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G = C2×C243C4order 128 = 27

Direct product of C2 and C243C4

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

Aliases: C2×C243C4, C254C4, C26.1C2, C24.163D4, C25.82C22, C24.526C23, C23.159C24, C2417(C2×C4), (C23×C4)⋊2C22, (C22×C4)⋊7C23, C237(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), C223(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

C1C22 — C2×C243C4
C1C2C22C23C24C25C26 — C2×C243C4
C1C22 — C2×C243C4
C1C24 — C2×C243C4
C1C23 — C2×C243C4

Generators and relations for C2×C243C4
 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 [×15], C2 [×24], C4 [×8], C22, C22 [×58], C22 [×296], C2×C4 [×40], C23, C23 [×98], C23 [×648], C22⋊C4 [×48], C22×C4 [×8], C22×C4 [×24], C24, C24 [×58], C24 [×296], C2×C22⋊C4 [×24], C2×C22⋊C4 [×24], C23×C4 [×4], C25 [×15], C25 [×24], C243C4 [×8], C22×C22⋊C4 [×6], C26, C2×C243C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×24], C23 [×15], C22⋊C4 [×48], C22×C4 [×14], C2×D4 [×36], C24, C2×C22⋊C4 [×36], C22≀C2 [×16], C23×C4, C22×D4 [×6], C243C4 [×8], C22×C22⋊C4 [×3], C2×C22≀C2 [×4], C2×C243C4

Smallest permutation representation of C2×C243C4
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×C243C4C243C4C22×C22⋊C4C26C25C24
# reps18611624

Matrix representation of C2×C243C4 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×C243C4 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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