Copied to
clipboard

## G = C2×M4(2)⋊4C4order 128 = 27

### Direct product of C2 and M4(2)⋊4C4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×M4(2)⋊4C4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×M4(2) — C2×M4(2)⋊4C4
 Lower central C1 — C2 — C22 — C2×M4(2)⋊4C4
 Upper central C1 — C2×C4 — C23×C4 — C2×M4(2)⋊4C4
 Jennings C1 — C2 — C2 — C22×C4 — C2×M4(2)⋊4C4

Generators and relations for C2×M4(2)⋊4C4
G = < a,b,c,d | a2=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b5, dbd-1=bc, dcd-1=b4c >

Subgroups: 308 in 194 conjugacy classes, 108 normal (72 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×8], C4 [×4], C22 [×7], C22 [×10], C8 [×8], C2×C4 [×28], C2×C4 [×8], C23 [×7], C23 [×2], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×10], M4(2) [×4], M4(2) [×14], C22×C4 [×14], C22×C4 [×2], C24, C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C42⋊C2 [×4], C42⋊C2 [×2], C22×C8 [×2], C22×C8, C2×M4(2) [×10], C2×M4(2) [×7], C23×C4, M4(2)⋊4C4 [×4], C2×C42⋊C2, C22×M4(2) [×2], C2×M4(2)⋊4C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C2.C42 [×8], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], M4(2)⋊4C4 [×2], C2×C2.C42, C2×M4(2)⋊4C4

Smallest permutation representation of C2×M4(2)⋊4C4
On 32 points
Generators in S32
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)
(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 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 23)(10 20)(11 17)(12 22)(13 19)(14 24)(15 21)(16 18)
(2 32 6 28)(3 7)(4 30 8 26)(9 13)(10 20 14 24)(12 18 16 22)(17 21)(27 31)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4J 4K ··· 4R 8A ··· 8P order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 type + + + + + - image C1 C2 C2 C2 C4 C4 C4 C4 D4 Q8 M4(2)⋊4C4 kernel C2×M4(2)⋊4C4 M4(2)⋊4C4 C2×C42⋊C2 C22×M4(2) C2×C22⋊C4 C42⋊C2 C22×C8 C2×M4(2) C22×C4 C22×C4 C2 # reps 1 4 1 2 4 4 4 12 6 2 4

Matrix representation of C2×M4(2)⋊4C4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 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 16 0 0 0 0 1 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 16 0 0 0 0 1 0 0 0
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 13 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 1 0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,13,0,0,0,0,0,0,4,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0] >;

C2×M4(2)⋊4C4 in GAP, Magma, Sage, TeX

C_2\times M_4(2)\rtimes_4C_4
% in TeX

G:=Group("C2xM4(2):4C4");
// GroupNames label

G:=SmallGroup(128,475);
// by ID

G=gap.SmallGroup(128,475);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,2019,1411,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^5,d*b*d^-1=b*c,d*c*d^-1=b^4*c>;
// generators/relations

׿
×
𝔽