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G = C23:3D8  order 128 = 27

2nd semidirect product of C23 and D8 acting via D8/C4=C22

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

Aliases: C23:3D8, C24.119D4, C4.12+ 1+4, C8:7D4:3C2, C22:D8:4C2, (C2xD8):2C22, C2.D8:3C22, (C22xC8):9C22, C22.23(C2xD8), C2.14(C22xD8), D4:C4:1C22, C4:C4.125C23, C4:D4:56C22, C22:C8:53C22, (C2xC4).384C24, (C2xC8).150C23, (C22xC4).482D4, C23.399(C2xD4), C22.D8:4C2, (C2xD4).137C23, (C22xD4):23C22, C2.65(C23:3D4), (C23xC4).564C22, C22.644(C22xD4), C2.47(D8:C22), (C22xC4).1062C23, (C2xC4:D4):49C2, (C2xC4:C4):50C22, (C2xC22:C8):22C2, (C2xC4).525(C2xD4), SmallGroup(128,1918)

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

Derived series
C1 — C2xC4 — C23:3D8
Chief series
C1 — C2 — C4 — C2xC4 — C22xC4 — C22xD4 — C2xC4:D4 — C23:3D8
Lower central
C1 — C2 — C2xC4 — C23:3D8
Upper central
C1 — C22 — C23xC4 — C23:3D8
Jennings
C1 — C2 — C2 — C2xC4 — C23:3D8

Generators and relations for C23:3D8
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, eae=ac=ca, ad=da, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 660 in 260 conjugacy classes, 94 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, C23, C23, C23, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, D8, C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C24, C24, C22:C8, D4:C4, C2.D8, C2xC22:C4, C2xC4:C4, C4:D4, C4:D4, C22xC8, C2xD8, C23xC4, C22xD4, C22xD4, C2xC22:C8, C22:D8, C8:7D4, C22.D8, C2xC4:D4, C23:3D8
Quotients: C1, C2, C22, D4, C23, D8, C2xD4, C24, C2xD8, C22xD4, 2+ 1+4, C23:3D4, C22xD8, D8:C22, C23:3D8

Smallest permutation representation of C23:3D8
►On 32 points
Generators in S32
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M4A4B4C4D4E4F4G4H4I4J8A···8H
order12222···2222244444444448···8
size11112···2888822224488884···4

32 irreducible representations

dim11111122244
type++++++++++
imageC1C2C2C2C2C2D4D4D82+ 1+4D8:C22
kernelC23:3D8C2xC22:C8C22:D8C8:7D4C22.D8C2xC4:D4C22xC4C24C23C4C2
# reps11444231822

Matrix representation of C23:3D8 ►in GL6(F17)

1600000
0160000
0001300
004000
000004
0000130
,
100000
010000
0016000
0001600
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
3140000
330000
000001
000010
001000
0001600
,
330000
3140000
000010
000001
001000
000100

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,1,0,0,0],[3,3,0,0,0,0,3,14,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C23:3D8 in GAP, Magma, Sage, TeX 

C_2^3\rtimes_3D_8
 
% in TeX
 
G:=Group("C2^3:3D8");
 
// GroupNames label
 
G:=SmallGroup(128,1918);
 
// by ID
 
G=gap.SmallGroup(128,1918);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,219,675,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,e*a*e=a*c=c*a,a*d=d*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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