2.C2≀C4
G = < a,b,c,d,e,f | a2=b2=d2=e2=1, c2=f4=a, cbc-1=ab=ba, ac=ca, ad=da, ae=ea, af=fa, bd=db, be=eb, fbf-1=bcde, cd=dc, ce=ec, fcf-1=cde, fdf-1=de=ed, ef=fe >
Subgroups: 268 in 84 conjugacy classes, 20 normal (all characteristic)
C1, C2 [×3], C2 [×4], C4 [×5], C22 [×3], C22 [×12], C8 [×3], C2×C4 [×2], C2×C4 [×9], D4 [×4], C23, C23 [×8], C22⋊C4 [×4], C4⋊C4, C2×C8 [×2], M4(2) [×3], C22×C4 [×3], C22×C4, C2×D4 [×3], C24, C2.C42, C22⋊C8, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×M4(2), C22×D4, C22.M4(2), C22.C42, C23.10D4, C2.C2≀C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, D8, SD16, C23⋊C4, D4⋊C4, C4≀C2, C22.SD16, C2≀C4, C42.C4, C2.C2≀C4
Character table of C2.C2≀C4
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -i | i | i | i | -i | -i | -i | i | linear of order 4 |
ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | i | i | i | -i | -i | -i | i | -i | linear of order 4 |
ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | i | -i | -i | -i | i | i | i | -i | linear of order 4 |
ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -i | -i | -i | i | i | i | -i | i | linear of order 4 |
ρ9 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ10 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ11 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -√2 | 0 | 0 | √2 | 0 | 0 | √2 | -√2 | orthogonal lifted from D8 |
ρ12 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | √2 | 0 | 0 | -√2 | 0 | 0 | -√2 | √2 | orthogonal lifted from D8 |
ρ13 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 1+i | -1-i | 0 | 1-i | -1+i | 0 | 0 | complex lifted from C4≀C2 |
ρ14 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 1-i | -1+i | 0 | 1+i | -1-i | 0 | 0 | complex lifted from C4≀C2 |
ρ15 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | -1+i | 1-i | 0 | -1-i | 1+i | 0 | 0 | complex lifted from C4≀C2 |
ρ16 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -√-2 | 0 | 0 | -√-2 | 0 | 0 | √-2 | √-2 | complex lifted from SD16 |
ρ17 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | √-2 | 0 | 0 | √-2 | 0 | 0 | -√-2 | -√-2 | complex lifted from SD16 |
ρ18 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | -1-i | 1+i | 0 | -1+i | 1-i | 0 | 0 | complex lifted from C4≀C2 |
ρ19 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2≀C4 |
ρ20 | 4 | -4 | -4 | 4 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2≀C4 |
ρ21 | 4 | 4 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
ρ22 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42.C4 |
ρ23 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42.C4 |
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)
(2 12)(3 28)(4 10)(6 16)(7 32)(8 14)(9 24)(11 15)(13 20)(17 25)(18 22)(19 31)(21 29)(23 27)
(1 11 5 15)(2 12 6 16)(3 20 7 24)(4 21 8 17)(9 32 13 28)(10 25 14 29)(18 26 22 30)(19 27 23 31)
(1 30)(3 32)(5 26)(7 28)(9 24)(11 18)(13 20)(15 22)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)
(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)
G:=sub<Sym(32)| (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (2,12)(3,28)(4,10)(6,16)(7,32)(8,14)(9,24)(11,15)(13,20)(17,25)(18,22)(19,31)(21,29)(23,27), (1,11,5,15)(2,12,6,16)(3,20,7,24)(4,21,8,17)(9,32,13,28)(10,25,14,29)(18,26,22,30)(19,27,23,31), (1,30)(3,32)(5,26)(7,28)(9,24)(11,18)(13,20)(15,22), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23), (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)>;
G:=Group( (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (2,12)(3,28)(4,10)(6,16)(7,32)(8,14)(9,24)(11,15)(13,20)(17,25)(18,22)(19,31)(21,29)(23,27), (1,11,5,15)(2,12,6,16)(3,20,7,24)(4,21,8,17)(9,32,13,28)(10,25,14,29)(18,26,22,30)(19,27,23,31), (1,30)(3,32)(5,26)(7,28)(9,24)(11,18)(13,20)(15,22), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23), (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) );
G=PermutationGroup([(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32)], [(2,12),(3,28),(4,10),(6,16),(7,32),(8,14),(9,24),(11,15),(13,20),(17,25),(18,22),(19,31),(21,29),(23,27)], [(1,11,5,15),(2,12,6,16),(3,20,7,24),(4,21,8,17),(9,32,13,28),(10,25,14,29),(18,26,22,30),(19,27,23,31)], [(1,30),(3,32),(5,26),(7,28),(9,24),(11,18),(13,20),(15,22)], [(1,30),(2,31),(3,32),(4,25),(5,26),(6,27),(7,28),(8,29),(9,24),(10,17),(11,18),(12,19),(13,20),(14,21),(15,22),(16,23)], [(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)])
Matrix representation of C2.C2≀C4 ►in GL6(𝔽17)
16 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
16 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 16 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 16 | 0 | 1 |
0 | 2 | 0 | 0 | 0 | 0 |
8 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 16 | 0 | 0 |
0 | 0 | 0 | 0 | 16 | 0 |
0 | 0 | 0 | 16 | 1 | 1 |
16 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 16 | 0 | 0 | 0 |
0 | 0 | 0 | 16 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 16 | 16 | 0 | 1 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 16 | 0 | 0 | 0 |
0 | 0 | 0 | 16 | 0 | 0 |
0 | 0 | 0 | 0 | 16 | 0 |
0 | 0 | 0 | 0 | 0 | 16 |
10 | 3 | 0 | 0 | 0 | 0 |
5 | 7 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 16 | 16 | 1 | 2 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 16 | 1 | 1 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,16,0,0,0,0,1,0,0,0,0,0,0,1],[0,8,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,16,0,0,0,0,16,1,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,16,0,0,0,16,0,16,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[10,5,0,0,0,0,3,7,0,0,0,0,0,0,0,16,0,0,0,0,0,16,1,16,0,0,1,1,0,1,0,0,0,2,0,1] >;
C2.C2≀C4 in GAP, Magma, Sage, TeX
C_2.C_2\wr C_4
% in TeX
G:=Group("C2.C2wrC4");
// GroupNames label
G:=SmallGroup(128,77);
// by ID
G=gap.SmallGroup(128,77);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,387,184,794,521,248,2804]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=d^2=e^2=1,c^2=f^4=a,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*d=d*b,b*e=e*b,f*b*f^-1=b*c*d*e,c*d=d*c,c*e=e*c,f*c*f^-1=c*d*e,f*d*f^-1=d*e=e*d,e*f=f*e>;
// generators/relations
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