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G = C24:9D4  order 128 = 27

4th semidirect product of C24 and D4 acting via D4/C2=C22

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

Aliases: C24:9D4, C25.50C22, C23.513C24, C24.359C23, C22.2922+ 1+4, (D4xC23):6C2, (C22xC4):31D4, C24:3C4:19C2, C23:2D4:22C2, C23.187(C2xD4), (C22xD4):8C22, C22.54C22wrC2, C23.10D4:52C2, C23.34D4:40C2, C2.21(C23:3D4), (C22xC4).851C23, (C23xC4).416C22, C22.338(C22xD4), C2.C42:28C22, C2.31(C22.29C24), (C2xC22wrC2):9C2, (C2xC4:C4):24C22, (C2xC4).373(C2xD4), C2.25(C2xC22wrC2), (C2xC22:C4):22C22, (C2xC22.D4):24C2, SmallGroup(128,1345)

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

Derived series
C1 — C23 — C24:9D4
Chief series
C1 — C2 — C22 — C23 — C24 — C22xD4 — C2xC22wrC2 — C24:9D4
Lower central
C1 — C23 — C24:9D4
Upper central
C1 — C23 — C24:9D4
Jennings
C1 — C23 — C24:9D4

Generators and relations for C24:9D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, ab=ba, faf=ac=ca, ad=da, eae-1=acd, ebe-1=bc=cb, bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1348 in 567 conjugacy classes, 116 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2xC4, C2xC4, D4, C23, C23, C23, C22:C4, C4:C4, C22xC4, C22xC4, C22xC4, C2xD4, C24, C24, C24, C2.C42, C2xC22:C4, C2xC22:C4, C2xC4:C4, C22wrC2, C22.D4, C23xC4, C22xD4, C22xD4, C22xD4, C25, C24:3C4, C23.34D4, C23:2D4, C23.10D4, C2xC22wrC2, C2xC22.D4, D4xC23, C24:9D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C24, C22wrC2, C22xD4, 2+ 1+4, C2xC22wrC2, C23:3D4, C22.29C24, C24:9D4

Smallest permutation representation of C24:9D4
►On 32 points
Generators in S32
(1 9)(2 17)(3 11)(4 19)(5 27)(6 29)(7 25)(8 31)(10 22)(12 24)(13 28)(14 30)(15 26)(16 32)(18 23)(20 21)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2S2T4A4B4C4D4E···4K
order12···222222···2244444···4
size11···122224···4844448···8

32 irreducible representations

dim11111111224
type+++++++++++
imageC1C2C2C2C2C2C2C2D4D42+ 1+4
kernelC24:9D4C24:3C4C23.34D4C23:2D4C23.10D4C2xC22wrC2C2xC22.D4D4xC23C22xC4C24C22
# reps12144211484

Matrix representation of C24:9D4 ►in GL8(Z)

-10000000
01000000
00-100000
00010000
00000010
00000001
00001000
00000100
,
-10000000
0-1000000
00100000
00010000
00001000
00000-100
00000010
0000000-1
,
10000000
01000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
-10000000
0-1000000
00-100000
000-10000
00001000
00000100
00000010
00000001
,
0-1000000
10000000
000-10000
00-100000
00000100
00001000
0000000-1
000000-10
,
10000000
0-1000000
00100000
00010000
00001000
00000100
000000-10
0000000-1

G:=sub<GL(8,Integers())| [-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[-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,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1],[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,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[-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,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1] >;
 

C24:9D4 in GAP, Magma, Sage, TeX 

C_2^4\rtimes_9D_4
 
% in TeX
 
G:=Group("C2^4:9D4");
 
// GroupNames label
 
G:=SmallGroup(128,1345);
 
// by ID
 
G=gap.SmallGroup(128,1345);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,185]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=f^2=1,a*b=b*a,f*a*f=a*c=c*a,a*d=d*a,e*a*e^-1=a*c*d,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=e^-1>;
 
// generators/relations
 

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