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G = C24.15Q8  order 128 = 27

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

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

Aliases: C24.15Q8, C25.65C22, C23.742C24, C24.467C23, C22.5152+ (1+4), C23.73(C2×Q8), C24⋊3C4.16C2, C23.4Q8⋊71C2, (C22×C4).253C23, C23.Q8⋊101C2, C2.26(C23⋊2Q8), C22.174(C22×Q8), C2.C42⋊44C22, C2.58(C22.54C24), (C2×C4⋊C4)⋊39C22, (C2×C22⋊C4).358C22, SmallGroup(128,1574)

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

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, C24⋊3C4 [×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], C23⋊2Q8 [×3], C22.54C24 [×4], C24.15Q8

Generators and relations
 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 >

Smallest permutation representation
►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 ►G ⊆ GL10(𝔽5)

1000000000
0100000000
0040000000
0001000000
0000400000
0000010000
0000004000
0000000100
0000000010
0000000004
,
4000000000
0400000000
0010000000
0001000000
0000400000
0000040000
0000001000
0000000400
0000000010
0000000004
,
1000000000
0100000000
0010000000
0001000000
0000100000
0000010000
0000004000
0000000400
0000000040
0000000004
,
1000000000
0100000000
0040000000
0004000000
0000400000
0000040000
0000004000
0000000400
0000000040
0000000004
,
0400000000
1000000000
0001000000
0040000000
0000040000
0000100000
0000000100
0000001000
0000000001
0000000010
,
0300000000
3000000000
0000100000
0000010000
0040000000
0004000000
0000000010
0000000001
0000004000
0000000400

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] >;
 

Character table of C24.15Q8

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M4A4B4C4D4E4F4G4H4I4J4K4L
 size 11111111444444888888888888
ρ111111111111111111111111111    trivial
ρ211111111-1-111-1-11111-1-1-1-111-1-1    linear of order 2
ρ31111111111-1-1-1-1-111-11-11-1-11-11    linear of order 2
ρ411111111-1-1-1-111-111-1-11-11-111-1    linear of order 2
ρ511111111111111-1-111-1-1-1-1-1-111    linear of order 2
ρ611111111-1-111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ71111111111-1-1-1-11-11-1-11-111-1-11    linear of order 2
ρ811111111-1-1-1-1111-11-11-11-11-11-1    linear of order 2
ρ91111111111111111-1-111-1-1-1-1-1-1    linear of order 2
ρ1011111111-1-111-1-111-1-1-1-111-1-111    linear of order 2
ρ111111111111-1-1-1-1-11-111-1-111-11-1    linear of order 2
ρ1211111111-1-1-1-111-11-11-111-11-1-11    linear of order 2
ρ1311111111111111-1-1-1-1-1-11111-1-1    linear of order 2
ρ1411111111-1-111-1-1-1-1-1-111-1-11111    linear of order 2
ρ151111111111-1-1-1-11-1-11-111-1-111-1    linear of order 2
ρ1611111111-1-1-1-1111-1-111-1-11-11-11    linear of order 2
ρ172-22-22-22-2-222-2-22000000000000    symplectic lifted from Q8, Schur index 2
ρ182-22-22-22-22-22-22-2000000000000    symplectic lifted from Q8, Schur index 2
ρ192-22-22-22-2-22-222-2000000000000    symplectic lifted from Q8, Schur index 2
ρ202-22-22-22-22-2-22-22000000000000    symplectic lifted from Q8, Schur index 2
ρ214-4-44-444-4000000000000000000    orthogonal lifted from 2+ (1+4)
ρ2244-444-4-4-4000000000000000000    orthogonal lifted from 2+ (1+4)
ρ234-444-4-4-44000000000000000000    orthogonal lifted from 2+ (1+4)
ρ24444-4-44-4-4000000000000000000    orthogonal lifted from 2+ (1+4)
ρ254-4-4-444-44000000000000000000    orthogonal lifted from 2+ (1+4)
ρ2644-4-4-4-444000000000000000000    orthogonal lifted from 2+ (1+4)

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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