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G = (C2×C4)⋊D8order 128 = 27

The semidirect product of C2×C4 and D8 acting via D8/C2=D4

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

Aliases: (C2×C4)⋊D8, C4⋊C4.2D4, C22⋊D82C2, C2.7C2≀C22, (C2×D4).10D4, (C22×C4).42D4, C22.14(C2×D8), C2.7(C22⋊D8), C23.515(C2×D4), C22.SD165C2, C4⋊D4.3C22, C22⋊C8.1C22, (C22×C4).4C23, C2.7(D4.8D4), C23.10D42C2, (C22×D4).5C22, C22.125C22≀C2, C22.39(C8⋊C22), C22.31C241C2, C22.M4(2)⋊3C2, C2.C42.13C22, (C2×C4).193(C2×D4), (C2×C4⋊C4).14C22, SmallGroup(128,330)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×C4)⋊D8
C1C2C22C23C22×C4C2×C4⋊C4C22.31C24 — (C2×C4)⋊D8
C1C22C22×C4 — (C2×C4)⋊D8
C1C22C22×C4 — (C2×C4)⋊D8
C1C2C22C22×C4 — (C2×C4)⋊D8

Generators and relations for (C2×C4)⋊D8
 G = < a,b,c,d | a2=b4=c8=d2=1, cbc-1=dbd=ab=ba, cac-1=ab2, ad=da, dcd=c-1 >

Subgroups: 452 in 155 conjugacy classes, 34 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×8], C22 [×3], C22 [×18], C8 [×2], C2×C4 [×4], C2×C4 [×14], D4 [×15], Q8 [×2], C23, C23 [×10], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], D8 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4 [×2], C2×D4 [×8], C2×Q8, C4○D4 [×4], C24, C2.C42, C22⋊C8 [×2], D4⋊C4 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×2], C4⋊D4 [×3], C22⋊Q8 [×2], C2×D8 [×2], C22×D4, C2×C4○D4, C22.M4(2), C22.SD16 [×2], C23.10D4, C22⋊D8 [×2], C22.31C24, (C2×C4)⋊D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], C22≀C2, C2×D8, C8⋊C22, C22⋊D8, D4.8D4, C2≀C22, (C2×C4)⋊D8

Character table of (C2×C4)⋊D8

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I8A8B8C8D
 size 11112288884444888888888
ρ111111111111111111111111    trivial
ρ2111111-111-11111-1111-1-1-1-1-1    linear of order 2
ρ31111111-111-111-1-1-11-1-1-11-11    linear of order 2
ρ4111111-1-11-1-111-11-11-111-11-1    linear of order 2
ρ5111111-1-1-1-11111-11-1-1-11111    linear of order 2
ρ61111111-1-11111111-1-11-1-1-1-1    linear of order 2
ρ7111111-11-1-1-111-11-1-111-11-11    linear of order 2
ρ811111111-11-111-1-1-1-11-11-11-1    linear of order 2
ρ92222-2-2002002-2000-2000000    orthogonal lifted from D4
ρ1022222200002-2-220-20000000    orthogonal lifted from D4
ρ112222-2-200-2002-20002000000    orthogonal lifted from D4
ρ122222-2-20-2000-220000200000    orthogonal lifted from D4
ρ132222220000-2-2-2-2020000000    orthogonal lifted from D4
ρ142222-2-202000-220000-200000    orthogonal lifted from D4
ρ1522-2-2-220000200-200000-222-2    orthogonal lifted from D8
ρ1622-2-2-220000-200200000-2-222    orthogonal lifted from D8
ρ1722-2-2-220000200-2000002-2-22    orthogonal lifted from D8
ρ1822-2-2-220000-20020000022-2-2    orthogonal lifted from D8
ρ194-44-400200-20000000000000    orthogonal lifted from C2≀C22
ρ204-44-400-20020000000000000    orthogonal lifted from C2≀C22
ρ2144-4-44-400000000000000000    orthogonal lifted from C8⋊C22
ρ224-4-4400000000002i000-2i0000    complex lifted from D4.8D4
ρ234-4-440000000000-2i0002i0000    complex lifted from D4.8D4

Smallest permutation representation of (C2×C4)⋊D8
On 32 points
Generators in S32
(1 32)(3 26)(5 28)(7 30)(10 17)(12 19)(14 21)(16 23)
(1 17 32 10)(2 11 25 18)(3 12 26 19)(4 20 27 13)(5 21 28 14)(6 15 29 22)(7 16 30 23)(8 24 31 9)
(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 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 16)(8 15)(17 28)(18 27)(19 26)(20 25)(21 32)(22 31)(23 30)(24 29)

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

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

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

Matrix representation of (C2×C4)⋊D8 in GL6(𝔽17)

100000
010000
00160614
0001600
000010
000001
,
100000
010000
00153148
004256
0000130
0000154
,
8110000
0150000
00116102
000081
008121012
004256
,
1600000
410000
00151284
004256
0000160
000081

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,6,0,1,0,0,0,14,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,4,0,0,0,0,3,2,0,0,0,0,14,5,13,15,0,0,8,6,0,4],[8,0,0,0,0,0,11,15,0,0,0,0,0,0,1,0,8,4,0,0,16,0,12,2,0,0,10,8,10,5,0,0,2,1,12,6],[16,4,0,0,0,0,0,1,0,0,0,0,0,0,15,4,0,0,0,0,12,2,0,0,0,0,8,5,16,8,0,0,4,6,0,1] >;

(C2×C4)⋊D8 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes D_8
% in TeX

G:=Group("(C2xC4):D8");
// GroupNames label

G:=SmallGroup(128,330);
// by ID

G=gap.SmallGroup(128,330);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,422,352,1123,570,521,136,1411]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,c*b*c^-1=d*b*d=a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,d*c*d=c^-1>;
// generators/relations

Export

Character table of (C2×C4)⋊D8 in TeX

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