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G = C22⋊D8order 64 = 26

The semidirect product of C22 and D8 acting via D8/D4=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: D42D4, C222D8, C23.41D4, (C2×D8)⋊1C2, C2.4(C2×D8), C4⋊D41C2, C22⋊C83C2, C4⋊C41C22, (C2×C8)⋊1C22, C4.19(C2×D4), (C2×C4).22D4, D4⋊C44C2, C2.8C22≀C2, (C2×D4)⋊1C22, (C22×D4)⋊2C2, C2.6(C8⋊C22), (C2×C4).81C23, C22.77(C2×D4), (C22×C4).42C22, SmallGroup(64,128)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C22⋊D8
C1C2C22C2×C4C22×C4C22×D4 — C22⋊D8
C1C2C2×C4 — C22⋊D8
C1C22C22×C4 — C22⋊D8
C1C2C2C2×C4 — C22⋊D8

Generators and relations for C22⋊D8
 G = < a,b,c,d | a2=b2=c8=d2=1, cac-1=dad=ab=ba, bc=cb, bd=db, dcd=c-1 >

Subgroups: 225 in 99 conjugacy classes, 31 normal (15 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×21], C8 [×2], C2×C4 [×2], C2×C4 [×3], D4 [×4], D4 [×10], C23, C23 [×11], C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×4], C22×C4, C2×D4, C2×D4 [×2], C2×D4 [×6], C24, C22⋊C8, D4⋊C4 [×2], C4⋊D4, C2×D8 [×2], C22×D4, C22⋊D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], C22≀C2, C2×D8, C8⋊C22, C22⋊D8

Character table of C22⋊D8

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D8A8B8C8D
 size 1111224444822484444
ρ11111111111111111111    trivial
ρ21111111111-1111-1-1-1-1-1    linear of order 2
ρ3111111-1-1-1-111111-1-1-1-1    linear of order 2
ρ4111111-1-1-1-1-1111-11111    linear of order 2
ρ51111-1-1-1-111111-1-11-1-11    linear of order 2
ρ61111-1-1-1-111-111-11-111-1    linear of order 2
ρ71111-1-111-1-1111-1-1-111-1    linear of order 2
ρ81111-1-111-1-1-111-111-1-11    linear of order 2
ρ92-22-200-220002-2000000    orthogonal lifted from D4
ρ102-22-2002-20002-2000000    orthogonal lifted from D4
ρ112222-2-200000-2-2200000    orthogonal lifted from D4
ρ122-22-200002-20-22000000    orthogonal lifted from D4
ρ132-22-20000-220-22000000    orthogonal lifted from D4
ρ1422222200000-2-2-200000    orthogonal lifted from D4
ρ152-2-22-220000000002-22-2    orthogonal lifted from D8
ρ162-2-222-200000000022-2-2    orthogonal lifted from D8
ρ172-2-22-22000000000-22-22    orthogonal lifted from D8
ρ182-2-222-2000000000-2-222    orthogonal lifted from D8
ρ1944-4-4000000000000000    orthogonal lifted from C8⋊C22

Permutation representations of C22⋊D8
On 16 points - transitive group 16T126
Generators in S16
(1 5)(2 13)(3 7)(4 15)(6 9)(8 11)(10 14)(12 16)
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 9)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)

G:=sub<Sym(16)| (1,5)(2,13)(3,7)(4,15)(6,9)(8,11)(10,14)(12,16), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,9)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)>;

G:=Group( (1,5)(2,13)(3,7)(4,15)(6,9)(8,11)(10,14)(12,16), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,9)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10) );

G=PermutationGroup([(1,5),(2,13),(3,7),(4,15),(6,9),(8,11),(10,14),(12,16)], [(1,16),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,9),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10)])

G:=TransitiveGroup(16,126);

C22⋊D8 is a maximal subgroup of
C23⋊D8  C4⋊C4.D4  C24.9D4  C24.103D4  C24.177D4  C24.105D4  C4○D4⋊D4  (C2×D4)⋊21D4  C42.225D4  C42.227D4  C42.232D4  C42.352C23  C42.356C23  C233D8  C24.121D4  C24.125D4  C24.127D4  C4.2+ 1+4  C4.142+ 1+4  C42.269D4  C42.271D4  C42.275D4  C42.406C23  C42.410C23  SD16⋊D4  SD167D4  SD161D4  D4×D8  SD1610D4  D44D8  C42.462C23  C42.41C23  C42.53C23  C42.54C23  C42.471C23  C42.474C23  D4⋊S4
 D4p⋊D4: D89D4  D85D4  D1213D4  D4⋊D12  D1216D4  D12⋊D4  D2013D4  D4⋊D20 ...
 (C2×C2p)⋊D8: (C2×C4)⋊D8  C42.221D4  C42.263D4  (C2×C6)⋊8D8  (C2×C10)⋊8D8  (C2×C14)⋊8D8 ...
C22⋊D8 is a maximal quotient of
C23⋊D8  C23.5D8  (C2×C4).5D8  D4⋊D8  Q8⋊D8  D43D8  Q83D8  D4.D8  Q8.D8  D4.7D8  D44Q16  C23.35D8  C23.37D8  C2.(C4×D8)  C23.38D8  C232D8  (C2×D4)⋊Q8  C24.83D4  C4⋊C47D4  C4⋊C4⋊Q8  Q167D4  D8.9D4  Q16.8D4  D8.10D4  D8.D4  Q16.10D4  Q16.D4  D8.3D4  D8.12D4
 D4p⋊D4: D87D4  D88D4  D8⋊D4  D1213D4  D4⋊D12  D1216D4  D12⋊D4  D2013D4 ...
 (C2×C2p)⋊D8: (C2×C4)⋊D8  (C2×C4)⋊9D8  (C2×C4)⋊2D8  (C2×C6)⋊8D8  (C2×C10)⋊8D8  (C2×C14)⋊8D8 ...

Matrix representation of C22⋊D8 in GL4(𝔽17) generated by

1000
0100
0010
00016
,
1000
0100
00160
00016
,
14300
141400
00016
00160
,
141400
14300
0001
0010
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[14,14,0,0,3,14,0,0,0,0,0,16,0,0,16,0],[14,14,0,0,14,3,0,0,0,0,0,1,0,0,1,0] >;

C22⋊D8 in GAP, Magma, Sage, TeX

C_2^2\rtimes D_8
% in TeX

G:=Group("C2^2:D8");
// GroupNames label

G:=SmallGroup(64,128);
// by ID

G=gap.SmallGroup(64,128);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,362,963,489,117]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^8=d^2=1,c*a*c^-1=d*a*d=a*b=b*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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Character table of C22⋊D8 in TeX

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