Copied to
clipboard

## G = C2×C32⋊M4(2)  order 288 = 25·32

### Direct product of C2 and C32⋊M4(2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×C32⋊M4(2)
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C32⋊2C8 — C2×C32⋊2C8 — C2×C32⋊M4(2)
 Lower central C32 — C3×C6 — C2×C32⋊M4(2)
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C32⋊M4(2)
G = < a,b,c,d,e | a2=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe=b-1, dcd-1=b-1c-1, ece=c-1, ede=d5 >

Subgroups: 576 in 122 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C4 [×2], C4 [×2], C22, C22 [×4], S3 [×8], C6 [×6], C8 [×4], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×4], C12 [×4], D6 [×12], C2×C6 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C3⋊S3 [×2], C3×C6, C3×C6 [×2], C4×S3 [×8], C2×Dic3 [×2], C2×C12 [×2], C22×S3 [×2], C2×M4(2), C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, S3×C2×C4 [×2], C322C8 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C32⋊M4(2) [×4], C2×C322C8 [×2], C2×C4×C3⋊S3, C2×C32⋊M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, M4(2) [×2], C22×C4, C2×M4(2), C32⋊C4, C2×C32⋊C4 [×3], C32⋊M4(2) [×2], C22×C32⋊C4, C2×C32⋊M4(2)

Smallest permutation representation of C2×C32⋊M4(2)
On 48 points
Generators in S48
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 46)(34 47)(35 48)(36 41)(37 42)(38 43)(39 44)(40 45)
(1 46 28)(3 30 48)(5 42 32)(7 26 44)(9 37 24)(11 18 39)(13 33 20)(15 22 35)
(1 46 28)(2 47 29)(3 30 48)(4 31 41)(5 42 32)(6 43 25)(7 26 44)(8 27 45)(9 37 24)(10 38 17)(11 18 39)(12 19 40)(13 33 20)(14 34 21)(15 22 35)(16 23 36)
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)
(2 6)(4 8)(10 14)(12 16)(17 34)(18 39)(19 36)(20 33)(21 38)(22 35)(23 40)(24 37)(25 47)(26 44)(27 41)(28 46)(29 43)(30 48)(31 45)(32 42)

G:=sub<Sym(48)| (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,46)(34,47)(35,48)(36,41)(37,42)(38,43)(39,44)(40,45), (1,46,28)(3,30,48)(5,42,32)(7,26,44)(9,37,24)(11,18,39)(13,33,20)(15,22,35), (1,46,28)(2,47,29)(3,30,48)(4,31,41)(5,42,32)(6,43,25)(7,26,44)(8,27,45)(9,37,24)(10,38,17)(11,18,39)(12,19,40)(13,33,20)(14,34,21)(15,22,35)(16,23,36), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (2,6)(4,8)(10,14)(12,16)(17,34)(18,39)(19,36)(20,33)(21,38)(22,35)(23,40)(24,37)(25,47)(26,44)(27,41)(28,46)(29,43)(30,48)(31,45)(32,42)>;

G:=Group( (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,46)(34,47)(35,48)(36,41)(37,42)(38,43)(39,44)(40,45), (1,46,28)(3,30,48)(5,42,32)(7,26,44)(9,37,24)(11,18,39)(13,33,20)(15,22,35), (1,46,28)(2,47,29)(3,30,48)(4,31,41)(5,42,32)(6,43,25)(7,26,44)(8,27,45)(9,37,24)(10,38,17)(11,18,39)(12,19,40)(13,33,20)(14,34,21)(15,22,35)(16,23,36), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (2,6)(4,8)(10,14)(12,16)(17,34)(18,39)(19,36)(20,33)(21,38)(22,35)(23,40)(24,37)(25,47)(26,44)(27,41)(28,46)(29,43)(30,48)(31,45)(32,42) );

G=PermutationGroup([(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,46),(34,47),(35,48),(36,41),(37,42),(38,43),(39,44),(40,45)], [(1,46,28),(3,30,48),(5,42,32),(7,26,44),(9,37,24),(11,18,39),(13,33,20),(15,22,35)], [(1,46,28),(2,47,29),(3,30,48),(4,31,41),(5,42,32),(6,43,25),(7,26,44),(8,27,45),(9,37,24),(10,38,17),(11,18,39),(12,19,40),(13,33,20),(14,34,21),(15,22,35),(16,23,36)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48)], [(2,6),(4,8),(10,14),(12,16),(17,34),(18,39),(19,36),(20,33),(21,38),(22,35),(23,40),(24,37),(25,47),(26,44),(27,41),(28,46),(29,43),(30,48),(31,45),(32,42)])

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 6A ··· 6F 8A ··· 8H 12A ··· 12H order 1 2 2 2 2 2 3 3 4 4 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 1 1 18 18 4 4 2 2 9 9 9 9 4 ··· 4 18 ··· 18 4 ··· 4

36 irreducible representations

 dim 1 1 1 1 1 1 1 2 4 4 4 4 type + + + + + + + image C1 C2 C2 C2 C4 C4 C4 M4(2) C32⋊C4 C2×C32⋊C4 C2×C32⋊C4 C32⋊M4(2) kernel C2×C32⋊M4(2) C32⋊M4(2) C2×C32⋊2C8 C2×C4×C3⋊S3 C4×C3⋊S3 C6×C12 C22×C3⋊S3 C3×C6 C2×C4 C4 C22 C2 # reps 1 4 2 1 4 2 2 4 2 4 2 8

Matrix representation of C2×C32⋊M4(2) in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 0 0 72 72 0 0 0 0 1 0
,
 26 46 0 0 0 0 72 47 0 0 0 0 0 0 0 0 72 0 0 0 0 0 1 1 0 0 72 0 0 0 0 0 0 72 0 0
,
 1 0 0 0 0 0 56 72 0 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 72 72

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[26,72,0,0,0,0,46,47,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,72,1,0,0,0,0,0,1,0,0],[1,56,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;

C2×C32⋊M4(2) in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes M_4(2)
% in TeX

G:=Group("C2xC3^2:M4(2)");
// GroupNames label

G:=SmallGroup(288,930);
// by ID

G=gap.SmallGroup(288,930);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,422,100,80,9413,362,12550,1203]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e=b^-1,d*c*d^-1=b^-1*c^-1,e*c*e=c^-1,e*d*e=d^5>;
// generators/relations

׿
×
𝔽