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## G = (C2×C8)⋊12D4order 128 = 27

### 8th 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)⋊12D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — (C2×C8)⋊12D4
 Lower central C1 — C2 — C2×C4 — (C2×C8)⋊12D4
 Upper central C1 — C22 — C2×C4○D4 — (C2×C8)⋊12D4
 Jennings C1 — C2 — C2 — C2×C4 — (C2×C8)⋊12D4

Generators and relations for (C2×C8)⋊12D4
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, cac-1=ab4, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 620 in 262 conjugacy classes, 100 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×28], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×8], D4 [×26], Q8 [×2], C23, C23 [×2], C23 [×12], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×6], C2×C8 [×4], M4(2) [×6], D8 [×8], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×6], C2×D4 [×18], C2×Q8, C4○D4 [×4], C24 [×2], D4⋊C4 [×8], C4.Q8 [×2], C2.D8 [×2], C42⋊C2 [×2], C22≀C2 [×4], C4⋊D4 [×8], C4.4D4 [×2], C41D4 [×2], C22×C8, C22×C8 [×2], C2×M4(2), C2×M4(2) [×2], C8○D4 [×4], C2×D8 [×4], C2×D8 [×4], C22×D4 [×2], C2×C4○D4, C23.37D4 [×2], C23.25D4, C87D4 [×4], C82D4 [×4], C22.29C24 [×2], C2×C8○D4, C22×D8, (C2×C8)⋊12D4
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○D8 [×2], (C2×C8)⋊12D4

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 C4○D4 D4○D8 kernel (C2×C8)⋊12D4 C23.37D4 C23.25D4 C8⋊7D4 C8⋊2D4 C22.29C24 C2×C8○D4 C22×D8 C2×C8 C2×D4 C2×Q8 C2×C4 C2 # reps 1 2 1 4 4 2 1 1 4 3 1 4 4

Matrix representation of (C2×C8)⋊12D4 in GL6(𝔽17)

 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 0 0 0 0 0 0 1
,
 0 13 0 0 0 0 13 0 0 0 0 0 0 0 3 14 0 0 0 0 3 3 0 0 0 0 0 0 3 14 0 0 0 0 3 3
,
 4 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 14 14 0 0 0 0 14 3 0 0 3 3 0 0 0 0 3 14 0 0
,
 0 13 0 0 0 0 4 0 0 0 0 0 0 0 3 3 0 0 0 0 3 14 0 0 0 0 0 0 14 14 0 0 0 0 14 3

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,0,0,0,0,3,3,0,0,0,0,14,3],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,3,3,0,0,0,0,3,14,0,0,14,14,0,0,0,0,14,3,0,0],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,3,3,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,14,3] >;

(C2×C8)⋊12D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)\rtimes_{12}D_4
% in TeX

G:=Group("(C2xC8):12D4");
// GroupNames label

G:=SmallGroup(128,1790);
// by ID

G=gap.SmallGroup(128,1790);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,521,2804,172]);
// 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^-1,d*c*d=c^-1>;
// generators/relations

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