direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4×C4○D4, D4○C42, Q8○C42, C22.10C24, C23.30C23, C42.87C22, C4○2(C4×D4), C4○2(C4×Q8), D4⋊6(C2×C4), Q8⋊6(C2×C4), C42○(C4×D4), C42○(C2×D4), (C4×D4)⋊23C2, C42○(C4×Q8), C42○(C2×Q8), C42○(C4⋊C4), (C2×C42)⋊8C2, (C4×Q8)⋊18C2, C2.6(C23×C4), C4⋊C4.79C22, C42○(C22⋊C4), C4.18(C22×C4), C4○2(C42⋊C2), C42⋊C2⋊19C2, (C2×C4).159C23, (C2×D4).76C22, C22.1(C22×C4), (C2×Q8).71C22, C42○(C42⋊C2), C22⋊C4.27C22, (C22×C4).121C22, (C2×C4)⋊8(C2×C4), C2.4(C2×C4○D4), C42○(C2×C4○D4), (C2×C4○D4).13C2, SmallGroup(64,198)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4×C4○D4
G = < a,b,c,d | a4=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >
Subgroups: 185 in 155 conjugacy classes, 125 normal (8 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C42, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, C4×C4○D4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C23×C4, C2×C4○D4, C4×C4○D4
►On 32 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)
(1 17 15 6)(2 18 16 7)(3 19 13 8)(4 20 14 5)(9 29 24 25)(10 30 21 26)(11 31 22 27)(12 32 23 28)
(1 8 15 19)(2 5 16 20)(3 6 13 17)(4 7 14 18)(9 31 24 27)(10 32 21 28)(11 29 22 25)(12 30 23 26)
(1 22)(2 23)(3 24)(4 21)(5 30)(6 31)(7 32)(8 29)(9 13)(10 14)(11 15)(12 16)(17 27)(18 28)(19 25)(20 26)
G:=sub<Sym(32)| (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), (1,17,15,6)(2,18,16,7)(3,19,13,8)(4,20,14,5)(9,29,24,25)(10,30,21,26)(11,31,22,27)(12,32,23,28), (1,8,15,19)(2,5,16,20)(3,6,13,17)(4,7,14,18)(9,31,24,27)(10,32,21,28)(11,29,22,25)(12,30,23,26), (1,22)(2,23)(3,24)(4,21)(5,30)(6,31)(7,32)(8,29)(9,13)(10,14)(11,15)(12,16)(17,27)(18,28)(19,25)(20,26)>;
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), (1,17,15,6)(2,18,16,7)(3,19,13,8)(4,20,14,5)(9,29,24,25)(10,30,21,26)(11,31,22,27)(12,32,23,28), (1,8,15,19)(2,5,16,20)(3,6,13,17)(4,7,14,18)(9,31,24,27)(10,32,21,28)(11,29,22,25)(12,30,23,26), (1,22)(2,23)(3,24)(4,21)(5,30)(6,31)(7,32)(8,29)(9,13)(10,14)(11,15)(12,16)(17,27)(18,28)(19,25)(20,26) );
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)], [(1,17,15,6),(2,18,16,7),(3,19,13,8),(4,20,14,5),(9,29,24,25),(10,30,21,26),(11,31,22,27),(12,32,23,28)], [(1,8,15,19),(2,5,16,20),(3,6,13,17),(4,7,14,18),(9,31,24,27),(10,32,21,28),(11,29,22,25),(12,30,23,26)], [(1,22),(2,23),(3,24),(4,21),(5,30),(6,31),(7,32),(8,29),(9,13),(10,14),(11,15),(12,16),(17,27),(18,28),(19,25),(20,26)]])
C4×C4○D4 is a maximal subgroup of
(C2×C42)⋊C4 D4.5C42 2- 1+4⋊5C4 C42.674C23 C42.260C23 C42.261C23 C42.678C23 C42.290C23 D4⋊6M4(2) Q8⋊6M4(2) C42.697C23 C42.698C23 D4⋊8M4(2) Q8⋊7M4(2) C22.14C25 C22.33C25 C22.44C25 C22.49C25 C22.50C25 C22.64C25 C22.69C25 C22.70C25 C22.71C25 C22.72C25 C22.87C25 C22.88C25 C22.89C25 C22.92C25 C22.93C25 C22.97C25 C22.98C25 C22.99C25 C22.100C25 C22.101C25 C22.103C25 C22.104C25 C22.106C25 C22.107C25 C22.110C25 C22.113C25
C42.D2p: C42.455D4 C42.397D4 C42.374D4 D4⋊4M4(2) D4.C42 C42.102D4 C42.426D4 C42.383D4 ...
C4×C4○D4 is a maximal quotient of
D4×C42 Q8×C42 C42⋊42D4 C43⋊9C2 C42⋊14Q8 C23.178C24 C23.179C24 C43⋊2C2 C23.214C24 C23.215C24 C24.203C23 C24.204C23 C23.218C24 C24.205C23 C24.549C23 C23.223C24 C23.225C24 C24.208C23 C23.229C24 C23.231C24 C23.233C24 C23.235C24 C23.238C24 C24.212C23 C23.241C24 C24.217C23 C24.218C23 C23.250C24 C23.252C24 C23.253C24 C24.221C23 C23.255C24 C24.223C23 C42.290C23 C42.291C23 C42.292C23 C42.293C23 C42.294C23 D4⋊6M4(2) Q8⋊6M4(2)
C4⋊C4.D2p: C23.244C24 C24.220C23 C42.188D6 C42.188D10 C42.188D14 ...
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | ··· | 4L | 4M | ··· | 4AD |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4○D4 |
| kernel | C4×C4○D4 | C2×C42 | C42⋊C2 | C4×D4 | C4×Q8 | C2×C4○D4 | C4○D4 | C4 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 16 | 8 |
Matrix representation of C4×C4○D4 ►in GL3(𝔽5) generated by
| 3 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 4 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 4 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 4 | 3 |
| 1 | 0 | 0 |
| 0 | 4 | 1 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(5))| [3,0,0,0,2,0,0,0,2],[4,0,0,0,2,0,0,0,2],[4,0,0,0,2,4,0,0,3],[1,0,0,0,4,0,0,1,1] >;
C4×C4○D4 in GAP, Magma, Sage, TeX
C_4\times C_4\circ D_4
% in TeX
G:=Group("C4xC4oD4"); // GroupNames label
G:=SmallGroup(64,198);
// by ID
G=gap.SmallGroup(64,198);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,192,217,158,69]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations