Copied to
clipboard

## G = C4×C23⋊C4order 128 = 27

### Direct product of C4 and C23⋊C4

direct product, p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C4×C23⋊C4
G = < a,b,c,d,e | a4=b2=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >

Subgroups: 436 in 206 conjugacy classes, 78 normal (26 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×15], C22 [×3], C22 [×4], C22 [×18], C2×C4 [×2], C2×C4 [×4], C2×C4 [×35], D4 [×8], C23, C23 [×8], C23 [×6], C42 [×6], C22⋊C4 [×8], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×5], C22×C4 [×4], C22×C4 [×10], C2×D4 [×4], C2×D4 [×4], C24 [×2], C2.C42 [×2], C23⋊C4 [×8], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4, C4×D4 [×4], C23×C4 [×2], C22×D4, C23.9D4 [×2], C4×C22⋊C4 [×2], C2×C23⋊C4 [×2], C2×C4×D4, C4×C23⋊C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×4], C23, C42 [×4], C22⋊C4 [×4], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], C23⋊C4 [×2], C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], C4×C22⋊C4, C2×C23⋊C4, C23.C23, C4×C23⋊C4

Smallest permutation representation of C4×C23⋊C4
On 32 points
Generators in S32
(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)
(1 30)(2 31)(3 32)(4 29)(5 22)(6 23)(7 24)(8 21)(9 20)(10 17)(11 18)(12 19)(13 28)(14 25)(15 26)(16 27)
(1 22)(2 23)(3 24)(4 21)(5 30)(6 31)(7 32)(8 29)(9 15)(10 16)(11 13)(12 14)(17 27)(18 28)(19 25)(20 26)
(1 20)(2 17)(3 18)(4 19)(5 15)(6 16)(7 13)(8 14)(9 30)(10 31)(11 32)(12 29)(21 25)(22 26)(23 27)(24 28)
(1 12 24 6)(2 9 21 7)(3 10 22 8)(4 11 23 5)(13 17 30 25)(14 18 31 26)(15 19 32 27)(16 20 29 28)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 4A 4B 4C 4D 4E ··· 4J 4K ··· 4AF order 1 2 2 2 2 ··· 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 size 1 1 1 1 2 ··· 2 4 4 1 1 1 1 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C4 C4 D4 C4○D4 C23⋊C4 C23.C23 kernel C4×C23⋊C4 C23.9D4 C4×C22⋊C4 C2×C23⋊C4 C2×C4×D4 C23⋊C4 C2×C42 C2×C22⋊C4 C2×C4⋊C4 C23×C4 C22×C4 C23 C4 C2 # reps 1 2 2 2 1 16 2 2 2 2 4 4 2 2

Matrix representation of C4×C23⋊C4 in GL6(𝔽5)

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

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

C4×C23⋊C4 in GAP, Magma, Sage, TeX

C_4\times C_2^3\rtimes C_4
% in TeX

G:=Group("C4xC2^3:C4");
// GroupNames label

G:=SmallGroup(128,486);
// by ID

G=gap.SmallGroup(128,486);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,100,1018,4037]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^2=c^2=d^2=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations

׿
×
𝔽