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G = D8.C8order 128 = 27

The non-split extension by D8 of C8 acting via C8/C4=C2

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

Aliases: D8.C8, Q16.C8, SD16.C8, C16.26D4, M5(2).25C22, C4≀C2.C4, C8.6(C2×C8), D4○C167C2, D4.C86C2, C4○D8.6C4, C8○D8.3C2, D4.4(C2×C8), C2.19(C8×D4), Q8.4(C2×C8), C165C411C2, C4.179(C4×D4), C8.142(C2×D4), C8.C813C2, C8.C4.6C4, C8.61(C4○D4), C4.16(C22×C8), (C2×C16).56C22, C42.172(C2×C4), (C2×C8).608C23, (C4×C8).160C22, C8○D4.18C22, C22.2(C8○D4), M4(2).24(C2×C4), (C2×C8).93(C2×C4), C4○D4.19(C2×C4), (C2×C4).447(C22×C4), SmallGroup(128,903)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — D8.C8
C1C2C4C8C2×C8C2×C16D4○C16 — D8.C8
C1C2C4 — D8.C8
C1C8C2×C16 — D8.C8
C1C2C2C2C2C4C4C2×C8 — D8.C8

Generators and relations for D8.C8
 G = < a,b,c | a8=b2=1, c8=a4, bab=a-1, cac-1=a3, cbc-1=a4b >

Subgroups: 104 in 72 conjugacy classes, 46 normal (24 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×3], C22, C22 [×2], C8 [×4], C8 [×2], C2×C4, C2×C4 [×3], D4 [×2], D4 [×2], Q8 [×2], C16 [×2], C16 [×3], C42, C2×C8 [×2], C2×C8 [×2], M4(2) [×2], M4(2) [×2], D8, SD16 [×2], Q16, C4○D4 [×2], C4×C8, C4≀C2 [×2], C8.C4, C2×C16 [×2], C2×C16 [×2], M5(2) [×2], M5(2) [×2], C8○D4 [×2], C4○D8, C165C4, D4.C8 [×2], C8.C8, C8○D8, D4○C16 [×2], D8.C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], C23, C2×C8 [×6], C22×C4, C2×D4, C4○D4, C4×D4, C22×C8, C8○D4, C8×D4, D8.C8

Smallest permutation representation of D8.C8
On 32 points
Generators in S32
(1 11 5 15 9 3 13 7)(2 16 14 12 10 8 6 4)(17 19 21 23 25 27 29 31)(18 24 30 20 26 32 22 28)
(1 22)(2 31)(3 24)(4 17)(5 26)(6 19)(7 28)(8 21)(9 30)(10 23)(11 32)(12 25)(13 18)(14 27)(15 20)(16 29)
(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)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G8A8B8C8D8E8F8G···8L16A···16H16I···16T
order1222244444448888888···816···1616···16
size1124411244441111224···42···24···4

44 irreducible representations

dim1111111111112224
type+++++++
imageC1C2C2C2C2C2C4C4C4C8C8C8D4C4○D4C8○D4D8.C8
kernelD8.C8C165C4D4.C8C8.C8C8○D8D4○C16C4≀C2C8.C4C4○D8D8SD16Q16C16C8C22C1
# reps1121124224842244

Matrix representation of D8.C8 in GL4(𝔽17) generated by

2000
01500
0090
0008
,
0090
0008
2000
01500
,
0008
0010
01500
4000
G:=sub<GL(4,GF(17))| [2,0,0,0,0,15,0,0,0,0,9,0,0,0,0,8],[0,0,2,0,0,0,0,15,9,0,0,0,0,8,0,0],[0,0,0,4,0,0,15,0,0,1,0,0,8,0,0,0] >;

D8.C8 in GAP, Magma, Sage, TeX

D_8.C_8
% in TeX

G:=Group("D8.C8");
// GroupNames label

G:=SmallGroup(128,903);
// by ID

G=gap.SmallGroup(128,903);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1430,100,2019,1018,248,102,124]);
// Polycyclic

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

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