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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

Aliases: C2×C4.Q8, C23.56D4, C22.12SD16, (C2×C8)⋊7C4, C88(C2×C4), C4.1(C2×Q8), (C2×C4).72D4, C4.13(C4⋊C4), (C2×C4).18Q8, C2.3(C2×SD16), C4⋊C4.46C22, C4.24(C22×C4), (C22×C8).13C2, (C2×C4).67C23, (C2×C8).90C22, C22.47(C2×D4), C22.19(C4⋊C4), (C22×C4).113C22, C2.11(C2×C4⋊C4), (C2×C4⋊C4).13C2, (C2×C4).72(C2×C4), SmallGroup(64,106)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C2×C4.Q8
C1C2C22C2×C4C22×C4C22×C8 — C2×C4.Q8
C1C2C4 — C2×C4.Q8
C1C23C22×C4 — C2×C4.Q8
C1C2C2C2×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 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H
 size 1111111122224444444422222222
ρ11111111111111111111111111111    trivial
ρ21-11-11-11-11-1-111-11-11-11-1-1-11-11-111    linear of order 2
ρ3111111111111-1-11111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ41-11-11-11-11-1-11-111-11-1-1111-11-11-1-1    linear of order 2
ρ511111111111111-1-1-1-111-1-1-1-1-1-1-1-1    linear of order 2
ρ61-11-11-11-11-1-11-11-11-11-11-1-11-11-111    linear of order 2
ρ71-11-11-11-11-1-111-1-11-111-111-11-11-1-1    linear of order 2
ρ8111111111111-1-1-1-1-1-1-1-111111111    linear of order 2
ρ911-111-1-1-1-1-111-ii-iii-ii-i1-11-1-11-11    linear of order 4
ρ101-1-1-111-11-11-11ii-i-iii-i-i1-1-1-1111-1    linear of order 4
ρ1111-111-1-1-1-1-111-iii-i-iii-i-11-111-11-1    linear of order 4
ρ121-1-1-111-11-11-11iiii-i-i-i-i-1111-1-1-11    linear of order 4
ρ131-1-1-111-11-11-11-i-i-i-iiiii-1111-1-1-11    linear of order 4
ρ1411-111-1-1-1-1-111i-i-iii-i-ii-11-111-11-1    linear of order 4
ρ151-1-1-111-11-11-11-i-iii-i-iii1-1-1-1111-1    linear of order 4
ρ1611-111-1-1-1-1-111i-ii-i-ii-ii1-11-1-11-11    linear of order 4
ρ1722222222-2-2-2-20000000000000000    orthogonal lifted from D4
ρ182-22-22-22-2-222-20000000000000000    orthogonal lifted from D4
ρ192-2-2-222-222-22-20000000000000000    symplectic lifted from Q8, Schur index 2
ρ2022-222-2-2-222-2-20000000000000000    symplectic lifted from Q8, Schur index 2
ρ212-222-2-2-22000000000000--2-2--2--2-2-2--2-2    complex lifted from SD16
ρ22222-2-22-2-2000000000000-2--2--2-2-2--2--2-2    complex lifted from SD16
ρ232-222-2-2-22000000000000-2--2-2-2--2--2-2--2    complex lifted from SD16
ρ24222-2-22-2-2000000000000--2-2-2--2--2-2-2--2    complex lifted from SD16
ρ252-2-22-222-2000000000000--2--2--2-2--2-2-2-2    complex lifted from SD16
ρ2622-2-2-2-222000000000000-2-2--2--2--2--2-2-2    complex lifted from SD16
ρ272-2-22-222-2000000000000-2-2-2--2-2--2--2--2    complex lifted from SD16
ρ2822-2-2-2-222000000000000--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)])

C2×C4.Q8 is a maximal subgroup of
C8.11C42  C8.C42  C8⋊C42  C24.133D4  C24.67D4  C4○D4.4Q8  C42.58Q8  C42.60Q8  C42.26Q8  C24.159D4  C24.71D4  D4⋊(C4⋊C4)  Q8⋊C4⋊C4  C4.Q89C4  C4.Q810C4  C4.67(C4×D4)  C4.68(C4×D4)  C87(C4⋊C4)  C4.(C4×Q8)  C8⋊(C4⋊C4)  C42.30Q8  C42.31Q8  M4(2).5Q8  (C2×C4)⋊9SD16  (C2×Q16)⋊10C4  (C2×D8)⋊10C4  C24.84D4  C24.85D4  (C2×C8)⋊Q8  C2.(C8⋊Q8)  C4⋊C4.106D4  (C2×Q8).8Q8  C24.89D4  (C2×C8).165D4  C2.(C83Q8)  (C2×C8).24Q8  C4.(C4⋊Q8)  M4(2).2Q8  (C2×C8).168D4  (C2×C8).169D4  (C2×C8).170D4  (C2×C8).171D4  C4○D4.7Q8  C2×C4×SD16  C42.281C23  (C2×C8)⋊13D4  C42.23C23  (C2×D4).304D4  M4(2)⋊3Q8  D49SD16  C42.486C23  C42.58C23  C42.59C23
C2×C4.Q8 is a maximal quotient of
C88M4(2)  C42.90D4  C24.133D4  C42.55Q8  C42.58Q8  C24.159D4  C42.30Q8  M5(2)⋊3C4

Matrix representation of C2×C4.Q8 in GL4(𝔽17) generated by

16000
01600
00160
00016
,
16000
0100
0001
00160
,
1000
01600
00512
0055
,
4000
01600
00613
001311
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

Export

Character table of C2×C4.Q8 in TeX

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