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G = C24.115D4  order 128 = 27

70th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.115D4, C2.D8:55C22, C4.Q8:64C22, C4:C4.390C23, (C2xC8).143C23, (C2xC4).289C24, (C2xD4).78C23, C4o(C22.D8), C23.241(C2xD4), (C22xC4).440D4, (C2xQ8).66C23, D4:C4:78C22, Q8:C4:82C22, C4o(C23.46D4), C4o(C23.47D4), C4o(C23.19D4), C4o(C23.48D4), C4o(C23.20D4), C22.28(C4oD8), C23.25D4:6C2, C22.D8:37C2, C23.48D4:37C2, C23.20D4:52C2, C23.46D4:35C2, C23.47D4:35C2, C23.24D4:16C2, C23.19D4:51C2, C4:D4.154C22, C22:C8.172C22, (C22xC8).183C22, (C23xC4).559C22, C22.549(C22xD4), C22:Q8.159C22, C2.21(D8:C22), C22.19C24.17C2, (C22xC4).1547C23, C42:C2.122C22, C4.144(C22.D4), C22.17(C22.D4), C2.22(C2xC4oD8), C4.99(C2xC4oD4), (C2xC22:C8):27C2, (C2xC4).485(C2xD4), (C2xC42:C2):45C2, (C2xC4).847(C4oD4), (C2xC4:C4).926C22, (C2xC4)o(C22.D8), (C2xC4oD4).136C22, (C2xC4)o(C23.47D4), (C2xC4)o(C23.46D4), (C2xC4)o(C23.20D4), (C2xC4)o(C23.19D4), C2.54(C2xC22.D4), SmallGroup(128,1823)

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

Derived series
C1 — C2xC4 — C24.115D4
Chief series
C1 — C2 — C4 — C2xC4 — C22xC4 — C42:C2 — C2xC42:C2 — C24.115D4
Lower central
C1 — C2 — C2xC4 — C24.115D4
Upper central
C1 — C2xC4 — C23xC4 — C24.115D4
Jennings
C1 — C2 — C2 — C2xC4 — C24.115D4

Generators and relations for C24.115D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, eae-1=faf=ac=ca, ad=da, bc=cb, fbf=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=cde3 >

Subgroups: 388 in 210 conjugacy classes, 94 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C42, C22:C4, C4:C4, C4:C4, C4:C4, C2xC8, C2xC8, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, C22:C8, C22:C8, D4:C4, Q8:C4, C4.Q8, C2.D8, C2xC42, C2xC22:C4, C2xC4:C4, C42:C2, C42:C2, C42:C2, C4xD4, C22wrC2, C4:D4, C22:Q8, C22.D4, C22xC8, C23xC4, C2xC4oD4, C2xC22:C8, C23.24D4, C23.25D4, C22.D8, C23.46D4, C23.19D4, C23.47D4, C23.48D4, C23.20D4, C2xC42:C2, C22.19C24, C24.115D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C4oD4, C24, C22.D4, C4oD8, C22xD4, C2xC4oD4, C2xC22.D4, C2xC4oD8, D8:C22, C24.115D4

Smallest permutation representation of C24.115D4
►On 32 points
Generators in S32
(2 27)(4 29)(6 31)(8 25)(9 22)(11 24)(13 18)(15 20)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I···4Q4R4S4T8A···8H
order1222222222444444444···44448···8
size1111222248111122224···48884···4

38 irreducible representations

dim11111111111122224
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4oD4C4oD8D8:C22
kernelC24.115D4C2xC22:C8C23.24D4C23.25D4C22.D8C23.46D4C23.19D4C23.47D4C23.48D4C23.20D4C2xC42:C2C22.19C24C22xC4C24C2xC4C22C2
# reps11221121121131882

Matrix representation of C24.115D4 ►in GL4(F17) generated by

1000
0100
0010
001616
,
1000
01600
00160
00016
,
1000
0100
00160
00016
,
16000
01600
0010
0001
,
15000
0800
00139
0044
,
0800
15000
001615
0001
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,16,0,0,0,16],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[15,0,0,0,0,8,0,0,0,0,13,4,0,0,9,4],[0,15,0,0,8,0,0,0,0,0,16,0,0,0,15,1] >;
 

C24.115D4 in GAP, Magma, Sage, TeX 

C_2^4._{115}D_4
 
% in TeX
 
G:=Group("C2^4.115D4");
 
// GroupNames label
 
G:=SmallGroup(128,1823);
 
// by ID
 
G=gap.SmallGroup(128,1823);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,100,248,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=f^2=1,e^4=d,a*b=b*a,e*a*e^-1=f*a*f=a*c=c*a,a*d=d*a,b*c=c*b,f*b*f=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=c*d*e^3>;
 
// generators/relations
 

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