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## G = C2×C24⋊3C4order 128 = 27

### Direct product of C2 and C24⋊3C4

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

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×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 ··· 2O 2P ··· 2AM 4A ··· 4P order 1 2 ··· 2 2 ··· 2 4 ··· 4 size 1 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 2 type + + + + + image C1 C2 C2 C2 C4 D4 kernel C2×C24⋊3C4 C24⋊3C4 C22×C22⋊C4 C26 C25 C24 # reps 1 8 6 1 16 24

Matrix representation of C2×C243C4 in GL6(𝔽5)

 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 0 0 0 0 0 2 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 4 0 0 0 0 1 0 0 0 0 0 0 0 3 2 0 0 0 0 0 2

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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