direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C4.Q8, C23.56D4, C22.12SD16, (C2×C8)⋊7C4, C8⋊8(C2×C4), C4.1(C2×Q8), (C2×C4).72D4, C4.13(C4⋊C4), (C2×C4).18Q8, C2.3(C2×SD16), C4⋊C4.46C22, C4.24(C22×C4), (C22×C8).13C2, (C2×C4).67C23, (C2×C8).90C22, C22.47(C2×D4), C22.19(C4⋊C4), (C22×C4).113C22, C2.11(C2×C4⋊C4), (C2×C4⋊C4).13C2, (C2×C4).72(C2×C4), SmallGroup(64,106)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C4.Q8
G = < a,b,c,d | a2=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
Subgroups: 97 in 65 conjugacy classes, 49 normal (11 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, C4.Q8, C2×C4⋊C4, C22×C8, C2×C4.Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, C4.Q8, C2×C4⋊C4, C2×SD16, C2×C4.Q8
Character table of C2×C4.Q8
| 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 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 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 | -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 | symplectic lifted from Q8, Schur index 2 |
| ρ20 | 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 |
| 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 | complex lifted from SD16 |
| 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 | complex lifted from SD16 |
| 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 | complex lifted from SD16 |
| 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 | complex lifted from SD16 |
| 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 |
►Regular action on 64 points
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 48)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(25 38)(26 39)(27 40)(28 33)(29 34)(30 35)(31 36)(32 37)(49 60)(50 61)(51 62)(52 63)(53 64)(54 57)(55 58)(56 59)
(1 45 5 41)(2 46 6 42)(3 47 7 43)(4 48 8 44)(9 18 13 22)(10 19 14 23)(11 20 15 24)(12 21 16 17)(25 54 29 50)(26 55 30 51)(27 56 31 52)(28 49 32 53)(33 60 37 64)(34 61 38 57)(35 62 39 58)(36 63 40 59)
(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 61 43 36)(2 64 44 39)(3 59 45 34)(4 62 46 37)(5 57 47 40)(6 60 48 35)(7 63 41 38)(8 58 42 33)(9 30 24 49)(10 25 17 52)(11 28 18 55)(12 31 19 50)(13 26 20 53)(14 29 21 56)(15 32 22 51)(16 27 23 54)
G:=sub<Sym(64)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,48)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(25,38)(26,39)(27,40)(28,33)(29,34)(30,35)(31,36)(32,37)(49,60)(50,61)(51,62)(52,63)(53,64)(54,57)(55,58)(56,59), (1,45,5,41)(2,46,6,42)(3,47,7,43)(4,48,8,44)(9,18,13,22)(10,19,14,23)(11,20,15,24)(12,21,16,17)(25,54,29,50)(26,55,30,51)(27,56,31,52)(28,49,32,53)(33,60,37,64)(34,61,38,57)(35,62,39,58)(36,63,40,59), (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), (1,61,43,36)(2,64,44,39)(3,59,45,34)(4,62,46,37)(5,57,47,40)(6,60,48,35)(7,63,41,38)(8,58,42,33)(9,30,24,49)(10,25,17,52)(11,28,18,55)(12,31,19,50)(13,26,20,53)(14,29,21,56)(15,32,22,51)(16,27,23,54)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,48)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(25,38)(26,39)(27,40)(28,33)(29,34)(30,35)(31,36)(32,37)(49,60)(50,61)(51,62)(52,63)(53,64)(54,57)(55,58)(56,59), (1,45,5,41)(2,46,6,42)(3,47,7,43)(4,48,8,44)(9,18,13,22)(10,19,14,23)(11,20,15,24)(12,21,16,17)(25,54,29,50)(26,55,30,51)(27,56,31,52)(28,49,32,53)(33,60,37,64)(34,61,38,57)(35,62,39,58)(36,63,40,59), (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), (1,61,43,36)(2,64,44,39)(3,59,45,34)(4,62,46,37)(5,57,47,40)(6,60,48,35)(7,63,41,38)(8,58,42,33)(9,30,24,49)(10,25,17,52)(11,28,18,55)(12,31,19,50)(13,26,20,53)(14,29,21,56)(15,32,22,51)(16,27,23,54) );
G=PermutationGroup([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,48),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(25,38),(26,39),(27,40),(28,33),(29,34),(30,35),(31,36),(32,37),(49,60),(50,61),(51,62),(52,63),(53,64),(54,57),(55,58),(56,59)], [(1,45,5,41),(2,46,6,42),(3,47,7,43),(4,48,8,44),(9,18,13,22),(10,19,14,23),(11,20,15,24),(12,21,16,17),(25,54,29,50),(26,55,30,51),(27,56,31,52),(28,49,32,53),(33,60,37,64),(34,61,38,57),(35,62,39,58),(36,63,40,59)], [(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)], [(1,61,43,36),(2,64,44,39),(3,59,45,34),(4,62,46,37),(5,57,47,40),(6,60,48,35),(7,63,41,38),(8,58,42,33),(9,30,24,49),(10,25,17,52),(11,28,18,55),(12,31,19,50),(13,26,20,53),(14,29,21,56),(15,32,22,51),(16,27,23,54)]])
C2×C4.Q8 is a maximal subgroup of
C8.11C42 C8.C42 C8⋊C42 C24.133D4 C24.67D4 C4○D4.4Q8 C42.58Q8 C42.60Q8 C42.26Q8 C24.159D4 C24.71D4 D4⋊(C4⋊C4) Q8⋊C4⋊C4 C4.Q8⋊9C4 C4.Q8⋊10C4 C4.67(C4×D4) C4.68(C4×D4) C8⋊7(C4⋊C4) C4.(C4×Q8) C8⋊(C4⋊C4) C42.30Q8 C42.31Q8 M4(2).5Q8 (C2×C4)⋊9SD16 (C2×Q16)⋊10C4 (C2×D8)⋊10C4 C24.84D4 C24.85D4 (C2×C8)⋊Q8 C2.(C8⋊Q8) C4⋊C4.106D4 (C2×Q8).8Q8 C24.89D4 (C2×C8).165D4 C2.(C8⋊3Q8) (C2×C8).24Q8 C4.(C4⋊Q8) M4(2).2Q8 (C2×C8).168D4 (C2×C8).169D4 (C2×C8).170D4 (C2×C8).171D4 C4○D4.7Q8 C2×C4×SD16 C42.281C23 (C2×C8)⋊13D4 C42.23C23 (C2×D4).304D4 M4(2)⋊3Q8 D4⋊9SD16 C42.486C23 C42.58C23 C42.59C23
C2×C4.Q8 is a maximal quotient of
C8⋊8M4(2) C42.90D4 C24.133D4 C42.55Q8 C42.58Q8 C24.159D4 C42.30Q8 M5(2)⋊3C4
Matrix representation of C2×C4.Q8 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 5 | 12 |
| 0 | 0 | 5 | 5 |
| 4 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 6 | 13 |
| 0 | 0 | 13 | 11 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[1,0,0,0,0,16,0,0,0,0,5,5,0,0,12,5],[4,0,0,0,0,16,0,0,0,0,6,13,0,0,13,11] >;
C2×C4.Q8 in GAP, Magma, Sage, TeX
C_2\times C_4.Q_8
% in TeX
G:=Group("C2xC4.Q8"); // GroupNames label
G:=SmallGroup(64,106);
// by ID
G=gap.SmallGroup(64,106);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,55,963,117]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=1,c^4=b^2,d^2=b^-1*c^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=c^3>;
// generators/relations
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