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## G = C2×C4⋊Q8order 64 = 26

### Direct product of C2 and C4⋊Q8

direct product, p-group, metabelian, nilpotent (class 2), monomial, rational

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C4⋊Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4⋊Q8
 Lower central C1 — C22 — C2×C4⋊Q8
 Upper central C1 — C23 — C2×C4⋊Q8
 Jennings C1 — C22 — C2×C4⋊Q8

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

Smallest permutation representation of C2×C4⋊Q8
Regular action on 64 points
Generators in S64
(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)]])

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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