direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C4.4D4, C42⋊17C22, C23.7C23, C22.20C24, C24.13C22, (C2×C42)⋊9C2, (C2×C4).85D4, C4.13(C2×D4), (C22×Q8)⋊4C2, C2.9(C22×D4), (C2×Q8)⋊10C22, C22.61(C2×D4), C22⋊C4⋊16C22, (C2×C4).129C23, (C22×D4).11C2, (C2×D4).61C22, C22.32(C4○D4), (C22×C4).99C22, C2.9(C2×C4○D4), (C2×C22⋊C4)⋊11C2, SmallGroup(64,207)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C4.4D4
G = < a,b,c,d | a2=b4=c4=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 265 in 165 conjugacy classes, 89 normal (9 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C2×C42, C2×C22⋊C4, C4.4D4, C22×D4, C22×Q8, C2×C4.4D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4.4D4, C22×D4, C2×C4○D4, C2×C4.4D4
Character table of C2×C4.4D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | |
| 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 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | 2 | -2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | 2 | -2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | -2 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►On 32 points
(1 25)(2 26)(3 27)(4 28)(5 23)(6 24)(7 21)(8 22)(9 29)(10 30)(11 31)(12 32)(13 17)(14 18)(15 19)(16 20)
(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 6 9 13)(2 7 10 14)(3 8 11 15)(4 5 12 16)(17 25 24 29)(18 26 21 30)(19 27 22 31)(20 28 23 32)
(1 15 3 13)(2 14 4 16)(5 10 7 12)(6 9 8 11)(17 25 19 27)(18 28 20 26)(21 32 23 30)(22 31 24 29)
G:=sub<Sym(32)| (1,25)(2,26)(3,27)(4,28)(5,23)(6,24)(7,21)(8,22)(9,29)(10,30)(11,31)(12,32)(13,17)(14,18)(15,19)(16,20), (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), (1,6,9,13)(2,7,10,14)(3,8,11,15)(4,5,12,16)(17,25,24,29)(18,26,21,30)(19,27,22,31)(20,28,23,32), (1,15,3,13)(2,14,4,16)(5,10,7,12)(6,9,8,11)(17,25,19,27)(18,28,20,26)(21,32,23,30)(22,31,24,29)>;
G:=Group( (1,25)(2,26)(3,27)(4,28)(5,23)(6,24)(7,21)(8,22)(9,29)(10,30)(11,31)(12,32)(13,17)(14,18)(15,19)(16,20), (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), (1,6,9,13)(2,7,10,14)(3,8,11,15)(4,5,12,16)(17,25,24,29)(18,26,21,30)(19,27,22,31)(20,28,23,32), (1,15,3,13)(2,14,4,16)(5,10,7,12)(6,9,8,11)(17,25,19,27)(18,28,20,26)(21,32,23,30)(22,31,24,29) );
G=PermutationGroup([[(1,25),(2,26),(3,27),(4,28),(5,23),(6,24),(7,21),(8,22),(9,29),(10,30),(11,31),(12,32),(13,17),(14,18),(15,19),(16,20)], [(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)], [(1,6,9,13),(2,7,10,14),(3,8,11,15),(4,5,12,16),(17,25,24,29),(18,26,21,30),(19,27,22,31),(20,28,23,32)], [(1,15,3,13),(2,14,4,16),(5,10,7,12),(6,9,8,11),(17,25,19,27),(18,28,20,26),(21,32,23,30),(22,31,24,29)]])
C2×C4.4D4 is a maximal subgroup of
C42.395D4 C42.407D4 C42.70D4 C24.23D4 C4.4D4⋊13C4 C42.433D4 C42.110D4 C42.115D4 C42.119D4 C42.129D4 C42⋊10D4 (C22×D8).C2 (C2×C8).41D4 C4⋊C4.94D4 C42.160D4 C24.205C23 C24.220C23 C24.221C23 C23.261C24 C24.259C23 C23.327C24 C24.263C23 C24.264C23 C23.335C24 C24.565C23 C24.271C23 C23.348C24 C23.359C24 C24.282C23 C23.372C24 C23.374C24 C23.391C24 C24.311C23 C42⋊19D4 C42⋊21D4 C42.168D4 C42.170D4 C42.171D4 C23.455C24 C23.457C24 C42.182D4 C42⋊26D4 C42⋊28D4 C42⋊29D4 C42.189D4 C42.193D4 C42⋊31D4 C42.196D4 C42⋊32D4 C23.570C24 C23.572C24 C23.574C24 C23.576C24 C23.584C24 C24.393C23 C23.600C24 C24.412C23 C23.612C24 C23.615C24 C23.617C24 C23.630C24 C23.631C24 C23.633C24 C42⋊34D4 C42.199D4 C42⋊46D4 C43⋊12C2 C43⋊14C2 C42.446D4 C42.242D4 M4(2)⋊9D4 C42.269D4 C42.271D4 C42.273D4 C22.89C25 C22.99C25 C22.103C25 C22.134C25 C22.147C25 C22.150C25
C2×C4.4D4 is a maximal quotient of
C42.163D4 C23.335C24 C24.565C23 C23.372C24 C23.388C24 C24.301C23 C23.390C24 C23.391C24 C23.392C24 C24.579C23 C23.404C24 C42.170D4 C42.171D4 C23.461C24 C42.172D4 C42.173D4 C42.177D4 C23.491C24 C42.182D4 C24.592C23 C42.193D4 C42.194D4 C42.195D4 C42⋊46D4 C43⋊12C2 C43⋊14C2 C42⋊18Q8 C42.355D4 C42.239D4 C42.366C23 C42.367C23 C42.240D4 C42.241D4 C42.242D4 C42.243D4 C42.244D4
Matrix representation of C2×C4.4D4 ►in GL5(𝔽5)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 2 |
| 0 | 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 2 |
| 0 | 0 | 0 | 1 | 3 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,0,3,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,2,3],[1,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,2,1,0,0,0,2,3] >;
C2×C4.4D4 in GAP, Magma, Sage, TeX
C_2\times C_4._4D_4
% in TeX
G:=Group("C2xC4.4D4"); // GroupNames label
G:=SmallGroup(64,207);
// by ID
G=gap.SmallGroup(64,207);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,217,199,650,86]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^4=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations
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