direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C4○D8, D8⋊6C22, C4.4C24, C8.13C23, Q16⋊6C22, D4.2C23, C23.29D4, Q8.2C23, SD16⋊5C22, (C2×C4)○D8, C4○(C2×D8), (C2×C4)○Q16, C4○(C2×Q16), C4○(C4○D8), (C2×C4)○SD16, C4○(C2×SD16), (C2×D8)⋊13C2, (C22×C8)⋊8C2, (C2×C4).90D4, C4.83(C2×D4), (C2×C8)⋊13C22, (C2×Q16)⋊13C2, C4○D4⋊3C22, (C2×SD16)⋊16C2, C2.26(C22×D4), C22.67(C2×D4), (C2×C4).138C23, (C2×D4).73C22, (C2×Q8).69C22, (C22×C4).132C22, (C2×C4)○(C2×D8), (C2×C4)○(C2×Q16), (C2×C4)○(C2×SD16), (C2×C4○D4)⋊10C2, SmallGroup(64,253)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C4○D8
G = < a,b,c,d | a2=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >
Subgroups: 201 in 133 conjugacy classes, 81 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C2×C8, C2×C8, D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C2×C4○D8
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C4○D8, C22×D4, C2×C4○D8
Character table of C2×C4○D8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 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 | -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 | 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 | -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 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -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 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | -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 |
| ρ8 | 1 | -1 | 1 | -1 | -1 | 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 |
| ρ9 | 1 | 1 | 1 | 1 | -1 | -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 |
| ρ10 | 1 | 1 | 1 | 1 | 1 | 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 |
| ρ11 | 1 | -1 | 1 | -1 | -1 | 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 |
| ρ12 | 1 | -1 | 1 | -1 | 1 | -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 |
| ρ13 | 1 | 1 | 1 | 1 | 1 | 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 |
| ρ14 | 1 | 1 | 1 | 1 | -1 | -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 |
| ρ15 | 1 | -1 | 1 | -1 | 1 | -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 |
| ρ16 | 1 | -1 | 1 | -1 | -1 | 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 |
| ρ17 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | √2 | √-2 | -√-2 | √2 | -√2 | -√-2 | -√2 | complex lifted from C4○D8 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √2 | -√-2 | √-2 | √2 | -√2 | √-2 | -√2 | complex lifted from C4○D8 |
| ρ23 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √2 | √-2 | √-2 | -√2 | -√2 | -√-2 | √2 | complex lifted from C4○D8 |
| ρ24 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | √2 | -√-2 | -√-2 | -√2 | -√2 | √-2 | √2 | complex lifted from C4○D8 |
| ρ25 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | -√2 | -√-2 | √-2 | -√2 | √2 | √-2 | √2 | complex lifted from C4○D8 |
| ρ26 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√2 | √-2 | -√-2 | -√2 | √2 | -√-2 | √2 | complex lifted from C4○D8 |
| ρ27 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√2 | -√-2 | -√-2 | √2 | √2 | √-2 | -√2 | complex lifted from C4○D8 |
| ρ28 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | -√2 | √-2 | √-2 | √2 | √2 | -√-2 | -√2 | complex lifted from C4○D8 |
►On 32 points
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)
(1 16 5 12)(2 9 6 13)(3 10 7 14)(4 11 8 15)(17 26 21 30)(18 27 22 31)(19 28 23 32)(20 29 24 25)
(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)
(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(17 19)(20 24)(21 23)(25 29)(26 28)(30 32)
G:=sub<Sym(32)| (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,17)(16,18), (1,16,5,12)(2,9,6,13)(3,10,7,14)(4,11,8,15)(17,26,21,30)(18,27,22,31)(19,28,23,32)(20,29,24,25), (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), (2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(17,19)(20,24)(21,23)(25,29)(26,28)(30,32)>;
G:=Group( (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,17)(16,18), (1,16,5,12)(2,9,6,13)(3,10,7,14)(4,11,8,15)(17,26,21,30)(18,27,22,31)(19,28,23,32)(20,29,24,25), (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), (2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(17,19)(20,24)(21,23)(25,29)(26,28)(30,32) );
G=PermutationGroup([[(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,19),(10,20),(11,21),(12,22),(13,23),(14,24),(15,17),(16,18)], [(1,16,5,12),(2,9,6,13),(3,10,7,14),(4,11,8,15),(17,26,21,30),(18,27,22,31),(19,28,23,32),(20,29,24,25)], [(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)], [(2,8),(3,7),(4,6),(9,15),(10,14),(11,13),(17,19),(20,24),(21,23),(25,29),(26,28),(30,32)]])
C2×C4○D8 is a maximal subgroup of
M4(2).43D4 C42.326D4 C42.116D4 M4(2).30D4 C23.24D8 C23.39D8 C23.20SD16 C23.13D8 C23.21SD16 D8.10D4 C42.383D4 C42.280C23 C42.281C23 M4(2)○D8 (C2×D4)⋊21D4 (C2×Q8)⋊17D4 C42.18C23 C42.19C23 M4(2).20D4 D8⋊10D4 SD16⋊8D4 SD16⋊10D4 D8⋊13D4 D16⋊C22 C8.C24
C4○(C8⋊pD4): C24.144D4 C24.110D4 C42.360D4 C42.247D4 ...
C8⋊pD4⋊C2: D8⋊8D4 (C2×C8)⋊13D4 (C2×C8)⋊14D4 M4(2)⋊10D4 M4(2)⋊11D4 SD16⋊7D4 Q16⋊10D4 D8⋊12D4 ...
C4○(D4.pD4): C24.103D4 C42.443D4 M4(2).10C23 ...
C2×C4○D8 is a maximal quotient of
C2×C4×D8 C2×C4×SD16 C2×C4×Q16 C24.103D4 C42.443D4 C24.144D4 C42.447D4 C24.115D4 C42.384D4 C42.225D4 C42.450D4 C42.451D4 C42.226D4 C42.355D4 C42.360D4 C42.364D4 C42.365D4 C42.308D4 C42.366D4 C42.367D4 C24.121D4 C24.123D4 C24.124D4 C42.265D4 C42.268D4 C42.269D4 C42.270D4 C42.280D4 C42.283D4 C42.284D4 C42.285D4 C42.295D4 C42.296D4 C42.298D4 D8⋊12D4 SD16⋊10D4 D8⋊13D4 SD16⋊11D4 Q16⋊12D4 Q16⋊13D4 C42.461C23 C42.462C23 C42.465C23 C42.466C23 C42.467C23 C42.468C23 C42.469C23 C42.470C23 C42.485C23 C42.486C23 C42.488C23 C42.489C23 C42.490C23 C42.491C23 C42.501C23 C42.502C23 C42.505C23 C42.506C23 D8⋊6Q8 SD16⋊4Q8 Q16⋊6Q8 C42.527C23 C42.528C23 C42.530C23
Matrix representation of C2×C4○D8 ►in GL3(𝔽17) generated by
| 16 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 1 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 13 |
| 16 | 0 | 0 |
| 0 | 3 | 14 |
| 0 | 3 | 3 |
| 16 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 16 |
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[1,0,0,0,13,0,0,0,13],[16,0,0,0,3,3,0,14,3],[16,0,0,0,1,0,0,0,16] >;
C2×C4○D8 in GAP, Magma, Sage, TeX
C_2\times C_4\circ D_8
% in TeX
G:=Group("C2xC4oD8"); // GroupNames label
G:=SmallGroup(64,253);
// by ID
G=gap.SmallGroup(64,253);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,-2,217,158,1444,730,88]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=d^2=1,c^4=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^3>;
// generators/relations
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