direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×Q8⋊C4, C23.55D4, C22.5Q16, C22.11SD16, Q8⋊3(C2×C4), (C2×Q8)⋊5C4, C4.48(C2×D4), (C2×C4).70D4, C2.1(C2×Q16), C4.2(C22×C4), (C22×C8).5C2, C2.2(C2×SD16), C4⋊C4.42C22, (C2×C4).60C23, (C2×C8).58C22, C22.42(C2×D4), (C22×Q8).4C2, C4.13(C22⋊C4), (C2×Q8).40C22, C22.33(C22⋊C4), (C22×C4).108C22, (C2×C4⋊C4).12C2, (C2×C4).44(C2×C4), C2.18(C2×C22⋊C4), SmallGroup(64,96)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×Q8⋊C4
G = < a,b,c,d | a2=b4=d4=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >
Subgroups: 121 in 81 conjugacy classes, 49 normal (15 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, Q8⋊C4, C2×C4⋊C4, C22×C8, C22×Q8, C2×Q8⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, SD16, Q16, C22×C4, C2×D4, Q8⋊C4, C2×C22⋊C4, C2×SD16, C2×Q16, C2×Q8⋊C4
Character table of C2×Q8⋊C4
| 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 | 1 | -1 | i | -i | i | -1 | 1 | -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 | -i | i | -i | -1 | 1 | 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 | -i | -i | i | 1 | 1 | 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 | i | i | -i | 1 | 1 | -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 | i | i | -i | -1 | -1 | -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 | -i | -i | i | -1 | -1 | 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 | -i | i | -i | 1 | -1 | 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 | i | -i | i | 1 | -1 | -i | -i | i | -i | i | i | -i | i | -i | 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 | orthogonal lifted from D4 |
| 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 | 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 | symplectic lifted from Q16, Schur index 2 |
| 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 | symplectic lifted from Q16, Schur index 2 |
| 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 | symplectic lifted from Q16, Schur index 2 |
| 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 | symplectic lifted from Q16, 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 | 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 20)(2 17)(3 18)(4 19)(5 54)(6 55)(7 56)(8 53)(9 16)(10 13)(11 14)(12 15)(21 35)(22 36)(23 33)(24 34)(25 31)(26 32)(27 29)(28 30)(37 52)(38 49)(39 50)(40 51)(41 47)(42 48)(43 45)(44 46)(57 64)(58 61)(59 62)(60 63)
(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 34 3 36)(2 33 4 35)(5 39 7 37)(6 38 8 40)(9 32 11 30)(10 31 12 29)(13 25 15 27)(14 28 16 26)(17 23 19 21)(18 22 20 24)(41 64 43 62)(42 63 44 61)(45 59 47 57)(46 58 48 60)(49 53 51 55)(50 56 52 54)
(1 49 11 48)(2 52 12 47)(3 51 9 46)(4 50 10 45)(5 28 64 24)(6 27 61 23)(7 26 62 22)(8 25 63 21)(13 43 19 39)(14 42 20 38)(15 41 17 37)(16 44 18 40)(29 58 33 55)(30 57 34 54)(31 60 35 53)(32 59 36 56)
G:=sub<Sym(64)| (1,20)(2,17)(3,18)(4,19)(5,54)(6,55)(7,56)(8,53)(9,16)(10,13)(11,14)(12,15)(21,35)(22,36)(23,33)(24,34)(25,31)(26,32)(27,29)(28,30)(37,52)(38,49)(39,50)(40,51)(41,47)(42,48)(43,45)(44,46)(57,64)(58,61)(59,62)(60,63), (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), (1,34,3,36)(2,33,4,35)(5,39,7,37)(6,38,8,40)(9,32,11,30)(10,31,12,29)(13,25,15,27)(14,28,16,26)(17,23,19,21)(18,22,20,24)(41,64,43,62)(42,63,44,61)(45,59,47,57)(46,58,48,60)(49,53,51,55)(50,56,52,54), (1,49,11,48)(2,52,12,47)(3,51,9,46)(4,50,10,45)(5,28,64,24)(6,27,61,23)(7,26,62,22)(8,25,63,21)(13,43,19,39)(14,42,20,38)(15,41,17,37)(16,44,18,40)(29,58,33,55)(30,57,34,54)(31,60,35,53)(32,59,36,56)>;
G:=Group( (1,20)(2,17)(3,18)(4,19)(5,54)(6,55)(7,56)(8,53)(9,16)(10,13)(11,14)(12,15)(21,35)(22,36)(23,33)(24,34)(25,31)(26,32)(27,29)(28,30)(37,52)(38,49)(39,50)(40,51)(41,47)(42,48)(43,45)(44,46)(57,64)(58,61)(59,62)(60,63), (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), (1,34,3,36)(2,33,4,35)(5,39,7,37)(6,38,8,40)(9,32,11,30)(10,31,12,29)(13,25,15,27)(14,28,16,26)(17,23,19,21)(18,22,20,24)(41,64,43,62)(42,63,44,61)(45,59,47,57)(46,58,48,60)(49,53,51,55)(50,56,52,54), (1,49,11,48)(2,52,12,47)(3,51,9,46)(4,50,10,45)(5,28,64,24)(6,27,61,23)(7,26,62,22)(8,25,63,21)(13,43,19,39)(14,42,20,38)(15,41,17,37)(16,44,18,40)(29,58,33,55)(30,57,34,54)(31,60,35,53)(32,59,36,56) );
G=PermutationGroup([[(1,20),(2,17),(3,18),(4,19),(5,54),(6,55),(7,56),(8,53),(9,16),(10,13),(11,14),(12,15),(21,35),(22,36),(23,33),(24,34),(25,31),(26,32),(27,29),(28,30),(37,52),(38,49),(39,50),(40,51),(41,47),(42,48),(43,45),(44,46),(57,64),(58,61),(59,62),(60,63)], [(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)], [(1,34,3,36),(2,33,4,35),(5,39,7,37),(6,38,8,40),(9,32,11,30),(10,31,12,29),(13,25,15,27),(14,28,16,26),(17,23,19,21),(18,22,20,24),(41,64,43,62),(42,63,44,61),(45,59,47,57),(46,58,48,60),(49,53,51,55),(50,56,52,54)], [(1,49,11,48),(2,52,12,47),(3,51,9,46),(4,50,10,45),(5,28,64,24),(6,27,61,23),(7,26,62,22),(8,25,63,21),(13,43,19,39),(14,42,20,38),(15,41,17,37),(16,44,18,40),(29,58,33,55),(30,57,34,54),(31,60,35,53),(32,59,36,56)]])
C2×Q8⋊C4 is a maximal subgroup of
Q8⋊C42 C24.155D4 C42.99D4 C42.101D4 Q8⋊(C4⋊C4) C24.160D4 (C2×SD16)⋊15C4 C24.135D4 C24.75D4 C4.68(C4×D4) C2.(C8⋊8D4) C2.(C8⋊D4) C42.431D4 C42.433D4 C42.110D4 C42.111D4 (C2×C4)⋊9SD16 (C2×C4)⋊6Q16 (C2×Q16)⋊10C4 C8⋊(C22⋊C4) C42.117D4 (C2×C4)⋊3SD16 (C2×C4)⋊2Q16 (C2×Q8)⋊Q8 C24.86D4 C4⋊C4.95D4 (C2×C4)⋊3Q16 (C2×Q8).8Q8 (C2×C4).19Q16 (C2×Q8).109D4 (C2×C8).60D4 (C2×C8).170D4 (C2×C8).171D4 2- 1+4⋊4C4 C2×C4×SD16 C2×C4×Q16 C42.276C23 (C2×Q8)⋊17D4 C42.19C23 M4(2)⋊17D4 C42.21C23 (C2×D4).302D4 C42.367C23 D4⋊8SD16 D4⋊5Q16 C42.465C23 C42.466C23 C42.43C23 C42.47C23 C42.51C23 C42.55C23
C2.(D4.pD4): C24.65D4 Q8⋊C4⋊C4 C24.73D4 (C2×SD16)⋊14C4 (C2×C4)⋊9Q16 M4(2).49D4 C2.(C4×Q16) C4.10D4⋊3C4 ...
C2×Q8⋊C4 is a maximal quotient of
C42.46D4 Q8⋊M4(2) C42.316D4 C42.404D4 C42.62D4 C24.61D4 C42.410D4 C42.415D4 C42.416D4 C42.79D4 C24.155D4 C42.99D4 C24.157D4 C24.160D4 C24.135D4 C42.431D4 C42.117D4 C42.121D4 C42.436D4
Matrix representation of C2×Q8⋊C4 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 1 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 9 |
| 0 | 0 | 0 | 13 |
| 13 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 10 | 7 |
| 0 | 0 | 5 | 7 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,15,16],[16,0,0,0,0,1,0,0,0,0,4,0,0,0,9,13],[13,0,0,0,0,16,0,0,0,0,10,5,0,0,7,7] >;
C2×Q8⋊C4 in GAP, Magma, Sage, TeX
C_2\times Q_8\rtimes C_4
% in TeX
G:=Group("C2xQ8:C4"); // GroupNames label
G:=SmallGroup(64,96);
// by ID
G=gap.SmallGroup(64,96);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,199,963,489,117]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=d^4=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c>;
// generators/relations
Export