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## G = C2×C24.4C4order 128 = 27

### Direct product of C2 and C24.4C4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C24.4C4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C24×C4 — C2×C24.4C4
 Lower central C1 — C22 — C2×C24.4C4
 Upper central C1 — C22×C4 — C2×C24.4C4
 Jennings C1 — C2 — C2 — C2×C4 — C2×C24.4C4

Generators and relations for C2×C24.4C4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=1, f4=e, 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: 812 in 504 conjugacy classes, 196 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×12], C4 [×8], C4 [×4], C22, C22 [×18], C22 [×68], C8 [×8], C2×C4 [×2], C2×C4 [×30], C2×C4 [×44], C23, C23 [×18], C23 [×68], C2×C8 [×8], C2×C8 [×8], M4(2) [×16], C22×C4 [×2], C22×C4 [×34], C22×C4 [×52], C24, C24 [×6], C24 [×12], C22⋊C8 [×16], C22×C8 [×4], C2×M4(2) [×8], C2×M4(2) [×8], C23×C4 [×2], C23×C4 [×12], C23×C4 [×8], C25, C2×C22⋊C8 [×4], C24.4C4 [×8], C22×M4(2) [×2], C24×C4, C2×C24.4C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], M4(2) [×8], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C2×M4(2) [×12], C23×C4, C22×D4 [×2], C24.4C4 [×4], C22×C22⋊C4, C22×M4(2) [×2], C2×C24.4C4

Smallest permutation representation of C2×C24.4C4
On 32 points
Generators in S32
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)
(2 28)(4 30)(6 32)(8 26)(9 22)(11 24)(13 18)(15 20)
(1 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 22)(10 19)(11 24)(12 21)(13 18)(14 23)(15 20)(16 17)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)
(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)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29), (2,28)(4,30)(6,32)(8,26)(9,22)(11,24)(13,18)(15,20), (1,31)(2,28)(3,25)(4,30)(5,27)(6,32)(7,29)(8,26)(9,22)(10,19)(11,24)(12,21)(13,18)(14,23)(15,20)(16,17), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (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( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29), (2,28)(4,30)(6,32)(8,26)(9,22)(11,24)(13,18)(15,20), (1,31)(2,28)(3,25)(4,30)(5,27)(6,32)(7,29)(8,26)(9,22)(10,19)(11,24)(12,21)(13,18)(14,23)(15,20)(16,17), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (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([(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,30),(10,31),(11,32),(12,25),(13,26),(14,27),(15,28),(16,29)], [(2,28),(4,30),(6,32),(8,26),(9,22),(11,24),(13,18),(15,20)], [(1,31),(2,28),(3,25),(4,30),(5,27),(6,32),(7,29),(8,26),(9,22),(10,19),(11,24),(12,21),(13,18),(14,23),(15,20),(16,17)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32)], [(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 ··· 2G 2H ··· 2S 4A ··· 4H 4I ··· 4T 8A ··· 8P order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 type + + + + + + image C1 C2 C2 C2 C2 C4 C4 D4 M4(2) kernel C2×C24.4C4 C2×C22⋊C8 C24.4C4 C22×M4(2) C24×C4 C23×C4 C25 C22×C4 C23 # reps 1 4 8 2 1 14 2 8 16

Matrix representation of C2×C24.4C4 in GL5(𝔽17)

 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 14 16
,
 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 14 16
,
 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 13 0 0 0 0 0 0 16 0 0 0 1 0 0 0 0 0 0 14 15 0 0 0 15 3

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,1,14,0,0,0,0,16],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,14,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[13,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,14,15,0,0,0,15,3] >;

C2×C24.4C4 in GAP, Magma, Sage, TeX

C_2\times C_2^4._4C_4
% in TeX

G:=Group("C2xC2^4.4C4");
// GroupNames label

G:=SmallGroup(128,1609);
// by ID

G=gap.SmallGroup(128,1609);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^2=1,f^4=e,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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