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## G = C2×C8.26D4order 128 = 27

### Direct product of C2 and C8.26D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×C8.26D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C2×C8○D4 — C2×C8.26D4
 Lower central C1 — C2 — C4 — C2×C8.26D4
 Upper central C1 — C2×C4 — C22×C8 — C2×C8.26D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×C8.26D4

Generators and relations for C2×C8.26D4
G = < a,b,c,d | a2=b8=c4=1, d2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b5, dcd-1=b2c-1 >

Subgroups: 348 in 232 conjugacy classes, 140 normal (28 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C2×C8, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C8⋊C4, C4≀C2, C8.C4, C2×C42, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C8○D4, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C2×C8⋊C4, C2×C4≀C2, C2×C8.C4, C8.26D4, C2×C8○D4, C2×C4○D8, C2×C8.26D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C23×C4, C22×D4, C2×C4○D4, C8.26D4, C2×C4×D4, C2×C8.26D4

Smallest permutation representation of C2×C8.26D4
On 32 points
Generators in S32
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)
(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)
(2 6)(4 8)(9 11 13 15)(10 16 14 12)(18 22)(20 24)(25 31 29 27)(26 28 30 32)
(1 30 3 32 5 26 7 28)(2 27 4 29 6 31 8 25)(9 19 11 21 13 23 15 17)(10 24 12 18 14 20 16 22)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G ··· 4N 8A ··· 8H 8I ··· 8T order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 1 1 2 2 4 4 4 4 1 1 1 1 2 2 4 ··· 4 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 C4○D4 C4○D4 C8.26D4 kernel C2×C8.26D4 C2×C8⋊C4 C2×C4≀C2 C2×C8.C4 C8.26D4 C2×C8○D4 C2×C4○D8 C2×D8 C2×SD16 C2×Q16 C4○D8 C2×C8 C2×C4 C23 C2 # reps 1 1 2 1 8 2 1 2 4 2 8 4 2 2 4

Matrix representation of C2×C8.26D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 2 0 0 0 0 2 0 0 0 0 0 13 0 4 2 0 0 8 4 11 13
,
 4 0 0 0 0 0 0 13 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 4 0 0 0 10 0 1 13
,
 0 13 0 0 0 0 4 0 0 0 0 0 0 0 0 0 13 0 0 0 9 0 8 4 0 0 16 0 0 0 0 0 2 1 9 0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,13,8,0,0,2,0,0,4,0,0,0,0,4,11,0,0,0,0,2,13],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,10,0,0,0,16,0,0,0,0,0,0,4,1,0,0,0,0,0,13],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,9,16,2,0,0,0,0,0,1,0,0,13,8,0,9,0,0,0,4,0,0] >;

C2×C8.26D4 in GAP, Magma, Sage, TeX

C_2\times C_8._{26}D_4
% in TeX

G:=Group("C2xC8.26D4");
// GroupNames label

G:=SmallGroup(128,1686);
// by ID

G=gap.SmallGroup(128,1686);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,184,2804,1411,172,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^2*c^-1>;
// generators/relations

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