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## G = (C22×Q8)⋊C4order 128 = 27

### 6th semidirect product of C22×Q8 and C4 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — (C22×Q8)⋊C4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C4○D4 — C2×2- 1+4 — (C22×Q8)⋊C4
 Lower central C1 — C2 — C2×C4 — (C22×Q8)⋊C4
 Upper central C1 — C2 — C22×C4 — (C22×Q8)⋊C4
 Jennings C1 — C2 — C2 — C22×C4 — (C22×Q8)⋊C4

Generators and relations for (C22×Q8)⋊C4
G = < a,b,c,d,e | a2=b2=c4=e4=1, d2=c2, ab=ba, ac=ca, ad=da, eae-1=bc2, ede-1=bc=cb, bd=db, ebe-1=a, dcd-1=c-1, ece-1=ac2d >

Subgroups: 508 in 268 conjugacy classes, 66 normal (20 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×9], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×33], D4 [×2], D4 [×19], Q8 [×6], Q8 [×17], C23, C23 [×2], C23, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4 [×6], C2×D4 [×2], C2×D4 [×2], C2×D4 [×3], C2×Q8 [×2], C2×Q8 [×6], C2×Q8 [×21], C4○D4 [×4], C4○D4 [×38], C23⋊C4 [×2], C4.D4, C4.10D4, Q8⋊C4 [×4], C4≀C2 [×4], C42⋊C2 [×2], C2×M4(2) [×2], C22×Q8, C22×Q8 [×2], C22×Q8, C2×C4○D4 [×2], C2×C4○D4 [×2], C2×C4○D4 [×3], 2- 1+4 [×8], C23.C23, M4(2).8C22, C23.38D4 [×2], C42⋊C22 [×2], C2×2- 1+4, (C22×Q8)⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×12], C23, C22⋊C4 [×12], C22×C4, C2×D4 [×6], C2×C22⋊C4 [×3], C22≀C2 [×4], C243C4, (C22×Q8)⋊C4

Character table of (C22×Q8)⋊C4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 8A 8B 8C 8D size 1 1 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 4 4 4 4 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 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 -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 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 -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 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 -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 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 -1 1 -1 linear of order 2 ρ9 1 1 -1 -1 1 -1 1 1 1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 -1 -i -i i i -i -i i i linear of order 4 ρ10 1 1 -1 -1 1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 i -i -i i -i i i -i linear of order 4 ρ11 1 1 -1 -1 1 -1 1 1 1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 -1 i i -i -i i i -i -i linear of order 4 ρ12 1 1 -1 -1 1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 -i i i -i i -i -i i linear of order 4 ρ13 1 1 -1 -1 1 1 -1 -1 -1 -1 1 -1 1 -1 -1 1 -1 1 1 1 1 i -i -i i i -i -i i linear of order 4 ρ14 1 1 -1 -1 1 -1 1 -1 -1 -1 1 -1 1 1 1 -1 1 -1 1 -1 1 -i -i i i i i -i -i linear of order 4 ρ15 1 1 -1 -1 1 1 -1 -1 -1 -1 1 -1 1 -1 -1 1 -1 1 1 1 1 -i i i -i -i i i -i linear of order 4 ρ16 1 1 -1 -1 1 -1 1 -1 -1 -1 1 -1 1 1 1 -1 1 -1 1 -1 1 i i -i -i -i -i i i linear of order 4 ρ17 2 2 -2 2 -2 0 0 -2 2 2 -2 -2 2 0 0 0 0 0 2 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 2 -2 0 0 0 0 -2 2 2 -2 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 2 -2 0 0 0 0 -2 2 2 -2 0 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 2 -2 2 0 0 -2 -2 -2 -2 -2 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 -2 -2 2 -2 -2 0 0 2 -2 2 -2 2 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ22 2 2 -2 2 -2 0 0 2 -2 2 -2 -2 2 0 0 0 0 0 -2 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ23 2 2 2 -2 -2 0 0 0 0 2 2 -2 -2 0 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ24 2 2 2 -2 -2 0 0 2 -2 -2 -2 2 2 0 0 0 0 0 2 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ25 2 2 2 2 2 2 -2 0 0 -2 -2 -2 -2 2 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ26 2 2 -2 -2 2 2 2 0 0 2 -2 2 -2 -2 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ27 2 2 2 -2 -2 0 0 0 0 2 2 -2 -2 0 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ28 2 2 2 -2 -2 0 0 -2 2 -2 -2 2 2 0 0 0 0 0 -2 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ29 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of (C22×Q8)⋊C4
On 32 points
Generators in S32
(1 19)(2 20)(3 17)(4 18)(5 16)(6 13)(7 14)(8 15)(9 32)(10 29)(11 30)(12 31)(21 27)(22 28)(23 25)(24 26)
(1 31)(2 32)(3 29)(4 30)(5 26)(6 27)(7 28)(8 25)(9 20)(10 17)(11 18)(12 19)(13 21)(14 22)(15 23)(16 24)
(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 7 3 5)(2 6 4 8)(9 21 11 23)(10 24 12 22)(13 18 15 20)(14 17 16 19)(25 32 27 30)(26 31 28 29)
(2 16)(4 14)(5 32 7 30)(6 23 8 21)(9 28)(10 12)(11 26)(13 15)(17 29 19 31)(18 22 20 24)

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

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

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

Matrix representation of (C22×Q8)⋊C4 in GL8(𝔽17)

 0 0 0 0 0 0 16 0 0 0 0 0 1 16 16 2 0 0 0 0 1 0 0 0 0 0 0 0 1 16 0 1 0 0 1 0 0 0 0 0 16 1 1 15 0 0 0 0 16 0 0 0 0 0 0 0 16 1 0 16 0 0 0 0
,
 0 0 0 0 16 1 1 15 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 16 0 1 1 16 16 2 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 1 0 16 0 0 0 0
,
 0 4 0 13 0 0 0 0 4 0 0 4 0 0 0 0 0 0 13 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 13 0 4 0 0 0 0 13 0 0 13 0 0 0 0 0 0 4 13 0 0 0 0 0 0 0 13
,
 0 0 0 0 0 13 0 4 0 0 0 0 13 0 0 13 0 0 0 0 0 0 4 13 0 0 0 0 0 0 0 13 0 13 0 4 0 0 0 0 13 0 0 13 0 0 0 0 0 0 4 13 0 0 0 0 0 0 0 13 0 0 0 0
,
 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16 16 1 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 1 1 15 0 0 0 0 0 1 1 16

G:=sub<GL(8,GF(17))| [0,0,0,0,0,16,16,16,0,0,0,0,0,1,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,15,0,16,0,1,1,1,0,0,0,0,0,16,0,16,0,0,0,0,16,16,0,0,0,0,0,0,0,2,0,1,0,0,0,0],[0,0,0,0,1,0,0,16,0,0,0,0,16,0,1,1,0,0,0,0,16,16,0,0,0,0,0,0,2,0,0,16,16,0,0,1,0,0,0,0,1,0,16,16,0,0,0,0,1,1,0,0,0,0,0,0,15,0,0,1,0,0,0,0],[0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,13,4,4,4,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,13,13,13],[0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,13,13,13,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,13,13,13,0,0,0,0],[1,0,0,0,0,0,0,0,0,16,0,16,0,0,0,0,0,0,16,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,1,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16] >;

(C22×Q8)⋊C4 in GAP, Magma, Sage, TeX

(C_2^2\times Q_8)\rtimes C_4
% in TeX

G:=Group("(C2^2xQ8):C4");
// GroupNames label

G:=SmallGroup(128,528);
// by ID

G=gap.SmallGroup(128,528);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,456,422,2019,1018,248,2804,1027]);
// Polycyclic

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

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