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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

Aliases: C2×C2.D8, C22.13D8, C23.57D4, C22.6Q16, C87(C2×C4), (C2×C8)⋊5C4, C2.2(C2×D8), C4.2(C2×Q8), (C2×C4).73D4, C4.14(C4⋊C4), (C2×C4).19Q8, C2.2(C2×Q16), (C22×C8).8C2, C4⋊C4.47C22, C4.25(C22×C4), (C2×C8).73C22, (C2×C4).68C23, C22.48(C2×D4), C22.20(C4⋊C4), (C22×C4).114C22, C2.12(C2×C4⋊C4), (C2×C4⋊C4).14C2, (C2×C4).73(C2×C4), SmallGroup(64,107)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C2×C2.D8
C1C2C22C2×C4C22×C4C22×C8 — C2×C2.D8
C1C2C4 — C2×C2.D8
C1C23C22×C4 — C2×C2.D8
C1C2C2C2×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 [×3], C2 [×4], 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], C2.D8 [×4], C2×C4⋊C4 [×2], C22×C8, C2×C2.D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], D8 [×2], Q16 [×2], C22×C4, C2×D4, C2×Q8, C2.D8 [×4], C2×C4⋊C4, C2×D8, C2×Q16, C2×C2.D8

Character table of C2×C2.D8

 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
ρ172-222-2-2-220000000000002-222-2-22-2    orthogonal lifted from D8
ρ1822222222-2-2-2-20000000000000000    orthogonal lifted from D4
ρ192-22-22-22-2-222-20000000000000000    orthogonal lifted from D4
ρ202-222-2-2-22000000000000-22-2-222-22    orthogonal lifted from D8
ρ21222-2-22-2-20000000000002-2-222-2-22    orthogonal lifted from D8
ρ22222-2-22-2-2000000000000-222-2-222-2    orthogonal lifted from D8
ρ232-2-2-222-222-22-20000000000000000    symplectic lifted from Q8, Schur index 2
ρ2422-222-2-2-222-2-20000000000000000    symplectic lifted from Q8, Schur index 2
ρ252-2-22-222-2000000000000222-22-2-2-2    symplectic lifted from Q16, Schur index 2
ρ2622-2-2-2-22200000000000022-2-2-2-222    symplectic lifted from Q16, Schur index 2
ρ2722-2-2-2-222000000000000-2-22222-2-2    symplectic lifted from Q16, Schur index 2
ρ282-2-22-222-2000000000000-2-2-22-2222    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)])

C2×C2.D8 is a maximal subgroup of
C8.7C42  C8.2C42  C8⋊C42  C23.22D8  C24.67D4  C4○D4.5Q8  C42.59Q8  C42.60Q8  C42.26Q8  C23.37D8  C24.71D4  C2.(C4×D8)  Q8⋊(C4⋊C4)  C2.D84C4  C2.D85C4  D4⋊C4⋊C4  C2.(C4×Q16)  C85(C4⋊C4)  C4.(C4×Q8)  C8⋊(C4⋊C4)  C42.29Q8  C42.31Q8  M4(2).6Q8  (C2×C4)⋊6Q16  (C2×C4)⋊6D8  C8⋊(C22⋊C4)  C24.83D4  C24.86D4  C4⋊C4⋊Q8  C2.(C8⋊Q8)  (C2×C4).23D8  (C2×C8).52D4  C23.12D8  C24.88D4  (C2×C8).55D4  (C2×C8).1Q8  (C2×C8).24Q8  (C2×C4).26D8  (C2×C4).21Q16  M4(2).Q8  (C2×C8).168D4  (C2×C4).27D8  (C2×C8).60D4  (C2×C8).171D4  C23.39D8  M5(2)⋊1C4  C22.D16  C23.49D8  C23.50D8  C23.51D8  C4○D4.8Q8  C2×C4×D8  C2×C4×Q16  C42.280C23  (C2×C8)⋊14D4  C42.22C23  (C2×D4).303D4  M4(2)⋊4Q8  D45D8  C42.485C23  D46Q16  C42.488C23  C42.57C23  C42.60C23
C2×C2.D8 is a maximal quotient of
C87M4(2)  C42.91D4  C23.22D8  C42.55Q8  C42.59Q8  C23.37D8  C42.29Q8  C23.25D8  M5(2)⋊1C4  M5(2).1C4

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

1000
01600
0010
0001
,
16000
0100
00160
00016
,
1000
0100
00011
00311
,
4000
01600
0007
00120
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

Export

Character table of C2×C2.D8 in TeX

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