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## G = (C2×D4)⋊2Q8order 128 = 27

### 2nd semidirect product of C2×D4 and Q8 acting via Q8/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×D4)⋊2Q8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×D4 — C24.3C22 — (C2×D4)⋊2Q8
 Lower central C1 — C2 — C22×C4 — (C2×D4)⋊2Q8
 Upper central C1 — C22 — C22×C4 — (C2×D4)⋊2Q8
 Jennings C1 — C2 — C2 — C22×C4 — (C2×D4)⋊2Q8

Generators and relations for (C2×D4)⋊2Q8
G = < a,b,c,d,e | a2=b4=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, eae-1=ab2, cbc=dbd-1=b-1, be=eb, dcd-1=ab2c, ece-1=abc, ede-1=d-1 >

Subgroups: 352 in 141 conjugacy classes, 42 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×8], C22 [×3], C22 [×12], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×12], D4 [×6], Q8 [×2], C23, C23 [×8], C42 [×4], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4, C2×D4 [×2], C2×D4 [×5], C2×Q8 [×2], C24, D4⋊C4 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C42⋊C2 [×2], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C22×D4, C426C4 [×2], C22.C42, C24.3C22, C23.37D4 [×2], C23.41C23, (C2×D4)⋊2Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, C2×D4 [×3], C2×Q8, C4○D4 [×3], C22≀C2, C22⋊Q8 [×3], C4.4D4 [×3], C23⋊Q8, D44D4, D4.8D4, (C2×D4)⋊2Q8

Character table of (C2×D4)⋊2Q8

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 8A 8B 8C 8D size 1 1 1 1 2 2 8 8 2 2 2 2 4 4 4 4 8 8 8 8 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 -1 -1 -1 1 -1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 -1 -1 1 -1 1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 1 -1 1 -1 -1 -1 1 -1 1 linear of order 2 ρ6 1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ7 1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ8 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 1 -1 1 -1 linear of order 2 ρ9 2 2 2 2 -2 -2 0 0 2 -2 2 -2 0 0 0 0 0 2 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 -2 0 0 2 -2 2 -2 0 0 0 0 0 -2 0 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 2 2 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 -2 -2 0 0 -2 2 -2 2 0 0 0 0 -2 0 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 2 2 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 -2 2 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 -2 -2 0 0 -2 2 -2 2 0 0 0 0 2 0 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 -2 -2 2 -2 2 -2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ16 2 -2 -2 2 -2 2 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ17 2 -2 -2 2 2 -2 0 0 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 2i 0 -2i complex lifted from C4○D4 ρ18 2 -2 -2 2 2 -2 0 0 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 -2i 0 2i complex lifted from C4○D4 ρ19 2 -2 -2 2 2 -2 0 0 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 2i 0 -2i 0 complex lifted from C4○D4 ρ20 2 -2 -2 2 -2 2 0 0 2 -2 -2 2 -2i 2i -2i 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ21 2 -2 -2 2 2 -2 0 0 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 -2i 0 2i 0 complex lifted from C4○D4 ρ22 2 -2 -2 2 -2 2 0 0 2 -2 -2 2 2i -2i 2i -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 4 -4 4 -4 0 0 0 0 0 0 0 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊4D4 ρ24 4 -4 4 -4 0 0 0 0 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊4D4 ρ25 4 4 -4 -4 0 0 0 0 0 0 0 0 2i 2i -2i -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from D4.8D4 ρ26 4 4 -4 -4 0 0 0 0 0 0 0 0 -2i -2i 2i 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from D4.8D4

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

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

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

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

Matrix representation of (C2×D4)⋊2Q8 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 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 1 0
,
 0 16 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 13 0 0 0 0 13 0 0 0 0 0 0 0 0 13 0 0 0 0 13 0 0 0 0 0 0 0 0 4 0 0 0 0 4 0
,
 4 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 16 0 0 0 0 0 0 16 0 0

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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C2×D4)⋊2Q8 in GAP, Magma, Sage, TeX

(C_2\times D_4)\rtimes_2Q_8
% in TeX

G:=Group("(C2xD4):2Q8");
// GroupNames label

G:=SmallGroup(128,759);
// by ID

G=gap.SmallGroup(128,759);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,64,422,387,2804,1411,718,172,4037]);
// Polycyclic

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

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