direct product, p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C2×C4⋊Q8, C22.25C24, C42.89C22, C23.73C23, (C2×C4)⋊4Q8, C4⋊1(C2×Q8), (C2×C4).86D4, C4.14(C2×D4), C2.5(C22×Q8), C4⋊C4.71C22, (C2×C42).19C2, C22.63(C2×D4), C2.11(C22×D4), (C22×Q8).8C2, C22.19(C2×Q8), (C2×C4).131C23, (C2×Q8).55C22, (C22×C4).124C22, (C2×C4⋊C4).19C2, SmallGroup(64,212)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C4⋊Q8
G = < a,b,c,d | a2=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 185 in 145 conjugacy classes, 105 normal (7 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2×C42, C2×C4⋊C4, C4⋊Q8, C22×Q8, C2×C4⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C4⋊Q8, C22×D4, C22×Q8, 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 | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 4T | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 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 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 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 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 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 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ22 | 2 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ23 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ24 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ25 | 2 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ26 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ27 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ28 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
►Regular action on 64 points
(1 12)(2 9)(3 10)(4 11)(5 24)(6 21)(7 22)(8 23)(13 18)(14 19)(15 20)(16 17)(25 64)(26 61)(27 62)(28 63)(29 44)(30 41)(31 42)(32 43)(33 54)(34 55)(35 56)(36 53)(37 57)(38 58)(39 59)(40 60)(45 50)(46 51)(47 52)(48 49)
(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 19 29 5)(2 20 30 6)(3 17 31 7)(4 18 32 8)(9 15 41 21)(10 16 42 22)(11 13 43 23)(12 14 44 24)(25 35 46 57)(26 36 47 58)(27 33 48 59)(28 34 45 60)(37 64 56 51)(38 61 53 52)(39 62 54 49)(40 63 55 50)
(1 26 29 47)(2 25 30 46)(3 28 31 45)(4 27 32 48)(5 36 19 58)(6 35 20 57)(7 34 17 60)(8 33 18 59)(9 64 41 51)(10 63 42 50)(11 62 43 49)(12 61 44 52)(13 39 23 54)(14 38 24 53)(15 37 21 56)(16 40 22 55)
G:=sub<Sym(64)| (1,12)(2,9)(3,10)(4,11)(5,24)(6,21)(7,22)(8,23)(13,18)(14,19)(15,20)(16,17)(25,64)(26,61)(27,62)(28,63)(29,44)(30,41)(31,42)(32,43)(33,54)(34,55)(35,56)(36,53)(37,57)(38,58)(39,59)(40,60)(45,50)(46,51)(47,52)(48,49), (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), (1,19,29,5)(2,20,30,6)(3,17,31,7)(4,18,32,8)(9,15,41,21)(10,16,42,22)(11,13,43,23)(12,14,44,24)(25,35,46,57)(26,36,47,58)(27,33,48,59)(28,34,45,60)(37,64,56,51)(38,61,53,52)(39,62,54,49)(40,63,55,50), (1,26,29,47)(2,25,30,46)(3,28,31,45)(4,27,32,48)(5,36,19,58)(6,35,20,57)(7,34,17,60)(8,33,18,59)(9,64,41,51)(10,63,42,50)(11,62,43,49)(12,61,44,52)(13,39,23,54)(14,38,24,53)(15,37,21,56)(16,40,22,55)>;
G:=Group( (1,12)(2,9)(3,10)(4,11)(5,24)(6,21)(7,22)(8,23)(13,18)(14,19)(15,20)(16,17)(25,64)(26,61)(27,62)(28,63)(29,44)(30,41)(31,42)(32,43)(33,54)(34,55)(35,56)(36,53)(37,57)(38,58)(39,59)(40,60)(45,50)(46,51)(47,52)(48,49), (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), (1,19,29,5)(2,20,30,6)(3,17,31,7)(4,18,32,8)(9,15,41,21)(10,16,42,22)(11,13,43,23)(12,14,44,24)(25,35,46,57)(26,36,47,58)(27,33,48,59)(28,34,45,60)(37,64,56,51)(38,61,53,52)(39,62,54,49)(40,63,55,50), (1,26,29,47)(2,25,30,46)(3,28,31,45)(4,27,32,48)(5,36,19,58)(6,35,20,57)(7,34,17,60)(8,33,18,59)(9,64,41,51)(10,63,42,50)(11,62,43,49)(12,61,44,52)(13,39,23,54)(14,38,24,53)(15,37,21,56)(16,40,22,55) );
G=PermutationGroup([[(1,12),(2,9),(3,10),(4,11),(5,24),(6,21),(7,22),(8,23),(13,18),(14,19),(15,20),(16,17),(25,64),(26,61),(27,62),(28,63),(29,44),(30,41),(31,42),(32,43),(33,54),(34,55),(35,56),(36,53),(37,57),(38,58),(39,59),(40,60),(45,50),(46,51),(47,52),(48,49)], [(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)], [(1,19,29,5),(2,20,30,6),(3,17,31,7),(4,18,32,8),(9,15,41,21),(10,16,42,22),(11,13,43,23),(12,14,44,24),(25,35,46,57),(26,36,47,58),(27,33,48,59),(28,34,45,60),(37,64,56,51),(38,61,53,52),(39,62,54,49),(40,63,55,50)], [(1,26,29,47),(2,25,30,46),(3,28,31,45),(4,27,32,48),(5,36,19,58),(6,35,20,57),(7,34,17,60),(8,33,18,59),(9,64,41,51),(10,63,42,50),(11,62,43,49),(12,61,44,52),(13,39,23,54),(14,38,24,53),(15,37,21,56),(16,40,22,55)]])
C2×C4⋊Q8 is a maximal subgroup of
C42.414D4 C42.415D4 C42.416D4 C42.83D4 C42.85D4 C4⋊Q8⋊15C4 C42.431D4 C42.111D4 C42.114D4 C42.117D4 C42.121D4 C42.122D4 C42.436D4 C42.125D4 M4(2)⋊8Q8 C42.130D4 (C2×C4)⋊3SD16 (C2×C4)⋊2Q16 (C2×D4)⋊Q8 (C2×Q8)⋊Q8 C4⋊C4.95D4 C4⋊C4⋊Q8 (C2×C8)⋊Q8 C42.161D4 C42⋊4Q8 C23.247C24 C23.251C24 C23.263C24 C23.329C24 C24.264C23 C23.334C24 C24.267C23 C24.568C23 C23.346C24 C23.351C24 C23.352C24 C23.391C24 C23.392C24 C42.167D4 C42.168D4 C42.169D4 C42.170D4 C42⋊6Q8 C42⋊7Q8 C42.173D4 C42.176D4 C42.177D4 C42.186D4 C42.187D4 C42.193D4 C42.195D4 C42.196D4 C42⋊10Q8 C23.574C24 C24.385C23 C23.613C24 C23.616C24 C24.421C23 C23.631C24 C23.634C24 C42.200D4 C42.440D4 C43⋊12C2 C42⋊18Q8 C42⋊19Q8 C42.445D4 C42.448D4 C42.241D4 C42.243D4 M4(2)⋊8D4 M4(2)⋊5Q8 C42.264D4 C42.267D4 C42.276D4 C42.278D4 C42.279D4 C42.281D4 C42.282D4 C42.290D4 C42.291D4 C2×D4×Q8 C2×Q82 C22.88C25 C22.92C25 C22.98C25 C22.100C25 C22.133C25 C22.141C25
C2×C4⋊Q8 is a maximal quotient of
C42⋊5Q8 C23.334C24 C24.568C23 C23.402C24 C42.169D4 C23.449C24 C42⋊6Q8 C42⋊7Q8 C42.35Q8 C42.176D4 C23.483C24 C42.181D4 C23.559C24 C42⋊10Q8 C42.440D4 C43.15C2 C42⋊18Q8 C42⋊19Q8 C42.364D4 C42.252D4 M4(2)⋊3Q8 M4(2)⋊4Q8 M4(2)⋊5Q8 M4(2)⋊6Q8
Matrix representation of C2×C4⋊Q8 ►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 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 4 | 0 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 1 |
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],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,1,0],[4,0,0,0,0,0,0,1,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,1] >;
C2×C4⋊Q8 in GAP, Magma, Sage, TeX
C_2\times C_4\rtimes Q_8
% in TeX
G:=Group("C2xC4:Q8"); // GroupNames label
G:=SmallGroup(64,212);
// by ID
G=gap.SmallGroup(64,212);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,96,217,103,650,158]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^4=1,d^2=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^-1>;
// generators/relations
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