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## G = (C2×C8).103D4order 128 = 27

### 71st non-split extension by C2×C8 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — (C2×C8).103D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — (C2×C8).103D4
 Lower central C1 — C2 — C2×C4 — (C2×C8).103D4
 Upper central C1 — C4 — C2×M4(2) — (C2×C8).103D4
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C8).103D4

Generators and relations for (C2×C8).103D4
G = < a,b,c,d | a2=b8=1, c4=b4, d2=ab2, ab=ba, ac=ca, dad-1=ab4, cbc-1=dbd-1=b-1, dcd-1=ab6c3 >

Subgroups: 212 in 126 conjugacy classes, 58 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4.D4, C4.10D4, C8.C4, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×C4○D4, C4.10C42, M4(2).8C22, C2×C8.C4, C2×C8○D4, (C2×C8).103D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C23.7Q8, (C2×C8).103D4

Smallest permutation representation of (C2×C8).103D4
On 32 points
Generators in S32
(1 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)
(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 15 22 28 5 11 18 32)(2 14 23 27 6 10 19 31)(3 13 24 26 7 9 20 30)(4 12 17 25 8 16 21 29)
(1 11 22 28 5 15 18 32)(2 10 23 27 6 14 19 31)(3 9 24 26 7 13 20 30)(4 16 17 25 8 12 21 29)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E ··· 8J 8K ··· 8R order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 8 8 8 8 8 ··· 8 8 ··· 8 size 1 1 2 2 2 4 4 1 1 2 2 2 4 4 2 2 2 2 4 ··· 4 8 ··· 8

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 D4 Q8 D4 C4○D4 (C2×C8).103D4 kernel (C2×C8).103D4 C4.10C42 M4(2).8C22 C2×C8.C4 C2×C8○D4 C22×C8 C2×M4(2) C2×C8 C2×D4 C2×D4 C2×Q8 C2×C4 C1 # reps 1 2 2 2 1 4 4 4 1 2 1 4 4

Matrix representation of (C2×C8).103D4 in GL4(𝔽17) generated by

 0 13 0 0 4 0 0 0 9 8 0 4 9 9 13 0
,
 2 0 0 0 0 2 0 0 14 0 9 0 3 0 0 9
,
 2 0 1 0 2 0 0 16 13 1 15 0 5 0 2 0
,
 2 0 1 0 15 0 0 1 13 1 15 0 3 0 2 0
G:=sub<GL(4,GF(17))| [0,4,9,9,13,0,8,9,0,0,0,13,0,0,4,0],[2,0,14,3,0,2,0,0,0,0,9,0,0,0,0,9],[2,2,13,5,0,0,1,0,1,0,15,2,0,16,0,0],[2,15,13,3,0,0,1,0,1,0,15,2,0,1,0,0] >;

(C2×C8).103D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)._{103}D_4
% in TeX

G:=Group("(C2xC8).103D4");
// GroupNames label

G:=SmallGroup(128,545);
// by ID

G=gap.SmallGroup(128,545);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,2019,248,2804,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=1,c^4=b^4,d^2=a*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^4,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a*b^6*c^3>;
// generators/relations

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