direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C42.C2, C22.21C24, C42.88C22, C23.71C23, C4.9(C2×Q8), (C2×C4).22Q8, C2.4(C22×Q8), C4⋊C4.69C22, (C2×C42).18C2, (C2×C4).14C23, C22.18(C2×Q8), C22.33(C4○D4), (C22×C4).59C22, (C2×C4⋊C4).18C2, C2.10(C2×C4○D4), SmallGroup(64,208)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C42.C2
G = < a,b,c,d | a2=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c >
Subgroups: 137 in 113 conjugacy classes, 89 normal (7 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C42, C4⋊C4, C22×C4, C22×C4, C2×C42, C2×C4⋊C4, C42.C2, C2×C42.C2
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C42.C2, C22×Q8, C2×C4○D4, C2×C42.C2
Character table of C2×C42.C2
| 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 | -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 |
| ρ18 | 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 |
| ρ19 | 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 |
| ρ20 | 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 |
| ρ21 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 2i | 0 | 0 | 0 | 0 | -2i | -2i | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 0 | 2i | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 2i | 2i | 0 | 0 | 0 | 0 | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 0 | -2i | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | -2i | 0 | 0 | 0 | 0 | 2i | 2i | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | -2i | -2i | 0 | 0 | 0 | 0 | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►Regular action on 64 points
(1 25)(2 26)(3 27)(4 28)(5 39)(6 40)(7 37)(8 38)(9 15)(10 16)(11 13)(12 14)(17 29)(18 30)(19 31)(20 32)(21 46)(22 47)(23 48)(24 45)(33 44)(34 41)(35 42)(36 43)(49 60)(50 57)(51 58)(52 59)(53 64)(54 61)(55 62)(56 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 31 11 5)(2 32 12 6)(3 29 9 7)(4 30 10 8)(13 39 25 19)(14 40 26 20)(15 37 27 17)(16 38 28 18)(21 57 54 34)(22 58 55 35)(23 59 56 36)(24 60 53 33)(41 46 50 61)(42 47 51 62)(43 48 52 63)(44 45 49 64)
(1 36 11 59)(2 60 12 33)(3 34 9 57)(4 58 10 35)(5 54 31 21)(6 22 32 55)(7 56 29 23)(8 24 30 53)(13 52 25 43)(14 44 26 49)(15 50 27 41)(16 42 28 51)(17 48 37 63)(18 64 38 45)(19 46 39 61)(20 62 40 47)
G:=sub<Sym(64)| (1,25)(2,26)(3,27)(4,28)(5,39)(6,40)(7,37)(8,38)(9,15)(10,16)(11,13)(12,14)(17,29)(18,30)(19,31)(20,32)(21,46)(22,47)(23,48)(24,45)(33,44)(34,41)(35,42)(36,43)(49,60)(50,57)(51,58)(52,59)(53,64)(54,61)(55,62)(56,63), (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), (1,31,11,5)(2,32,12,6)(3,29,9,7)(4,30,10,8)(13,39,25,19)(14,40,26,20)(15,37,27,17)(16,38,28,18)(21,57,54,34)(22,58,55,35)(23,59,56,36)(24,60,53,33)(41,46,50,61)(42,47,51,62)(43,48,52,63)(44,45,49,64), (1,36,11,59)(2,60,12,33)(3,34,9,57)(4,58,10,35)(5,54,31,21)(6,22,32,55)(7,56,29,23)(8,24,30,53)(13,52,25,43)(14,44,26,49)(15,50,27,41)(16,42,28,51)(17,48,37,63)(18,64,38,45)(19,46,39,61)(20,62,40,47)>;
G:=Group( (1,25)(2,26)(3,27)(4,28)(5,39)(6,40)(7,37)(8,38)(9,15)(10,16)(11,13)(12,14)(17,29)(18,30)(19,31)(20,32)(21,46)(22,47)(23,48)(24,45)(33,44)(34,41)(35,42)(36,43)(49,60)(50,57)(51,58)(52,59)(53,64)(54,61)(55,62)(56,63), (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), (1,31,11,5)(2,32,12,6)(3,29,9,7)(4,30,10,8)(13,39,25,19)(14,40,26,20)(15,37,27,17)(16,38,28,18)(21,57,54,34)(22,58,55,35)(23,59,56,36)(24,60,53,33)(41,46,50,61)(42,47,51,62)(43,48,52,63)(44,45,49,64), (1,36,11,59)(2,60,12,33)(3,34,9,57)(4,58,10,35)(5,54,31,21)(6,22,32,55)(7,56,29,23)(8,24,30,53)(13,52,25,43)(14,44,26,49)(15,50,27,41)(16,42,28,51)(17,48,37,63)(18,64,38,45)(19,46,39,61)(20,62,40,47) );
G=PermutationGroup([[(1,25),(2,26),(3,27),(4,28),(5,39),(6,40),(7,37),(8,38),(9,15),(10,16),(11,13),(12,14),(17,29),(18,30),(19,31),(20,32),(21,46),(22,47),(23,48),(24,45),(33,44),(34,41),(35,42),(36,43),(49,60),(50,57),(51,58),(52,59),(53,64),(54,61),(55,62),(56,63)], [(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)], [(1,31,11,5),(2,32,12,6),(3,29,9,7),(4,30,10,8),(13,39,25,19),(14,40,26,20),(15,37,27,17),(16,38,28,18),(21,57,54,34),(22,58,55,35),(23,59,56,36),(24,60,53,33),(41,46,50,61),(42,47,51,62),(43,48,52,63),(44,45,49,64)], [(1,36,11,59),(2,60,12,33),(3,34,9,57),(4,58,10,35),(5,54,31,21),(6,22,32,55),(7,56,29,23),(8,24,30,53),(13,52,25,43),(14,44,26,49),(15,50,27,41),(16,42,28,51),(17,48,37,63),(18,64,38,45),(19,46,39,61),(20,62,40,47)]])
C2×C42.C2 is a maximal subgroup of
C42.396D4 C42.408D4 C42.71D4 C42.123D4 C42.437D4 C42.124D4 C42.128D4 C4⋊C4.84D4 C4⋊C4.85D4 C2.(C8⋊Q8) C42.33Q8 C23.218C24 C23.252C24 C23.253C24 C23.264C24 C24.268C23 C24.569C23 C23.353C24 C23.354C24 C23.360C24 C23.362C24 C24.572C23 C23.375C24 C24.301C23 C23.390C24 C23.406C24 C23.419C24 C42⋊6Q8 C42.35Q8 C23.456C24 C23.458C24 C42.172D4 C42.174D4 C42.175D4 C42.181D4 C42.185D4 C42.188D4 C42.190D4 C42.191D4 C42.194D4 C42.195D4 C42⋊10Q8 C42.198D4 C23.590C24 C24.401C23 C24.408C23 C23.607C24 C23.611C24 C23.613C24 C23.619C24 C23.620C24 C23.621C24 C23.625C24 C23.626C24 C42.199D4 C42.201D4 C42.439D4 C43.15C2 C43⋊14C2 C42.449D4 C42.244D4 M4(2)⋊6Q8 C42.284D4 C42.286D4 C42.288D4 C22.93C25 C22.101C25 C22.104C25 C22.142C25 C22.148C25 C22.152C25
C2×C42.C2 is a maximal quotient of
C42.34Q8 C24.569C23 C24.572C23 C23.405C24 C23.406C24 C23.407C24 C23.408C24 C23.409C24 C42.35Q8 C24.584C23 C42.36Q8 C42.37Q8 C23.546C24 C42.39Q8 C42.439D4 C43.15C2 C43.18C2
Matrix representation of C2×C42.C2 ►in GL5(𝔽5)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 4 | 0 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3],[1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,4,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,4,0] >;
C2×C42.C2 in GAP, Magma, Sage, TeX
C_2\times C_4^2.C_2
% in TeX
G:=Group("C2xC4^2.C2"); // GroupNames label
G:=SmallGroup(64,208);
// by ID
G=gap.SmallGroup(64,208);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,96,217,199,650,86]);
// 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*c^2,d*c*d^-1=b^2*c>;
// generators/relations
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