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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

Aliases: C2×C4⋊Q8, C22.25C24, C42.89C22, C23.73C23, (C2×C4)⋊4Q8, C41(C2×Q8), (C2×C4).86D4, C4.14(C2×D4), C2.5(C22×Q8), C4⋊C4.71C22, (C2×C42).19C2, C22.63(C2×D4), C2.11(C22×D4), (C22×Q8).8C2, C22.19(C2×Q8), (C2×C4).131C23, (C2×Q8).55C22, (C22×C4).124C22, (C2×C4⋊C4).19C2, SmallGroup(64,212)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C4⋊Q8
C1C2C22C23C22×C4C2×C42 — C2×C4⋊Q8
C1C22 — C2×C4⋊Q8
C1C23 — C2×C4⋊Q8
C1C22 — 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 [×6], C4 [×12], C4 [×8], C22, C22 [×6], C2×C4 [×26], C2×C4 [×8], Q8 [×16], C23, C42 [×4], C4⋊C4 [×16], C22×C4, C22×C4 [×6], C2×Q8 [×8], C2×Q8 [×8], C2×C42, C2×C4⋊C4 [×4], C4⋊Q8 [×8], C22×Q8 [×2], C2×C4⋊Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C24, C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], C2×C4⋊Q8

Character table of C2×C4⋊Q8

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R4S4T
 size 1111111122222222222244444444
ρ11111111111111111111111111111    trivial
ρ2111111111-1-1-111-1-1-1-11-111-1-1-1-111    linear of order 2
ρ311111111-1-111-1-1-1-111-1-1-1-111-1-111    linear of order 2
ρ411111111-11-1-1-1-111-1-1-11-1-1-1-11111    linear of order 2
ρ511111111-1-111-1-1-1-111-1-111-1-111-1-1    linear of order 2
ρ6111111111-1-1-111-1-1-1-11-1-1-11111-1-1    linear of order 2
ρ711111111-11-1-1-1-111-1-1-111111-1-1-1-1    linear of order 2
ρ811111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ91-1111-1-1-1-1-11-11-11-11-1111-11-11-11-1    linear of order 2
ρ101-1111-1-1-11-1-11-111-1-11-11-11-111-11-1    linear of order 2
ρ111-1111-1-1-1-11-111-1-11-111-11-1-11-111-1    linear of order 2
ρ121-1111-1-1-1111-1-11-111-1-1-1-111-1-111-1    linear of order 2
ρ131-1111-1-1-11-1-11-111-1-11-111-11-1-11-11    linear of order 2
ρ141-1111-1-1-1-1-11-11-11-11-111-11-11-11-11    linear of order 2
ρ151-1111-1-1-1111-1-11-111-1-1-11-1-111-1-11    linear of order 2
ρ161-1111-1-1-1-11-111-1-11-111-1-111-11-1-11    linear of order 2
ρ172-2-22-2-22200220000-2-20000000000    orthogonal lifted from D4
ρ1822-22-22-2-2002-20000-220000000000    orthogonal lifted from D4
ρ1922-22-22-2-200-2200002-20000000000    orthogonal lifted from D4
ρ202-2-22-2-22200-2-20000220000000000    orthogonal lifted from D4
ρ2122-2-22-2-222000-2-200002000000000    symplectic lifted from Q8, Schur index 2
ρ222-2-2-2222-2-2000-2200002000000000    symplectic lifted from Q8, Schur index 2
ρ232-22-2-22-220-2000022000-200000000    symplectic lifted from Q8, Schur index 2
ρ24222-2-2-22-20-20000-22000200000000    symplectic lifted from Q8, Schur index 2
ρ252-2-2-2222-220002-20000-2000000000    symplectic lifted from Q8, Schur index 2
ρ2622-2-22-2-22-2000220000-2000000000    symplectic lifted from Q8, Schur index 2
ρ27222-2-2-22-20200002-2000-200000000    symplectic lifted from Q8, Schur index 2
ρ282-22-2-22-22020000-2-2000200000000    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 55 46 57)(26 56 47 58)(27 53 48 59)(28 54 45 60)(33 50 40 63)(34 51 37 64)(35 52 38 61)(36 49 39 62)
(1 26 29 47)(2 25 30 46)(3 28 31 45)(4 27 32 48)(5 56 19 58)(6 55 20 57)(7 54 17 60)(8 53 18 59)(9 64 41 51)(10 63 42 50)(11 62 43 49)(12 61 44 52)(13 39 23 36)(14 38 24 35)(15 37 21 34)(16 40 22 33)

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

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

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

C2×C4⋊Q8 is a maximal subgroup of
C42.414D4  C42.415D4  C42.416D4  C42.83D4  C42.85D4  C4⋊Q815C4  C42.431D4  C42.111D4  C42.114D4  C42.117D4  C42.121D4  C42.122D4  C42.436D4  C42.125D4  M4(2)⋊8Q8  C42.130D4  (C2×C4)⋊3SD16  (C2×C4)⋊2Q16  (C2×D4)⋊Q8  (C2×Q8)⋊Q8  C4⋊C4.95D4  C4⋊C4⋊Q8  (C2×C8)⋊Q8  C42.161D4  C424Q8  C23.247C24  C23.251C24  C23.263C24  C23.329C24  C24.264C23  C23.334C24  C24.267C23  C24.568C23  C23.346C24  C23.351C24  C23.352C24  C23.391C24  C23.392C24  C42.167D4  C42.168D4  C42.169D4  C42.170D4  C426Q8  C427Q8  C42.173D4  C42.176D4  C42.177D4  C42.186D4  C42.187D4  C42.193D4  C42.195D4  C42.196D4  C4210Q8  C23.574C24  C24.385C23  C23.613C24  C23.616C24  C24.421C23  C23.631C24  C23.634C24  C42.200D4  C42.440D4  C4312C2  C4218Q8  C4219Q8  C42.445D4  C42.448D4  C42.241D4  C42.243D4  M4(2)⋊8D4  M4(2)⋊5Q8  C42.264D4  C42.267D4  C42.276D4  C42.278D4  C42.279D4  C42.281D4  C42.282D4  C42.290D4  C42.291D4  C2×D4×Q8  C2×Q82  C22.88C25  C22.92C25  C22.98C25  C22.100C25  C22.133C25  C22.141C25
C2×C4⋊Q8 is a maximal quotient of
C425Q8  C23.334C24  C24.568C23  C23.402C24  C42.169D4  C23.449C24  C426Q8  C427Q8  C42.35Q8  C42.176D4  C23.483C24  C42.181D4  C23.559C24  C4210Q8  C42.440D4  C43.15C2  C4218Q8  C4219Q8  C42.364D4  C42.252D4  M4(2)⋊3Q8  M4(2)⋊4Q8  M4(2)⋊5Q8  M4(2)⋊6Q8

Matrix representation of C2×C4⋊Q8 in GL5(𝔽5)

40000
04000
00400
00040
00004
,
10000
01000
00100
00001
00040
,
40000
00400
01000
00040
00004
,
10000
03000
00200
00040
00001

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

Export

Character table of C2×C4⋊Q8 in TeX

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