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## G = C2×C8.Q8order 128 = 27

### Direct product of C2 and C8.Q8

direct product, p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C2×C8.Q8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C2×M5(2) — C2×C8.Q8
 Lower central C1 — C2 — C4 — C8 — C2×C8.Q8
 Upper central C1 — C22 — C22×C4 — C22×C8 — C2×C8.Q8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C2×C8.Q8

Generators and relations for C2×C8.Q8
G = < a,b,c,d | a2=b8=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b3, dcd-1=b4c3 >

Subgroups: 140 in 76 conjugacy classes, 52 normal (32 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C16, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C4.Q8, C4.Q8, C8.C4, C8.C4, C2×C16, M5(2), C2×C4⋊C4, C22×C8, C2×M4(2), C8.Q8, C2×C4.Q8, C2×C8.C4, C2×M5(2), C2×C8.Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, C4.Q8, C2×C4⋊C4, C2×SD16, C8.Q8, C2×C4.Q8, C2×C8.Q8

Smallest permutation representation of C2×C8.Q8
On 32 points
Generators in S32
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 17)(15 18)(16 19)
(1 3 5 7 9 11 13 15)(2 12 6 16 10 4 14 8)(17 27 21 31 25 19 29 23)(18 20 22 24 26 28 30 32)
(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 28)(2 31 10 23)(3 18)(4 21 12 29)(5 24)(6 27 14 19)(7 30)(8 17 16 25)(9 20)(11 26)(13 32)(15 22)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 16A ··· 16H order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 8 8 8 8 8 8 8 8 8 8 16 ··· 16 size 1 1 1 1 2 2 2 2 2 2 8 8 8 8 2 2 2 2 4 4 8 8 8 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + - + image C1 C2 C2 C2 C2 C4 C4 D4 Q8 D4 SD16 SD16 C8.Q8 kernel C2×C8.Q8 C8.Q8 C2×C4.Q8 C2×C8.C4 C2×M5(2) C2×C16 M5(2) C2×C8 C2×C8 C22×C4 C2×C4 C23 C2 # reps 1 4 1 1 1 4 4 1 2 1 6 2 4

Matrix representation of C2×C8.Q8 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 5 12 0 0 0 0 5 5 0 0 0 0 0 0 12 5 0 0 0 0 12 12
,
 7 15 0 0 0 0 8 10 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 5 12 0 0 0 0 5 5 0 0
,
 13 0 0 0 0 0 6 4 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 12 5 0 0 0 0 5 5

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,12,12,0,0,0,0,5,12],[7,8,0,0,0,0,15,10,0,0,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,1,0,0,0,0,0,0,1,0,0],[13,6,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,12,5,0,0,0,0,5,5] >;

C2×C8.Q8 in GAP, Magma, Sage, TeX

C_2\times C_8.Q_8
% in TeX

G:=Group("C2xC8.Q8");
// GroupNames label

G:=SmallGroup(128,886);
// by ID

G=gap.SmallGroup(128,886);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,64,1123,136,1411,4037,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=1,c^4=b^2,d^2=b^-1*c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^3,d*c*d^-1=b^4*c^3>;
// generators/relations

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