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## G = (C2×Q8)⋊2Q8order 128 = 27

### 2nd semidirect product of C2×Q8 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×Q8)⋊2Q8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×Q8 — C23.67C23 — (C2×Q8)⋊2Q8
 Lower central C1 — C2 — C22×C4 — (C2×Q8)⋊2Q8
 Upper central C1 — C22 — C22×C4 — (C2×Q8)⋊2Q8
 Jennings C1 — C2 — C2 — C22×C4 — (C2×Q8)⋊2Q8

Generators and relations for (C2×Q8)⋊2Q8
G = < a,b,c,d,e | a2=b4=d4=1, c2=b2, e2=d2, ab=ba, ac=ca, ad=da, eae-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=ab2c, ece-1=ab-1c, ede-1=d-1 >

Subgroups: 256 in 123 conjugacy classes, 42 normal (18 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×10], C22 [×3], C22 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×16], Q8 [×8], C23, C42 [×4], C22⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×Q8 [×2], C2×Q8 [×7], C2.C42 [×2], Q8⋊C4 [×4], C2×C42, C2×C4⋊C4, C42⋊C2 [×2], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C22×Q8, C426C4 [×2], C22.C42, C23.67C23, C23.38D4 [×2], C23.41C23, (C2×Q8)⋊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, D4.9D4, D4.10D4, (C2×Q8)⋊2Q8

Character table of (C2×Q8)⋊2Q8

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 8A 8B 8C 8D size 1 1 1 1 2 2 2 2 2 2 4 4 4 4 8 8 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 -2 -2 -2 -2 0 0 0 0 0 0 -2 0 0 0 0 2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 -2 2 -2 -2 2 0 0 0 0 0 0 0 2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 -2 2 2 -2 0 0 0 0 -2 0 0 0 0 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 -2 -2 2 -2 -2 2 0 0 0 0 0 0 0 -2 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 2 2 -2 -2 -2 -2 0 0 0 0 0 0 2 0 0 0 0 -2 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 -2 -2 -2 2 2 -2 0 0 0 0 2 0 0 0 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 -2 -2 2 2 -2 -2 -2 2 2 0 0 0 0 0 2 0 0 0 0 -2 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ16 2 -2 -2 2 2 -2 -2 -2 2 2 0 0 0 0 0 -2 0 0 0 0 2 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ17 2 -2 -2 2 -2 2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 2i -2i 0 0 complex lifted from C4○D4 ρ18 2 -2 -2 2 -2 2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 -2i 2i 0 0 complex lifted from C4○D4 ρ19 2 -2 -2 2 -2 2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2i 2i complex lifted from C4○D4 ρ20 2 -2 -2 2 2 -2 2 2 -2 -2 -2i 2i -2i 2i 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ21 2 -2 -2 2 -2 2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2i -2i complex lifted from C4○D4 ρ22 2 -2 -2 2 2 -2 2 2 -2 -2 2i -2i 2i -2i 0 0 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 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.10D4, Schur index 2 ρ24 4 4 -4 -4 0 0 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.10D4, Schur index 2 ρ25 4 -4 4 -4 0 0 0 0 0 0 2i 2i -2i -2i 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D4.9D4 ρ26 4 -4 4 -4 0 0 0 0 0 0 -2i -2i 2i 2i 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D4.9D4

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

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

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

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

Matrix representation of (C2×Q8)⋊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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 13
,
 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 13 0 0 0 0 13 0
,
 4 0 0 0 0 0 0 13 0 0 0 0 0 0 0 1 0 0 0 0 1 0 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 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,13,0,0,0,0,13,0],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C2×Q8)⋊2Q8 in GAP, Magma, Sage, TeX

(C_2\times Q_8)\rtimes_2Q_8
% in TeX

G:=Group("(C2xQ8):2Q8");
// GroupNames label

G:=SmallGroup(128,760);
// by ID

G=gap.SmallGroup(128,760);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,64,422,387,352,2804,1411,718,172,4037]);
// Polycyclic

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

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