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## G = C2×C4.Q8order 64 = 26

### Direct product of C2 and C4.Q8

direct product, p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×C4.Q8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C2×C4.Q8
 Lower central C1 — C2 — C4 — C2×C4.Q8
 Upper central C1 — C23 — C22×C4 — C2×C4.Q8
 Jennings C1 — C2 — C2 — C2×C4 — C2×C4.Q8

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 [×6], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], C23, C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×6], C22×C4, C22×C4 [×2], C4.Q8 [×4], C2×C4⋊C4 [×2], C22×C8, C2×C4.Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], SD16 [×4], C22×C4, C2×D4, C2×Q8, C4.Q8 [×4], C2×C4⋊C4, C2×SD16 [×2], 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

Smallest permutation representation of C2×C4.Q8
Regular action on 64 points
Generators in S64
(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 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)(49 62)(50 63)(51 64)(52 57)(53 58)(54 59)(55 60)(56 61)
(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 52 29 56)(26 53 30 49)(27 54 31 50)(28 55 32 51)(33 62 37 58)(34 63 38 59)(35 64 39 60)(36 57 40 61)
(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 34)(2 64 44 37)(3 59 45 40)(4 62 46 35)(5 57 47 38)(6 60 48 33)(7 63 41 36)(8 58 42 39)(9 30 24 55)(10 25 17 50)(11 28 18 53)(12 31 19 56)(13 26 20 51)(14 29 21 54)(15 32 22 49)(16 27 23 52)

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)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35)(49,62)(50,63)(51,64)(52,57)(53,58)(54,59)(55,60)(56,61), (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)(25,52,29,56)(26,53,30,49)(27,54,31,50)(28,55,32,51)(33,62,37,58)(34,63,38,59)(35,64,39,60)(36,57,40,61), (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), (1,61,43,34)(2,64,44,37)(3,59,45,40)(4,62,46,35)(5,57,47,38)(6,60,48,33)(7,63,41,36)(8,58,42,39)(9,30,24,55)(10,25,17,50)(11,28,18,53)(12,31,19,56)(13,26,20,51)(14,29,21,54)(15,32,22,49)(16,27,23,52)>;

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)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35)(49,62)(50,63)(51,64)(52,57)(53,58)(54,59)(55,60)(56,61), (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)(25,52,29,56)(26,53,30,49)(27,54,31,50)(28,55,32,51)(33,62,37,58)(34,63,38,59)(35,64,39,60)(36,57,40,61), (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), (1,61,43,34)(2,64,44,37)(3,59,45,40)(4,62,46,35)(5,57,47,38)(6,60,48,33)(7,63,41,36)(8,58,42,39)(9,30,24,55)(10,25,17,50)(11,28,18,53)(12,31,19,56)(13,26,20,51)(14,29,21,54)(15,32,22,49)(16,27,23,52) );

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),(25,36),(26,37),(27,38),(28,39),(29,40),(30,33),(31,34),(32,35),(49,62),(50,63),(51,64),(52,57),(53,58),(54,59),(55,60),(56,61)], [(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),(25,52,29,56),(26,53,30,49),(27,54,31,50),(28,55,32,51),(33,62,37,58),(34,63,38,59),(35,64,39,60),(36,57,40,61)], [(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)], [(1,61,43,34),(2,64,44,37),(3,59,45,40),(4,62,46,35),(5,57,47,38),(6,60,48,33),(7,63,41,36),(8,58,42,39),(9,30,24,55),(10,25,17,50),(11,28,18,53),(12,31,19,56),(13,26,20,51),(14,29,21,54),(15,32,22,49),(16,27,23,52)])

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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