direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×D4⋊C4, C22.12D8, C23.54D4, C22.10SD16, D4⋊3(C2×C4), (C2×D4)⋊7C4, C2.1(C2×D8), C4⋊C4⋊7C22, (C22×C8)⋊3C2, C4.47(C2×D4), (C2×C4).69D4, (C2×C8)⋊11C22, C4.1(C22×C4), C2.1(C2×SD16), (C2×C4).59C23, (C22×D4).6C2, C22.41(C2×D4), C4.12(C22⋊C4), (C2×D4).46C22, C22.32(C22⋊C4), (C22×C4).107C22, (C2×C4⋊C4)⋊9C2, (C2×C4).43(C2×C4), C2.17(C2×C22⋊C4), SmallGroup(64,95)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×D4⋊C4
G = < a,b,c,d | a2=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >
Subgroups: 201 in 101 conjugacy classes, 49 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, D4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C2×D4⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, C2×D4⋊C4
Character table of C2×D4⋊C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 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 | -i | -i | i | i | i | -i | i | -i | -i | i | -i | i | linear of order 4 |
| ρ10 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | i | -i | -i | i | -i | -i | -i | -i | i | i | i | i | linear of order 4 |
| ρ11 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | i | i | -i | i | i | i | i | -i | -i | -i | -i | linear of order 4 |
| ρ12 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | i | i | -i | -i | -i | i | -i | i | i | -i | i | -i | linear of order 4 |
| ρ13 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -i | i | i | -i | -i | -i | -i | -i | i | i | i | i | linear of order 4 |
| ρ14 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | i | i | -i | -i | i | -i | i | -i | -i | i | -i | i | linear of order 4 |
| ρ15 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | -i | i | i | -i | i | -i | i | i | -i | i | -i | linear of order 4 |
| ρ16 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | i | -i | -i | i | i | i | i | i | -i | -i | -i | -i | linear of order 4 |
| ρ17 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 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 | -2 | 2 | 0 | 0 | 0 | 0 | -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 | 2 | 2 | 0 | 0 | 0 | 0 | -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 | 2 | -2 | 0 | 0 | 0 | 0 | 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 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | -√2 | -√2 | √2 | √2 | orthogonal lifted from D8 |
| ρ22 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | -√2 | √2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ23 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | √2 | -√2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ24 | 2 | -2 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | √2 | √2 | -√2 | -√2 | orthogonal lifted from D8 |
| ρ25 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | -√-2 | -√-2 | √-2 | √-2 | complex lifted from SD16 |
| ρ26 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | √-2 | -√-2 | -√-2 | √-2 | complex lifted from SD16 |
| ρ27 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | -√-2 | √-2 | √-2 | -√-2 | complex lifted from SD16 |
| ρ28 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | √-2 | √-2 | -√-2 | -√-2 | complex lifted from SD16 |
►On 32 points
(1 20)(2 17)(3 18)(4 19)(5 22)(6 23)(7 24)(8 21)(9 14)(10 15)(11 16)(12 13)(25 29)(26 30)(27 31)(28 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 4)(2 3)(5 7)(9 12)(10 11)(13 14)(15 16)(17 18)(19 20)(22 24)(26 28)(30 32)
(1 7 11 30)(2 6 12 29)(3 5 9 32)(4 8 10 31)(13 25 17 23)(14 28 18 22)(15 27 19 21)(16 26 20 24)
G:=sub<Sym(32)| (1,20)(2,17)(3,18)(4,19)(5,22)(6,23)(7,24)(8,21)(9,14)(10,15)(11,16)(12,13)(25,29)(26,30)(27,31)(28,32), (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), (1,4)(2,3)(5,7)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(22,24)(26,28)(30,32), (1,7,11,30)(2,6,12,29)(3,5,9,32)(4,8,10,31)(13,25,17,23)(14,28,18,22)(15,27,19,21)(16,26,20,24)>;
G:=Group( (1,20)(2,17)(3,18)(4,19)(5,22)(6,23)(7,24)(8,21)(9,14)(10,15)(11,16)(12,13)(25,29)(26,30)(27,31)(28,32), (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), (1,4)(2,3)(5,7)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(22,24)(26,28)(30,32), (1,7,11,30)(2,6,12,29)(3,5,9,32)(4,8,10,31)(13,25,17,23)(14,28,18,22)(15,27,19,21)(16,26,20,24) );
G=PermutationGroup([[(1,20),(2,17),(3,18),(4,19),(5,22),(6,23),(7,24),(8,21),(9,14),(10,15),(11,16),(12,13),(25,29),(26,30),(27,31),(28,32)], [(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)], [(1,4),(2,3),(5,7),(9,12),(10,11),(13,14),(15,16),(17,18),(19,20),(22,24),(26,28),(30,32)], [(1,7,11,30),(2,6,12,29),(3,5,9,32),(4,8,10,31),(13,25,17,23),(14,28,18,22),(15,27,19,21),(16,26,20,24)]])
C2×D4⋊C4 is a maximal subgroup of
D4⋊C42 C23.35D8 C24.65D4 C42.98D4 C42.100D4 C2.(C4×D8) D4⋊(C4⋊C4) C23.38D8 C24.74D4 (C2×SD16)⋊14C4 (C2×C4)⋊9D8 (C2×SD16)⋊15C4 M4(2).48D4 D4⋊C4⋊C4 C4.67(C4×D4) C4.D4⋊3C4 C42.433D4 C42.118D4 C42.119D4 C23⋊3SD16 (C2×C8).41D4 (C2×D4)⋊Q8 C4⋊C4.84D4 C4⋊C4⋊7D4 C4⋊C4.94D4 M4(2).10D4 (C2×C4)⋊5SD16 M4(2).12D4 (C2×C4).24D8 C42⋊8C4⋊C2 (C2×C4).27D8 2+ 1+4⋊5C4 C2×C4×D8 C2×C4×SD16 C42.275C23 (C2×D4)⋊21D4 C42.18C23 M4(2)⋊16D4 C42.20C23 (C2×D4).301D4 C42.366C23 D4⋊4D8 D4⋊7SD16 C42.461C23 C42.462C23 C42.41C23 C42.45C23 C42.49C23 C42.53C23
C2.(C8⋊pD4): C23.23D8 C24.76D4 C2.(C8⋊7D4) C2.(C8⋊2D4) C42.432D4 C42.110D4 C42.112D4 (C2×C4)⋊9SD16 ...
C2×D4⋊C4 is a maximal quotient of
C42.45D4 D4⋊M4(2) C42.315D4 C42.403D4 C42.61D4 C24.60D4 C42.409D4 C42.413D4 C42.414D4 C42.78D4 C23.35D8 C42.98D4 C23.36D8 C23.38D8 C23.23D8 C42.432D4 C42.118D4 C42.122D4 C42.436D4 C23.24D8 C23.39D8 C23.40D8 C23.41D8 C23.20SD16 C23.13D8 C23.21SD16
Matrix representation of C2×D4⋊C4 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 12 | 5 |
| 0 | 0 | 5 | 5 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[16,0,0,0,0,16,0,0,0,0,0,1,0,0,1,0],[4,0,0,0,0,13,0,0,0,0,12,5,0,0,5,5] >;
C2×D4⋊C4 in GAP, Magma, Sage, TeX
C_2\times D_4\rtimes C_4
% in TeX
G:=Group("C2xD4:C4"); // GroupNames label
G:=SmallGroup(64,95);
// by ID
G=gap.SmallGroup(64,95);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,963,489,117]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b*c>;
// generators/relations
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