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## G = C2×C2.D8order 64 = 26

### Direct product of C2 and C2.D8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×C2.D8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C2×C2.D8
 Lower central C1 — C2 — C4 — C2×C2.D8
 Upper central C1 — C23 — C22×C4 — C2×C2.D8
 Jennings C1 — C2 — C2 — C2×C4 — C2×C2.D8

Generators and relations for C2×C2.D8
G = < a,b,c,d | a2=b2=c8=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 97 in 65 conjugacy classes, 49 normal (13 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, C2.D8, C2×C4⋊C4, C22×C8, C2×C2.D8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C2.D8, C2×C4⋊C4, C2×D8, C2×Q16, C2×C2.D8

Character table of C2×C2.D8

 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 0 0 0 0 0 0 0 0 0 0 0 0 √2 -√2 √2 √2 -√2 -√2 √2 -√2 orthogonal lifted from D8 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 0 0 0 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 -√2 √2 √2 -√2 √2 orthogonal lifted from D8 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 orthogonal lifted from D8 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 orthogonal lifted from D8 ρ23 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 ρ24 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 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 symplectic lifted from Q16, Schur index 2 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 symplectic lifted from Q16, Schur index 2 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 symplectic lifted from Q16, Schur index 2 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 symplectic lifted from Q16, Schur index 2

Smallest permutation representation of C2×C2.D8
Regular action on 64 points
Generators in S64
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(25 39)(26 40)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 57)
(1 47)(2 48)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)
(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 60 47 35)(2 59 48 34)(3 58 41 33)(4 57 42 40)(5 64 43 39)(6 63 44 38)(7 62 45 37)(8 61 46 36)(9 30 22 52)(10 29 23 51)(11 28 24 50)(12 27 17 49)(13 26 18 56)(14 25 19 55)(15 32 20 54)(16 31 21 53)

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

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

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

Matrix representation of C2×C2.D8 in GL4(𝔽17) generated by

 1 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 16 0 0 0 0 1 0 0 0 0 16 0 0 0 0 16
,
 1 0 0 0 0 1 0 0 0 0 0 11 0 0 3 11
,
 4 0 0 0 0 16 0 0 0 0 0 7 0 0 12 0
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,0,3,0,0,11,11],[4,0,0,0,0,16,0,0,0,0,0,12,0,0,7,0] >;

C2×C2.D8 in GAP, Magma, Sage, TeX

C_2\times C_2.D_8
% in TeX

G:=Group("C2xC2.D8");
// GroupNames label

G:=SmallGroup(64,107);
// by ID

G=gap.SmallGroup(64,107);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,247,963,117]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^8=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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