direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C2.D8, C22.13D8, C23.57D4, C22.6Q16, C8⋊7(C2×C4), (C2×C8)⋊5C4, C2.2(C2×D8), C4.2(C2×Q8), (C2×C4).73D4, C4.14(C4⋊C4), (C2×C4).19Q8, C2.2(C2×Q16), (C22×C8).8C2, C4⋊C4.47C22, C4.25(C22×C4), (C2×C8).73C22, (C2×C4).68C23, C22.48(C2×D4), C22.20(C4⋊C4), (C22×C4).114C22, C2.12(C2×C4⋊C4), (C2×C4⋊C4).14C2, (C2×C4).73(C2×C4), SmallGroup(64,107)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C2.D8
G = < a,b,c,d | a2=b2=c8=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 97 in 65 conjugacy classes, 49 normal (13 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2.D8, C2×C4⋊C4, C22×C8, C2×C2.D8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C2.D8, C2×C4⋊C4, C2×D8, C2×Q16, C2×C2.D8
Character table of C2×C2.D8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | -i | i | -i | i | i | -i | i | -i | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 4 |
| ρ10 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | i | i | -i | -i | i | i | -i | -i | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | linear of order 4 |
| ρ11 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | i | i | -i | -i | i | i | -i | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ12 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | i | i | i | i | -i | -i | -i | -i | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | linear of order 4 |
| ρ13 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -i | -i | -i | -i | i | i | i | i | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | linear of order 4 |
| ρ14 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | i | -i | -i | i | i | -i | -i | i | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ15 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -i | -i | i | i | -i | -i | i | i | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | linear of order 4 |
| ρ16 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | i | -i | i | -i | -i | i | -i | i | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 4 |
| ρ17 | 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 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 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 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 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 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | -√2 | √2 | √2 | -√2 | √2 | orthogonal lifted from D8 |
| 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 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ24 | 2 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| 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 | symplectic lifted from Q16, Schur index 2 |
| 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 | symplectic lifted from Q16, Schur index 2 |
| 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 | symplectic lifted from Q16, Schur index 2 |
| 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 | symplectic lifted from Q16, Schur index 2 |
►Regular action on 64 points
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(25 39)(26 40)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 57)
(1 47)(2 48)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 60 47 35)(2 59 48 34)(3 58 41 33)(4 57 42 40)(5 64 43 39)(6 63 44 38)(7 62 45 37)(8 61 46 36)(9 30 22 52)(10 29 23 51)(11 28 24 50)(12 27 17 49)(13 26 18 56)(14 25 19 55)(15 32 20 54)(16 31 21 53)
G:=sub<Sym(64)| (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(25,39)(26,40)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,57), (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,57), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,60,47,35)(2,59,48,34)(3,58,41,33)(4,57,42,40)(5,64,43,39)(6,63,44,38)(7,62,45,37)(8,61,46,36)(9,30,22,52)(10,29,23,51)(11,28,24,50)(12,27,17,49)(13,26,18,56)(14,25,19,55)(15,32,20,54)(16,31,21,53)>;
G:=Group( (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(25,39)(26,40)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,57), (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,57), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,60,47,35)(2,59,48,34)(3,58,41,33)(4,57,42,40)(5,64,43,39)(6,63,44,38)(7,62,45,37)(8,61,46,36)(9,30,22,52)(10,29,23,51)(11,28,24,50)(12,27,17,49)(13,26,18,56)(14,25,19,55)(15,32,20,54)(16,31,21,53) );
G=PermutationGroup([[(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,46),(10,47),(11,48),(12,41),(13,42),(14,43),(15,44),(16,45),(25,39),(26,40),(27,33),(28,34),(29,35),(30,36),(31,37),(32,38),(49,58),(50,59),(51,60),(52,61),(53,62),(54,63),(55,64),(56,57)], [(1,47),(2,48),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21),(25,55),(26,56),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,58),(34,59),(35,60),(36,61),(37,62),(38,63),(39,64),(40,57)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,60,47,35),(2,59,48,34),(3,58,41,33),(4,57,42,40),(5,64,43,39),(6,63,44,38),(7,62,45,37),(8,61,46,36),(9,30,22,52),(10,29,23,51),(11,28,24,50),(12,27,17,49),(13,26,18,56),(14,25,19,55),(15,32,20,54),(16,31,21,53)]])
C2×C2.D8 is a maximal subgroup of
C8.7C42 C8.2C42 C8⋊C42 C23.22D8 C24.67D4 C4○D4.5Q8 C42.59Q8 C42.60Q8 C42.26Q8 C23.37D8 C24.71D4 C2.(C4×D8) Q8⋊(C4⋊C4) C2.D8⋊4C4 C2.D8⋊5C4 D4⋊C4⋊C4 C2.(C4×Q16) C8⋊5(C4⋊C4) C4.(C4×Q8) C8⋊(C4⋊C4) C42.29Q8 C42.31Q8 M4(2).6Q8 (C2×C4)⋊6Q16 (C2×C4)⋊6D8 C8⋊(C22⋊C4) C24.83D4 C24.86D4 C4⋊C4⋊Q8 C2.(C8⋊Q8) (C2×C4).23D8 (C2×C8).52D4 C23.12D8 C24.88D4 (C2×C8).55D4 (C2×C8).1Q8 (C2×C8).24Q8 (C2×C4).26D8 (C2×C4).21Q16 M4(2).Q8 (C2×C8).168D4 (C2×C4).27D8 (C2×C8).60D4 (C2×C8).171D4 C23.39D8 M5(2)⋊1C4 C22.D16 C23.49D8 C23.50D8 C23.51D8 C4○D4.8Q8 C2×C4×D8 C2×C4×Q16 C42.280C23 (C2×C8)⋊14D4 C42.22C23 (C2×D4).303D4 M4(2)⋊4Q8 D4⋊5D8 C42.485C23 D4⋊6Q16 C42.488C23 C42.57C23 C42.60C23
C2×C2.D8 is a maximal quotient of
C8⋊7M4(2) C42.91D4 C23.22D8 C42.55Q8 C42.59Q8 C23.37D8 C42.29Q8 C23.25D8 M5(2)⋊1C4 M5(2).1C4
Matrix representation of C2×C2.D8 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 3 | 11 |
| 4 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 7 |
| 0 | 0 | 12 | 0 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,0,3,0,0,11,11],[4,0,0,0,0,16,0,0,0,0,0,12,0,0,7,0] >;
C2×C2.D8 in GAP, Magma, Sage, TeX
C_2\times C_2.D_8
% in TeX
G:=Group("C2xC2.D8"); // GroupNames label
G:=SmallGroup(64,107);
// by ID
G=gap.SmallGroup(64,107);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,247,963,117]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^8=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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