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G = (C2xC8):D4order 128 = 27

3rd semidirect product of C2xC8 and D4 acting faithfully

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

Aliases: (C2xC8):3D4, (C2xD4).77D4, (C2xQ8).72D4, C22wrC2.2C4, Q8oM4(2):7C2, C24.58(C2xC4), C22.40(C4xD4), (C22xC4).66D4, C4.127C22wrC2, C4.132(C4:D4), C24.4C4:26C2, C23.69(C22xC4), C22.D4.2C4, M4(2):4C4:15C2, C23.12(C22:C4), C22.19C24.6C2, (C22xC4).687C23, (C23xC4).258C22, C42:C2.24C22, C4.109(C22.D4), C2.30(C23.23D4), (C2xM4(2)).184C22, M4(2).8C22:10C2, (C2xD4).80(C2xC4), C22:C4.5(C2xC4), (C2xC4).1330(C2xD4), (C2xC4).321(C4oD4), (C2xC4).16(C22:C4), (C22xC4).120(C2xC4), (C2xC4oD4).20C22, C22.46(C2xC22:C4), SmallGroup(128,623)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — (C2xC8):D4
C1C2C4C2xC4C22xC4C2xC4oD4Q8oM4(2) — (C2xC8):D4
C1C2C23 — (C2xC8):D4
C1C4C22xC4 — (C2xC8):D4
C1C2C2C22xC4 — (C2xC8):D4

Generators and relations for (C2xC8):D4
 G = < a,b,c,d | a2=b8=c4=d2=1, cbc-1=dbd=ab=ba, cac-1=ab4, ad=da, dcd=c-1 >

Subgroups: 340 in 171 conjugacy classes, 54 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C23, C42, C22:C4, C22:C4, C4:C4, C2xC8, C2xC8, M4(2), C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, C22:C8, C4.D4, C4.10D4, C42:C2, C4xD4, C22wrC2, C4:D4, C22:Q8, C22.D4, C2xM4(2), C2xM4(2), C8oD4, C23xC4, C2xC4oD4, M4(2):4C4, C24.4C4, M4(2).8C22, C22.19C24, Q8oM4(2), (C2xC8):D4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, C22xC4, C2xD4, C4oD4, C2xC22:C4, C4xD4, C22wrC2, C4:D4, C22.D4, C23.23D4, (C2xC8):D4

Permutation representations of (C2xC8):D4
On 16 points - transitive group 16T209
Generators in S16
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 7 5 3)(2 14)(4 16)(6 10)(8 12)(9 11 13 15)
(1 15)(2 6)(3 9)(4 8)(5 11)(7 13)(10 14)(12 16)

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

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

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

G:=TransitiveGroup(16,209);

On 16 points - transitive group 16T279
Generators in S16
(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 11 5 15)(2 12)(3 13 7 9)(4 14)(6 16)(8 10)
(9 13)(11 15)

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

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

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

G:=TransitiveGroup(16,279);

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K8A···8H8I8J8K8L
order122222222444444444448···88888
size112224444112224444884···48888

32 irreducible representations

dim11111111222224
type++++++++++
imageC1C2C2C2C2C2C4C4D4D4D4D4C4oD4(C2xC8):D4
kernel(C2xC8):D4M4(2):4C4C24.4C4M4(2).8C22C22.19C24Q8oM4(2)C22wrC2C22.D4C2xC8C22xC4C2xD4C2xQ8C2xC4C1
# reps12211144421144

Matrix representation of (C2xC8):D4 in GL4(F5) generated by

0433
0310
0220
2400
,
0342
1004
0031
0042
,
0120
2340
4114
0121
,
0013
0100
0010
2030
G:=sub<GL(4,GF(5))| [0,0,0,2,4,3,2,4,3,1,2,0,3,0,0,0],[0,1,0,0,3,0,0,0,4,0,3,4,2,4,1,2],[0,2,4,0,1,3,1,1,2,4,1,2,0,0,4,1],[0,0,0,2,0,1,0,0,1,0,1,3,3,0,0,0] >;

(C2xC8):D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)\rtimes D_4
% in TeX

G:=Group("(C2xC8):D4");
// GroupNames label

G:=SmallGroup(128,623);
// by ID

G=gap.SmallGroup(128,623);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,521,2804,1411,1027,124]);
// Polycyclic

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

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