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G = C2×C4.4D4order 64 = 26

Direct product of C2 and C4.4D4

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

Aliases: C2×C4.4D4, C4217C22, C23.7C23, C22.20C24, C24.13C22, (C2×C42)⋊9C2, (C2×C4).85D4, C4.13(C2×D4), (C22×Q8)⋊4C2, C2.9(C22×D4), (C2×Q8)⋊10C22, C22.61(C2×D4), C22⋊C416C22, (C2×C4).129C23, (C22×D4).11C2, (C2×D4).61C22, C22.32(C4○D4), (C22×C4).99C22, C2.9(C2×C4○D4), (C2×C22⋊C4)⋊11C2, SmallGroup(64,207)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C4.4D4
C1C2C22C23C22×C4C2×C42 — C2×C4.4D4
C1C22 — C2×C4.4D4
C1C23 — C2×C4.4D4
C1C22 — C2×C4.4D4

Generators and relations for C2×C4.4D4
 G = < a,b,c,d | a2=b4=c4=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 265 in 165 conjugacy classes, 89 normal (9 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×4], C4 [×8], C22, C22 [×6], C22 [×20], C2×C4 [×14], C2×C4 [×8], D4 [×8], Q8 [×8], C23, C23 [×4], C23 [×12], C42 [×4], C22⋊C4 [×16], C22×C4, C22×C4 [×4], C2×D4 [×4], C2×D4 [×4], C2×Q8 [×4], C2×Q8 [×4], C24 [×2], C2×C42, C2×C22⋊C4 [×4], C4.4D4 [×8], C22×D4, C22×Q8, C2×C4.4D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], C2×C4.4D4

Character table of C2×C4.4D4

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 1111111144442222222222224444
ρ11111111111111111111111111111    trivial
ρ21-1-1111-1-1-1-1111-11-111-11-1-11-111-1-1    linear of order 2
ρ31-1-1111-1-111-1-111-1-11-11-111-1-111-1-1    linear of order 2
ρ411111111-1-1-1-11-1-111-1-1-1-1-1-111111    linear of order 2
ρ51-1-1111-1-111-1-11-11-111-11-1-11-1-1-111    linear of order 2
ρ61-1-1111-1-1-1-11111-1-11-11-111-1-1-1-111    linear of order 2
ρ711111111-1-1-1-1111111111111-1-1-1-1    linear of order 2
ρ81111111111111-1-111-1-1-1-1-1-11-1-1-1-1    linear of order 2
ρ91-1-1111-1-1-11-11-1-111-1-11-11-1111-11-1    linear of order 2
ρ101-1-1111-1-11-11-1-11-11-11-11-11-111-11-1    linear of order 2
ρ11111111111-1-11-111-1-1-1-1-1-111-11-1-11    linear of order 2
ρ1211111111-111-1-1-1-1-1-11111-1-1-11-1-11    linear of order 2
ρ131-1-1111-1-11-11-1-1-111-1-11-11-111-11-11    linear of order 2
ρ141-1-1111-1-1-11-11-11-11-11-11-11-11-11-11    linear of order 2
ρ1511111111-111-1-111-1-1-1-1-1-111-1-111-1    linear of order 2
ρ16111111111-1-11-1-1-1-1-11111-1-1-1-111-1    linear of order 2
ρ17222-2-22-2-20000200-2-200000020000    orthogonal lifted from D4
ρ18222-2-22-2-20000-20022000000-20000    orthogonal lifted from D4
ρ192-2-2-2-222200002002-2000000-20000    orthogonal lifted from D4
ρ202-2-2-2-22220000-200-2200000020000    orthogonal lifted from D4
ρ212-222-2-22-20000000002i2i-2i-2i0000000    complex lifted from C4○D4
ρ2222-22-2-2-22000000000-2i2i2i-2i0000000    complex lifted from C4○D4
ρ2322-22-2-2-220000000002i-2i-2i2i0000000    complex lifted from C4○D4
ρ2422-2-22-22-2000002i-2i000000-2i2i00000    complex lifted from C4○D4
ρ2522-2-22-22-200000-2i2i0000002i-2i00000    complex lifted from C4○D4
ρ262-222-2-22-2000000000-2i-2i2i2i0000000    complex lifted from C4○D4
ρ272-22-22-2-22000002i2i000000-2i-2i00000    complex lifted from C4○D4
ρ282-22-22-2-2200000-2i-2i0000002i2i00000    complex lifted from C4○D4

Smallest permutation representation of C2×C4.4D4
On 32 points
Generators in S32
(1 25)(2 26)(3 27)(4 28)(5 23)(6 24)(7 21)(8 22)(9 29)(10 30)(11 31)(12 32)(13 17)(14 18)(15 19)(16 20)
(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)
(1 6 9 13)(2 7 10 14)(3 8 11 15)(4 5 12 16)(17 25 24 29)(18 26 21 30)(19 27 22 31)(20 28 23 32)
(1 15 3 13)(2 14 4 16)(5 10 7 12)(6 9 8 11)(17 25 19 27)(18 28 20 26)(21 32 23 30)(22 31 24 29)

G:=sub<Sym(32)| (1,25)(2,26)(3,27)(4,28)(5,23)(6,24)(7,21)(8,22)(9,29)(10,30)(11,31)(12,32)(13,17)(14,18)(15,19)(16,20), (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), (1,6,9,13)(2,7,10,14)(3,8,11,15)(4,5,12,16)(17,25,24,29)(18,26,21,30)(19,27,22,31)(20,28,23,32), (1,15,3,13)(2,14,4,16)(5,10,7,12)(6,9,8,11)(17,25,19,27)(18,28,20,26)(21,32,23,30)(22,31,24,29)>;

G:=Group( (1,25)(2,26)(3,27)(4,28)(5,23)(6,24)(7,21)(8,22)(9,29)(10,30)(11,31)(12,32)(13,17)(14,18)(15,19)(16,20), (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), (1,6,9,13)(2,7,10,14)(3,8,11,15)(4,5,12,16)(17,25,24,29)(18,26,21,30)(19,27,22,31)(20,28,23,32), (1,15,3,13)(2,14,4,16)(5,10,7,12)(6,9,8,11)(17,25,19,27)(18,28,20,26)(21,32,23,30)(22,31,24,29) );

G=PermutationGroup([(1,25),(2,26),(3,27),(4,28),(5,23),(6,24),(7,21),(8,22),(9,29),(10,30),(11,31),(12,32),(13,17),(14,18),(15,19),(16,20)], [(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)], [(1,6,9,13),(2,7,10,14),(3,8,11,15),(4,5,12,16),(17,25,24,29),(18,26,21,30),(19,27,22,31),(20,28,23,32)], [(1,15,3,13),(2,14,4,16),(5,10,7,12),(6,9,8,11),(17,25,19,27),(18,28,20,26),(21,32,23,30),(22,31,24,29)])

C2×C4.4D4 is a maximal subgroup of
C42.395D4  C42.407D4  C42.70D4  C24.23D4  C4.4D413C4  C42.433D4  C42.110D4  C42.115D4  C42.119D4  C42.129D4  C4210D4  (C22×D8).C2  (C2×C8).41D4  C4⋊C4.94D4  C42.160D4  C24.205C23  C24.220C23  C24.221C23  C23.261C24  C24.259C23  C23.327C24  C24.263C23  C24.264C23  C23.335C24  C24.565C23  C24.271C23  C23.348C24  C23.359C24  C24.282C23  C23.372C24  C23.374C24  C23.391C24  C24.311C23  C4219D4  C4221D4  C42.168D4  C42.170D4  C42.171D4  C23.455C24  C23.457C24  C42.182D4  C4226D4  C4228D4  C4229D4  C42.189D4  C42.193D4  C4231D4  C42.196D4  C4232D4  C23.570C24  C23.572C24  C23.574C24  C23.576C24  C23.584C24  C24.393C23  C23.600C24  C24.412C23  C23.612C24  C23.615C24  C23.617C24  C23.630C24  C23.631C24  C23.633C24  C4234D4  C42.199D4  C4246D4  C4312C2  C4314C2  C42.446D4  C42.242D4  M4(2)⋊9D4  C42.269D4  C42.271D4  C42.273D4  C22.89C25  C22.99C25  C22.103C25  C22.134C25  C22.147C25  C22.150C25
C2×C4.4D4 is a maximal quotient of
C42.163D4  C23.335C24  C24.565C23  C23.372C24  C23.388C24  C24.301C23  C23.390C24  C23.391C24  C23.392C24  C24.579C23  C23.404C24  C42.170D4  C42.171D4  C23.461C24  C42.172D4  C42.173D4  C42.177D4  C23.491C24  C42.182D4  C24.592C23  C42.193D4  C42.194D4  C42.195D4  C4246D4  C4312C2  C4314C2  C4218Q8  C42.355D4  C42.239D4  C42.366C23  C42.367C23  C42.240D4  C42.241D4  C42.242D4  C42.243D4  C42.244D4

Matrix representation of C2×C4.4D4 in GL5(𝔽5)

40000
04000
00400
00040
00004
,
40000
00300
03000
00010
00001
,
40000
00100
01000
00022
00003
,
10000
00100
04000
00022
00013

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],[4,0,0,0,0,0,0,3,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,2,3],[1,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,2,1,0,0,0,2,3] >;

C2×C4.4D4 in GAP, Magma, Sage, TeX

C_2\times C_4._4D_4
% in TeX

G:=Group("C2xC4.4D4");
// GroupNames label

G:=SmallGroup(64,207);
// by ID

G=gap.SmallGroup(64,207);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,199,650,86]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^4=1,d^2=b^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=b^2*c^-1>;
// generators/relations

Export

Character table of C2×C4.4D4 in TeX

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