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## G = C24.15Q8order 128 = 27

### 14th non-split extension by C24 of Q8 acting via Q8/C2=C22

p-group, metabelian, nilpotent (class 2), monomial, rational

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C24.15Q8
 Chief series C1 — C2 — C22 — C23 — C24 — C2×C22⋊C4 — C23.4Q8 — C24.15Q8
 Lower central C1 — C23 — C24.15Q8
 Upper central C1 — C23 — C24.15Q8
 Jennings C1 — C23 — C24.15Q8

Generators and relations for C24.15Q8
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=ce2, ab=ba, faf-1=ac=ca, eae-1=ad=da, ebe-1=bc=cb, bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 676 in 274 conjugacy classes, 92 normal (6 characteristic)
C1, C2 [×7], C2 [×6], C4 [×12], C22, C22 [×6], C22 [×46], C2×C4 [×36], C23, C23 [×6], C23 [×46], C22⋊C4 [×18], C4⋊C4 [×12], C22×C4 [×12], C24, C24 [×6], C24 [×6], C2.C42 [×4], C2×C22⋊C4 [×18], C2×C4⋊C4 [×12], C25, C243C4 [×3], C23.Q8 [×8], C23.4Q8 [×4], C24.15Q8
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C24, C22×Q8, 2+ 1+4 [×6], C232Q8 [×3], C22.54C24 [×4], C24.15Q8

Character table of C24.15Q8

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L size 1 1 1 1 1 1 1 1 4 4 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 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 ρ10 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 ρ11 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 ρ12 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 ρ13 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 ρ14 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 ρ15 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 ρ16 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 ρ17 2 -2 2 -2 2 -2 2 -2 -2 2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ18 2 -2 2 -2 2 -2 2 -2 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ19 2 -2 2 -2 2 -2 2 -2 -2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ20 2 -2 2 -2 2 -2 2 -2 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ21 4 -4 -4 4 -4 4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ22 4 4 -4 4 4 -4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ23 4 -4 4 4 -4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ24 4 4 4 -4 -4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ25 4 -4 -4 -4 4 4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ26 4 4 -4 -4 -4 -4 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4

Smallest permutation representation of C24.15Q8
On 32 points
Generators in S32
(1 15)(3 13)(6 18)(8 20)(9 24)(10 32)(11 22)(12 30)(21 27)(23 25)(26 31)(28 29)
(2 7)(4 5)(9 24)(10 27)(11 22)(12 25)(14 17)(16 19)(21 32)(23 30)(26 31)(28 29)
(1 6)(2 7)(3 8)(4 5)(9 31)(10 32)(11 29)(12 30)(13 20)(14 17)(15 18)(16 19)(21 27)(22 28)(23 25)(24 26)
(1 15)(2 16)(3 13)(4 14)(5 17)(6 18)(7 19)(8 20)(9 26)(10 27)(11 28)(12 25)(21 32)(22 29)(23 30)(24 31)
(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 26 8 22)(2 25 5 21)(3 28 6 24)(4 27 7 23)(9 20 29 15)(10 19 30 14)(11 18 31 13)(12 17 32 16)

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

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

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

Matrix representation of C24.15Q8 in GL10(𝔽5)

 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4
,
 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
,
 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4 0 0

G:=sub<GL(10,GF(5))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4],[0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0] >;

C24.15Q8 in GAP, Magma, Sage, TeX

C_2^4._{15}Q_8
% in TeX

G:=Group("C2^4.15Q8");
// GroupNames label

G:=SmallGroup(128,1574);
// by ID

G=gap.SmallGroup(128,1574);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,568,758,723,352,794,185]);
// Polycyclic

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

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