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## G = (C2×Q8)⋊16D4order 128 = 27

### 12nd semidirect product of C2×Q8 and 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×Q8)⋊16D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2×2- 1+4 — (C2×Q8)⋊16D4
 Lower central C1 — C2 — C2×C4 — (C2×Q8)⋊16D4
 Upper central C1 — C22 — C2×C4○D4 — (C2×Q8)⋊16D4
 Jennings C1 — C2 — C2 — C2×C4 — (C2×Q8)⋊16D4

Generators and relations for (C2×Q8)⋊16D4
G = < a,b,c,d,e | a2=b4=d4=e2=1, c2=b2, ab=ba, ac=ca, dad-1=ab2, ae=ea, cbc-1=b-1, dbd-1=abc, ebe=ab-1c, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 732 in 366 conjugacy classes, 108 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×22], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×33], D4 [×2], D4 [×29], Q8 [×6], Q8 [×17], C23, C23 [×2], C23 [×7], C42, C22⋊C4 [×5], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], D8 [×4], SD16 [×12], C22×C4, C22×C4 [×2], C22×C4 [×6], C2×D4 [×2], C2×D4 [×4], C2×D4 [×12], C2×Q8 [×2], C2×Q8 [×6], C2×Q8 [×21], C4○D4 [×4], C4○D4 [×38], C24, C22⋊C8 [×4], D4⋊C4 [×2], Q8⋊C4 [×6], C42⋊C2, C22≀C2 [×2], C4⋊D4 [×4], C4.4D4, C41D4, C22×C8, C2×M4(2), C2×D8 [×2], C2×SD16 [×6], C2×SD16 [×4], C8⋊C22 [×4], C22×D4, C22×Q8, C22×Q8 [×2], C22×Q8, C2×C4○D4 [×2], C2×C4○D4 [×2], C2×C4○D4 [×3], 2- 1+4 [×8], (C22×C8)⋊C2, C23.24D4, C23.38D4, Q8⋊D4 [×4], D4⋊D4 [×4], C22.29C24, C22×SD16, C2×C8⋊C22, C2×2- 1+4, (C2×Q8)⋊16D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2, D4○SD16 [×2], (C2×Q8)⋊16D4

Smallest permutation representation of (C2×Q8)⋊16D4
On 32 points
Generators in S32
(1 5)(2 6)(3 7)(4 8)(9 19)(10 20)(11 17)(12 18)(13 22)(14 23)(15 24)(16 21)(25 29)(26 30)(27 31)(28 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 21 3 23)(2 24 4 22)(5 16 7 14)(6 15 8 13)(9 29 11 31)(10 32 12 30)(17 27 19 25)(18 26 20 28)
(1 29 7 25)(2 20 8 10)(3 31 5 27)(4 18 6 12)(9 16 19 23)(11 14 17 21)(13 32 24 28)(15 30 22 26)
(2 13)(4 15)(6 22)(8 24)(9 17)(10 32)(11 19)(12 30)(14 16)(18 26)(20 28)(21 23)(25 29)(27 31)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E ··· 4L 4M 4N 8A 8B 8C 8D 8E 8F order 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 8 8 8 8 8 8 size 1 1 1 1 2 2 4 4 4 4 8 8 2 2 2 2 4 ··· 4 8 8 4 4 4 4 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4○SD16 kernel (C2×Q8)⋊16D4 (C22×C8)⋊C2 C23.24D4 C23.38D4 Q8⋊D4 D4⋊D4 C22.29C24 C22×SD16 C2×C8⋊C22 C2×2- 1+4 C2×D4 C2×Q8 C4○D4 C2 # reps 1 1 1 1 4 4 1 1 1 1 3 5 4 4

Matrix representation of (C2×Q8)⋊16D4 in GL6(𝔽17)

 1 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 16 0 0 0 0 0 0 16
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 12 12 0 0 0 0 12 5 0 0 0 0 0 0 5 5 0 0 0 0 5 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 16 0 0
,
 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,12,12,0,0,0,0,12,5,0,0,0,0,0,0,5,5,0,0,0,0,5,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1] >;

(C2×Q8)⋊16D4 in GAP, Magma, Sage, TeX

(C_2\times Q_8)\rtimes_{16}D_4
% in TeX

G:=Group("(C2xQ8):16D4");
// GroupNames label

G:=SmallGroup(128,1742);
// by ID

G=gap.SmallGroup(128,1742);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,521,2804,1411,172]);
// Polycyclic

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

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