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G = C24⋊5Q8  order 128 = 27

4th semidirect product of C24 and Q8 acting via Q8/C2=C22

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

Aliases: C24⋊5Q8, C25.53C22, C23.526C24, C24.590C23, C22.3032+ 1+4, C23.66(C2×Q8), C23⋊Q8⋊28C2, (C22×C4).404D4, C23.626(C2×D4), (C22×Q8)⋊6C22, C24⋊3C4.12C2, C23.Q8⋊38C2, C23.4Q8⋊28C2, C23.8Q8⋊84C2, C23.7Q8⋊79C2, C2.8(C23⋊2Q8), C23.242(C4○D4), C2.25(C23⋊3D4), (C23×C4).428C22, (C22×C4).136C23, C22.351(C22×D4), C22.42(C22⋊Q8), C22.131(C22×Q8), C2.C42⋊31C22, C2.38(C22.32C24), C2.38(C22.29C24), (C2×C4⋊C4)⋊27C22, (C2×C4).385(C2×D4), (C2×C22⋊Q8)⋊28C2, C2.41(C2×C22⋊Q8), C22.398(C2×C4○D4), (C22×C22⋊C4).26C2, (C2×C22⋊C4).519C22, SmallGroup(128,1358)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C23 — C24⋊5Q8
Chief series
C1 — C2 — C22 — C23 — C24 — C2×C22⋊C4 — C24⋊3C4 — C24⋊5Q8
Lower central
C1 — C23 — C24⋊5Q8
Upper central
C1 — C23 — C24⋊5Q8
Jennings
C1 — C23 — C24⋊5Q8

Generators and relations for C24⋊5Q8
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=e2, ab=ba, faf-1=ac=ca, eae-1=ad=da, ebe-1=bc=cb, bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 756 in 334 conjugacy classes, 108 normal (20 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C24, C24, C24, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C22×Q8, C25, C24⋊3C4, C23.7Q8, C23.8Q8, C23⋊Q8, C23.Q8, C23.4Q8, C22×C22⋊C4, C2×C22⋊Q8, C24⋊5Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, C2×C22⋊Q8, C23⋊3D4, C22.29C24, C22.32C24, C23⋊2Q8, C24⋊5Q8

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

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

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4H4I···4P
order12···2222222224···44···4
size11···1222244444···48···8

32 irreducible representations

dim1111111112224
type++++++++++-+
imageC1C2C2C2C2C2C2C2C2D4Q8C4○D42+ 1+4
kernelC24⋊5Q8C24⋊3C4C23.7Q8C23.8Q8C23⋊Q8C23.Q8C23.4Q8C22×C22⋊C4C2×C22⋊Q8C22×C4C24C23C22
# reps1212242114444

Matrix representation of C24⋊5Q8 ►in GL8(𝔽5)

40000000
01000000
00400000
00040000
00001000
00000100
00000040
00004404
,
10000000
01000000
00400000
00040000
00001000
00000400
00000010
00004044
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00001000
00000100
00000010
00000001
,
04000000
10000000
00300000
00320000
00000100
00001000
00004443
00000001
,
30000000
02000000
00420000
00410000
00000010
00004443
00001000
00000001

G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,1,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,1,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,1],[3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,3,0,1] >;
 

C24⋊5Q8 in GAP, Magma, Sage, TeX 

C_2^4\rtimes_5Q_8
 
% in TeX
 
G:=Group("C2^4:5Q8");
 
// GroupNames label
 
G:=SmallGroup(128,1358);
 
// by ID
 
G=gap.SmallGroup(128,1358);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,184,185]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=1,f^2=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,b*f=f*b,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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