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G = C2×C42.C2order 64 = 26

Direct product of C2 and C42.C2

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

Aliases: C2×C42.C2, C22.21C24, C42.88C22, C23.71C23, C4.9(C2×Q8), (C2×C4).22Q8, C2.4(C22×Q8), C4⋊C4.69C22, (C2×C42).18C2, (C2×C4).14C23, C22.18(C2×Q8), C22.33(C4○D4), (C22×C4).59C22, (C2×C4⋊C4).18C2, C2.10(C2×C4○D4), SmallGroup(64,208)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C42.C2
C1C2C22C23C22×C4C2×C42 — C2×C42.C2
C1C22 — C2×C42.C2
C1C23 — C2×C42.C2
C1C22 — C2×C42.C2

Generators and relations for C2×C42.C2
 G = < a,b,c,d | a2=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c >

Subgroups: 137 in 113 conjugacy classes, 89 normal (7 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×12], C22, C22 [×6], C2×C4 [×18], C2×C4 [×12], C23, C42 [×4], C4⋊C4 [×24], C22×C4, C22×C4 [×6], C2×C42, C2×C4⋊C4 [×6], C42.C2 [×8], C2×C42.C2
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×4], C24, C42.C2 [×4], C22×Q8, C2×C4○D4 [×2], C2×C42.C2

Character table of C2×C42.C2

 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
ρ172-2-2-2222-2-2000-2200002000000000    symplectic lifted from Q8, Schur index 2
ρ1822-2-22-2-222000-2-200002000000000    symplectic lifted from Q8, Schur index 2
ρ1922-2-22-2-22-2000220000-2000000000    symplectic lifted from Q8, Schur index 2
ρ202-2-2-2222-220002-20000-2000000000    symplectic lifted from Q8, Schur index 2
ρ212-22-2-22-2202i0000-2i-2i0002i00000000    complex lifted from C4○D4
ρ22222-2-2-22-202i00002i-2i000-2i00000000    complex lifted from C4○D4
ρ232-2-22-2-222002i2i0000-2i-2i0000000000    complex lifted from C4○D4
ρ2422-22-22-2-200-2i2i00002i-2i0000000000    complex lifted from C4○D4
ρ2522-22-22-2-2002i-2i0000-2i2i0000000000    complex lifted from C4○D4
ρ26222-2-2-22-20-2i0000-2i2i0002i00000000    complex lifted from C4○D4
ρ272-22-2-22-220-2i00002i2i000-2i00000000    complex lifted from C4○D4
ρ282-2-22-2-22200-2i-2i00002i2i0000000000    complex lifted from C4○D4

Smallest permutation representation of C2×C42.C2
Regular action on 64 points
Generators in S64
(1 25)(2 26)(3 27)(4 28)(5 39)(6 40)(7 37)(8 38)(9 15)(10 16)(11 13)(12 14)(17 29)(18 30)(19 31)(20 32)(21 46)(22 47)(23 48)(24 45)(33 44)(34 41)(35 42)(36 43)(49 60)(50 57)(51 58)(52 59)(53 64)(54 61)(55 62)(56 63)
(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 31 11 5)(2 32 12 6)(3 29 9 7)(4 30 10 8)(13 39 25 19)(14 40 26 20)(15 37 27 17)(16 38 28 18)(21 57 54 34)(22 58 55 35)(23 59 56 36)(24 60 53 33)(41 46 50 61)(42 47 51 62)(43 48 52 63)(44 45 49 64)
(1 36 11 59)(2 60 12 33)(3 34 9 57)(4 58 10 35)(5 54 31 21)(6 22 32 55)(7 56 29 23)(8 24 30 53)(13 52 25 43)(14 44 26 49)(15 50 27 41)(16 42 28 51)(17 48 37 63)(18 64 38 45)(19 46 39 61)(20 62 40 47)

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

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

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

C2×C42.C2 is a maximal subgroup of
C42.396D4  C42.408D4  C42.71D4  C42.123D4  C42.437D4  C42.124D4  C42.128D4  C4⋊C4.84D4  C4⋊C4.85D4  C2.(C8⋊Q8)  C42.33Q8  C23.218C24  C23.252C24  C23.253C24  C23.264C24  C24.268C23  C24.569C23  C23.353C24  C23.354C24  C23.360C24  C23.362C24  C24.572C23  C23.375C24  C24.301C23  C23.390C24  C23.406C24  C23.419C24  C426Q8  C42.35Q8  C23.456C24  C23.458C24  C42.172D4  C42.174D4  C42.175D4  C42.181D4  C42.185D4  C42.188D4  C42.190D4  C42.191D4  C42.194D4  C42.195D4  C4210Q8  C42.198D4  C23.590C24  C24.401C23  C24.408C23  C23.607C24  C23.611C24  C23.613C24  C23.619C24  C23.620C24  C23.621C24  C23.625C24  C23.626C24  C42.199D4  C42.201D4  C42.439D4  C43.15C2  C4314C2  C42.449D4  C42.244D4  M4(2)⋊6Q8  C42.284D4  C42.286D4  C42.288D4  C22.93C25  C22.101C25  C22.104C25  C22.142C25  C22.148C25  C22.152C25
C2×C42.C2 is a maximal quotient of
C42.34Q8  C24.569C23  C24.572C23  C23.405C24  C23.406C24  C23.407C24  C23.408C24  C23.409C24  C42.35Q8  C24.584C23  C42.36Q8  C42.37Q8  C23.546C24  C42.39Q8  C42.439D4  C43.15C2  C43.18C2

Matrix representation of C2×C42.C2 in GL5(𝔽5)

40000
04000
00400
00010
00001
,
10000
04000
00100
00030
00003
,
10000
03000
00300
00040
00001
,
10000
00100
04000
00004
00040

G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3],[1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,4,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,4,0] >;

C2×C42.C2 in GAP, Magma, Sage, TeX

C_2\times C_4^2.C_2
% in TeX

G:=Group("C2xC4^2.C2");
// GroupNames label

G:=SmallGroup(64,208);
// by ID

G=gap.SmallGroup(64,208);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,96,217,199,650,86]);
// 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*c^2,d*c*d^-1=b^2*c>;
// generators/relations

Export

Character table of C2×C42.C2 in TeX

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