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G = C2×C422C2order 64 = 26

Direct product of C2 and C422C2

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

Aliases: C2×C422C2, C4214C22, C22.22C24, C23.72C23, C24.14C22, (C2×C42)⋊4C2, C4⋊C413C22, (C2×C4).53C23, C22.34(C4○D4), C22⋊C4.12C22, (C22×C4).60C22, (C2×C4⋊C4)⋊17C2, C2.11(C2×C4○D4), (C2×C22⋊C4).12C2, SmallGroup(64,209)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C422C2
C1C2C22C23C22×C4C2×C42 — C2×C422C2
C1C22 — C2×C422C2
C1C23 — C2×C422C2
C1C22 — C2×C422C2

Generators and relations for C2×C422C2
 G = < a,b,c,d | a2=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, dcd=b2c-1 >

Subgroups: 177 in 123 conjugacy classes, 81 normal (6 characteristic)
C1, C2 [×7], C2 [×2], C4 [×12], C22, C22 [×6], C22 [×10], C2×C4 [×12], C2×C4 [×12], C23, C23 [×2], C23 [×6], C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×6], C24, C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C422C2 [×8], C2×C422C2
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×6], C24, C422C2 [×4], C2×C4○D4 [×3], C2×C422C2

Character table of C2×C422C2

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R
 size 1111111144222222222222444444
ρ11111111111111111111111111111    trivial
ρ21-1111-1-1-1-11-111-1-11-11-111-11-11-1-11    linear of order 2
ρ311111111111-111-1-1-1-1-1-11-1-1-1-1-111    linear of order 2
ρ41-1111-1-1-1-11-1-11-11-11-11-111-11-11-11    linear of order 2
ρ511111111-1-1-1-1-1-11111-1-1-1-1-1-11111    linear of order 2
ρ61-1111-1-1-11-11-1-11-11-111-1-11-111-1-11    linear of order 2
ρ711111111-1-1-11-1-1-1-1-1-111-1111-1-111    linear of order 2
ρ81-1111-1-1-11-111-111-11-1-11-1-11-1-11-11    linear of order 2
ρ91-1111-1-1-1-1111-111-11-1-11-1-1-111-11-1    linear of order 2
ρ101111111111-11-1-1-1-1-1-111-11-1-111-1-1    linear of order 2
ρ111-1111-1-1-1-111-1-11-11-111-1-111-1-111-1    linear of order 2
ρ121111111111-1-1-1-11111-1-1-1-111-1-1-1-1    linear of order 2
ρ131-1111-1-1-11-1-1-11-11-11-11-1111-11-11-1    linear of order 2
ρ1411111111-1-11-111-1-1-1-1-1-11-11111-1-1    linear of order 2
ρ151-1111-1-1-11-1-111-1-11-11-111-1-11-111-1    linear of order 2
ρ1611111111-1-1111111111111-1-1-1-1-1-1    linear of order 2
ρ172-2-2-2222-2002i02i-2i000000-2i0000000    complex lifted from C4○D4
ρ1822-2-22-2-22002i0-2i-2i0000002i0000000    complex lifted from C4○D4
ρ1922-22-22-2-2000-2i000000-2i2i02i000000    complex lifted from C4○D4
ρ202-22-2-22-22000000-2i-2i2i2i0000000000    complex lifted from C4○D4
ρ212-22-2-22-220000002i2i-2i-2i0000000000    complex lifted from C4○D4
ρ222-2-22-2-2220002i000000-2i-2i02i000000    complex lifted from C4○D4
ρ2322-2-22-2-2200-2i02i2i000000-2i0000000    complex lifted from C4○D4
ρ242-2-2-2222-200-2i0-2i2i0000002i0000000    complex lifted from C4○D4
ρ2522-22-22-2-20002i0000002i-2i0-2i000000    complex lifted from C4○D4
ρ26222-2-2-22-2000000-2i2i2i-2i0000000000    complex lifted from C4○D4
ρ272-2-22-2-222000-2i0000002i2i0-2i000000    complex lifted from C4○D4
ρ28222-2-2-22-20000002i-2i-2i2i0000000000    complex lifted from C4○D4

Smallest permutation representation of C2×C422C2
On 32 points
Generators in S32
(1 25)(2 26)(3 27)(4 28)(5 16)(6 13)(7 14)(8 15)(9 19)(10 20)(11 17)(12 18)(21 31)(22 32)(23 29)(24 30)
(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 17 13 31)(2 18 14 32)(3 19 15 29)(4 20 16 30)(5 24 28 10)(6 21 25 11)(7 22 26 12)(8 23 27 9)
(1 27)(2 5)(3 25)(4 7)(6 15)(8 13)(9 29)(10 20)(11 31)(12 18)(14 28)(16 26)(17 21)(19 23)(22 32)(24 30)

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

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

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

C2×C422C2 is a maximal subgroup of
C42.372D4  C24.203C23  C23.255C24  C24.230C23  C24.286C23  C23.367C24  C23.368C24  C23.369C24  C24.295C23  C23.379C24  C23.380C24  C24.573C23  C23.396C24  C23.412C24  C23.419C24  C24.311C23  C24.326C23  C24.327C23  C24.331C23  C24.332C23  C4223D4  C4224D4  C42.184D4  C4230D4  C42.192D4  C4232D4  C42.198D4  C23.585C24  C23.589C24  C23.591C24  C23.595C24  C24.405C23  C23.602C24  C23.605C24  C24.413C23  C23.615C24  C23.616C24  C23.618C24  C23.621C24  C23.622C24  C24.418C23  C23.625C24  C23.627C24  C4233D4  C42.200D4  C4235D4  C4243D4  C4313C2  C22.110C25  C22.149C25  C22.153C25
C2×C422C2 is a maximal quotient of
C23.301C24  C23.380C24  C24.573C23  C24.577C23  C24.304C23  C23.395C24  C23.396C24  C23.397C24  C23.410C24  C23.411C24  C23.412C24  C23.413C24  C23.414C24  C23.543C24  C23.544C24  C23.545C24  C4243D4  C4313C2  C4215Q8

Matrix representation of C2×C422C2 in GL5(𝔽5)

40000
01000
00100
00010
00001
,
40000
00400
01000
00044
00001
,
10000
02000
00200
00033
00002
,
10000
04000
00100
00010
00034

G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,0,1,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,4,1],[1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,3,2],[1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,3,0,0,0,0,4] >;

C2×C422C2 in GAP, Magma, Sage, TeX

C_2\times C_4^2\rtimes_2C_2
% in TeX

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

G:=SmallGroup(64,209);
// by ID

G=gap.SmallGroup(64,209);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,295,650,86]);
// Polycyclic

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

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Character table of C2×C422C2 in TeX

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