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G = C2×C24⋊3C4  order 128 = 27

Direct product of C2 and C24⋊3C4

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

Aliases: C2×C24⋊3C4, C25⋊4C4, C26.1C2, C24.163D4, C25.82C22, C24.526C23, C23.159C24, C24⋊17(C2×C4), (C23×C4)⋊2C22, (C22×C4)⋊7C23, C23⋊7(C22⋊C4), C23.596(C2×D4), C22.50(C23×C4), C22.59(C22×D4), C22.105C22≀C2, C23.205(C22×C4), C2.1(C2×C22≀C2), C22⋊3(C2×C22⋊C4), (C22×C22⋊C4)⋊3C2, C2.4(C22×C22⋊C4), (C2×C22⋊C4)⋊68C22, SmallGroup(128,1009)

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

Derived series
C1 — C22 — C2×C24⋊3C4
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C26 — C2×C24⋊3C4
Lower central
C1 — C22 — C2×C24⋊3C4
Upper central
C1 — C24 — C2×C24⋊3C4
Jennings
C1 — C23 — C2×C24⋊3C4

Generators and relations for C2×C24⋊3C4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, fcf-1=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 3212 in 1732 conjugacy classes, 300 normal (6 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C23, C23, C23, C22⋊C4, C22×C4, C22×C4, C24, C24, C24, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C25, C25, C24⋊3C4, C22×C22⋊C4, C26, C2×C24⋊3C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C24, C2×C22⋊C4, C22≀C2, C23×C4, C22×D4, C24⋊3C4, C22×C22⋊C4, C2×C22≀C2, C2×C24⋊3C4

Smallest permutation representation of C2×C24⋊3C4
►On 32 points
Generators in S32
(1 5)(2 6)(3 7)(4 8)(9 21)(10 22)(11 23)(12 24)(13 32)(14 29)(15 30)(16 31)(17 27)(18 28)(19 25)(20 26)

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

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

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

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

(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)| (1,5)(2,6)(3,7)(4,8)(9,21)(10,22)(11,23)(12,24)(13,32)(14,29)(15,30)(16,31)(17,27)(18,28)(19,25)(20,26), (1,31)(2,26)(3,29)(4,28)(5,16)(6,20)(7,14)(8,18)(9,25)(10,32)(11,27)(12,30)(13,22)(15,24)(17,23)(19,21), (1,9)(2,26)(3,11)(4,28)(5,21)(6,20)(7,23)(8,18)(10,32)(12,30)(13,22)(14,17)(15,24)(16,19)(25,31)(27,29), (1,9)(2,10)(3,11)(4,12)(5,21)(6,22)(7,23)(8,24)(13,20)(14,17)(15,18)(16,19)(25,31)(26,32)(27,29)(28,30), (1,31)(2,32)(3,29)(4,30)(5,16)(6,13)(7,14)(8,15)(9,25)(10,26)(11,27)(12,28)(17,23)(18,24)(19,21)(20,22), (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( (1,5)(2,6)(3,7)(4,8)(9,21)(10,22)(11,23)(12,24)(13,32)(14,29)(15,30)(16,31)(17,27)(18,28)(19,25)(20,26), (1,31)(2,26)(3,29)(4,28)(5,16)(6,20)(7,14)(8,18)(9,25)(10,32)(11,27)(12,30)(13,22)(15,24)(17,23)(19,21), (1,9)(2,26)(3,11)(4,28)(5,21)(6,20)(7,23)(8,18)(10,32)(12,30)(13,22)(14,17)(15,24)(16,19)(25,31)(27,29), (1,9)(2,10)(3,11)(4,12)(5,21)(6,22)(7,23)(8,24)(13,20)(14,17)(15,18)(16,19)(25,31)(26,32)(27,29)(28,30), (1,31)(2,32)(3,29)(4,30)(5,16)(6,13)(7,14)(8,15)(9,25)(10,26)(11,27)(12,28)(17,23)(18,24)(19,21)(20,22), (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([[(1,5),(2,6),(3,7),(4,8),(9,21),(10,22),(11,23),(12,24),(13,32),(14,29),(15,30),(16,31),(17,27),(18,28),(19,25),(20,26)], [(1,31),(2,26),(3,29),(4,28),(5,16),(6,20),(7,14),(8,18),(9,25),(10,32),(11,27),(12,30),(13,22),(15,24),(17,23),(19,21)], [(1,9),(2,26),(3,11),(4,28),(5,21),(6,20),(7,23),(8,18),(10,32),(12,30),(13,22),(14,17),(15,24),(16,19),(25,31),(27,29)], [(1,9),(2,10),(3,11),(4,12),(5,21),(6,22),(7,23),(8,24),(13,20),(14,17),(15,18),(16,19),(25,31),(26,32),(27,29),(28,30)], [(1,31),(2,32),(3,29),(4,30),(5,16),(6,13),(7,14),(8,15),(9,25),(10,26),(11,27),(12,28),(17,23),(18,24),(19,21),(20,22)], [(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···2O2P···2AM4A···4P
order12···22···24···4
size11···12···24···4

56 irreducible representations

dim111112
type+++++
imageC1C2C2C2C4D4
kernelC2×C24⋊3C4C24⋊3C4C22×C22⋊C4C26C25C24
# reps18611624

Matrix representation of C2×C24⋊3C4 ►in GL6(𝔽5)

400000
040000
004000
000400
000040
000004
,
400000
040000
004000
000400
000010
000024
,
400000
040000
001000
000400
000040
000004
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
200000
010000
000400
001000
000032
000002

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,2,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,0,0,0,2,2] >;
 

C2×C24⋊3C4 in GAP, Magma, Sage, TeX 

C_2\times C_2^4\rtimes_3C_4
 
% in TeX
 
G:=Group("C2xC2^4:3C4");
 
// GroupNames label
 
G:=SmallGroup(128,1009);
 
// by ID
 
G=gap.SmallGroup(128,1009);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^2=f^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,f*c*f^-1=c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
 
// generators/relations
 

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