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G = C24:3C8  order 128 = 27

1st semidirect product of C24 and C8 acting via C8/C4=C2

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

Aliases: C24:3C8, C25.6C4, C23.30M4(2), (C24xC4).2C2, (C23xC4).29C4, C23.31(C2xC8), (C22xC8):1C22, C4.110C22wrC2, C22:2(C22:C8), C24.111(C2xC4), (C22xC4).674D4, C2.2(C24:3C4), C22.33(C22xC8), C2.3(C24.4C4), C23.262(C22xC4), (C23xC4).632C22, C22.44(C2xM4(2)), C23.192(C22:C4), (C22xC4).1615C23, (C2xC22:C8):9C2, C2.16(C2xC22:C8), (C2xC4).1511(C2xD4), (C22xC4).440(C2xC4), (C2xC4).329(C22:C4), C22.123(C2xC22:C4), SmallGroup(128,511)

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

Derived series
C1 — C22 — C24:3C8
Chief series
C1 — C2 — C4 — C2xC4 — C22xC4 — C23xC4 — C24xC4 — C24:3C8
Lower central
C1 — C22 — C24:3C8
Upper central
C1 — C22xC4 — C24:3C8
Jennings
C1 — C2 — C2 — C22xC4 — C24:3C8

Generators and relations for C24:3C8
 G = < a,b,c,d,e | a2=b2=c2=d2=e8=1, ab=ba, eae-1=ac=ca, ad=da, bc=cb, ebe-1=bd=db, cd=dc, ce=ec, de=ed >

Subgroups: 756 in 426 conjugacy classes, 104 normal (8 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C23, C23, C23, C2xC8, C22xC4, C22xC4, C22xC4, C24, C24, C22:C8, C22xC8, C23xC4, C23xC4, C25, C2xC22:C8, C24xC4, C24:3C8
Quotients: C1, C2, C4, C22, C8, C2xC4, D4, C23, C22:C4, C2xC8, M4(2), C22xC4, C2xD4, C22:C8, C2xC22:C4, C22wrC2, C22xC8, C2xM4(2), C24:3C4, C2xC22:C8, C24.4C4, C24:3C8

Smallest permutation representation of C24:3C8
►On 32 points
Generators in S32
(1 23)(2 28)(3 17)(4 30)(5 19)(6 32)(7 21)(8 26)(9 24)(10 29)(11 18)(12 31)(13 20)(14 25)(15 22)(16 27)

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

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

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

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

56 conjugacy classes

class 1 2A···2G2H···2S4A···4H4I···4T8A···8P
order12···22···24···44···48···8
size11···12···21···12···24···4

56 irreducible representations

dim11111122
type++++
imageC1C2C2C4C4C8D4M4(2)
kernelC24:3C8C2xC22:C8C24xC4C23xC4C25C24C22xC4C23
# reps16162161212

Matrix representation of C24:3C8 ►in GL6(F17)

100000
0160000
0016000
000100
000010
0000016
,
1600000
0160000
0016000
0001600
0000160
000001
,
1600000
0160000
0016000
0001600
0000160
0000016
,
100000
010000
001000
000100
0000160
0000016
,
010000
1300000
000100
001000
000001
0000160

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

C24:3C8 in GAP, Magma, Sage, TeX 

C_2^4\rtimes_3C_8
 
% in TeX
 
G:=Group("C2^4:3C8");
 
// GroupNames label
 
G:=SmallGroup(128,511);
 
// by ID
 
G=gap.SmallGroup(128,511);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^8=1,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*c=c*b,e*b*e^-1=b*d=d*b,c*d=d*c,c*e=e*c,d*e=e*d>;
 
// generators/relations
 

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