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## G = (C2×C8)⋊11D4order 128 = 27

### 7th semidirect product of C2×C8 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×C8)⋊11D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — (C2×C8)⋊11D4
 Lower central C1 — C2 — C2×C4 — (C2×C8)⋊11D4
 Upper central C1 — C22 — C2×C4○D4 — (C2×C8)⋊11D4
 Jennings C1 — C2 — C2 — C2×C4 — (C2×C8)⋊11D4

Generators and relations for (C2×C8)⋊11D4
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, cac-1=ab4, ad=da, cbc-1=dbd=b3, dcd=c-1 >

Subgroups: 508 in 246 conjugacy classes, 100 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×18], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×12], D4 [×16], Q8 [×8], C23, C23 [×2], C23 [×6], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×6], C2×C8 [×4], M4(2) [×6], SD16 [×8], C22×C4, C22×C4 [×2], C22×C4, C2×D4, C2×D4 [×4], C2×D4 [×9], C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C24, D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C42⋊C2 [×2], C22≀C2 [×2], C4⋊D4 [×4], C22⋊Q8 [×4], C22.D4 [×2], C4.4D4 [×2], C41D4, C4⋊Q8, C22×C8, C22×C8 [×2], C2×M4(2), C2×M4(2) [×2], C8○D4 [×4], C2×SD16 [×4], C2×SD16 [×4], C22×D4, C22×Q8, C2×C4○D4, C23.37D4, C23.38D4, C23.25D4, C88D4 [×4], C8⋊D4 [×4], C22.29C24, C23.38C23, C2×C8○D4, C22×SD16, (C2×C8)⋊11D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, D4○SD16 [×2], (C2×C8)⋊11D4

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 C4○D4 D4○SD16 kernel (C2×C8)⋊11D4 C23.37D4 C23.38D4 C23.25D4 C8⋊8D4 C8⋊D4 C22.29C24 C23.38C23 C2×C8○D4 C22×SD16 C2×C8 C2×D4 C2×Q8 C2×C4 C2 # reps 1 1 1 1 4 4 1 1 1 1 4 3 1 4 4

Matrix representation of (C2×C8)⋊11D4 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 1 16 0 16
,
 0 4 0 0 0 0 4 0 0 0 0 0 0 0 12 5 0 0 0 0 12 12 0 0 0 0 5 12 0 7 0 0 5 0 5 7
,
 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 16 1 16 2 0 0 1 0 0 0 0 0 1 0 1 16
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 1 0 1 16

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,16,0,0,0,0,16,0,0,0,0,0,0,16],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,12,12,5,5,0,0,5,12,12,0,0,0,0,0,0,5,0,0,0,0,7,7],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,1,1,0,0,0,1,0,0,0,0,1,16,0,1,0,0,0,2,0,16],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,16] >;

(C2×C8)⋊11D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)\rtimes_{11}D_4
% in TeX

G:=Group("(C2xC8):11D4");
// GroupNames label

G:=SmallGroup(128,1789);
// by ID

G=gap.SmallGroup(128,1789);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,521,2804,172]);
// Polycyclic

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

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