Copied to
clipboard

## G = C8×C32⋊C4order 288 = 25·32

### Direct product of C8 and C32⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C8×C32⋊C4
 Chief series C1 — C32 — C3×C6 — C2×C3⋊S3 — C4×C3⋊S3 — C4×C32⋊C4 — C8×C32⋊C4
 Lower central C32 — C8×C32⋊C4
 Upper central C1 — C8

Generators and relations for C8×C32⋊C4
G = < a,b,c,d | a8=b3=c3=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >

Subgroups: 288 in 66 conjugacy classes, 26 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, C32, Dic3, C12, D6, C42, C2×C8, C3⋊S3, C3×C6, C3⋊C8, C24, C4×S3, C4×C8, C3⋊Dic3, C3×C12, C32⋊C4, C2×C3⋊S3, S3×C8, C324C8, C3×C24, C322C8, C4×C3⋊S3, C2×C32⋊C4, C8×C3⋊S3, C3⋊S33C8, C4×C32⋊C4, C8×C32⋊C4
Quotients: C1, C2, C4, C22, C8, C2×C4, C42, C2×C8, C4×C8, C32⋊C4, C2×C32⋊C4, C4×C32⋊C4, C8×C32⋊C4

Smallest permutation representation of C8×C32⋊C4
On 48 points
Generators in S48
(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)
(1 30 35)(2 31 36)(3 32 37)(4 25 38)(5 26 39)(6 27 40)(7 28 33)(8 29 34)(9 41 20)(10 42 21)(11 43 22)(12 44 23)(13 45 24)(14 46 17)(15 47 18)(16 48 19)
(9 20 41)(10 21 42)(11 22 43)(12 23 44)(13 24 45)(14 17 46)(15 18 47)(16 19 48)
(1 10 5 14)(2 11 6 15)(3 12 7 16)(4 13 8 9)(17 35 42 26)(18 36 43 27)(19 37 44 28)(20 38 45 29)(21 39 46 30)(22 40 47 31)(23 33 48 32)(24 34 41 25)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C ··· 4L 6A 6B 8A 8B 8C 8D 8E ··· 8P 12A 12B 12C 12D 24A ··· 24H order 1 2 2 2 3 3 4 4 4 ··· 4 6 6 8 8 8 8 8 ··· 8 12 12 12 12 24 ··· 24 size 1 1 9 9 4 4 1 1 9 ··· 9 4 4 1 1 1 1 9 ··· 9 4 4 4 4 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + + + image C1 C2 C2 C2 C4 C4 C4 C4 C8 C32⋊C4 C2×C32⋊C4 C4×C32⋊C4 C8×C32⋊C4 kernel C8×C32⋊C4 C8×C3⋊S3 C3⋊S3⋊3C8 C4×C32⋊C4 C32⋊4C8 C3×C24 C32⋊2C8 C2×C32⋊C4 C32⋊C4 C8 C4 C2 C1 # reps 1 1 1 1 2 2 4 4 16 2 2 4 8

Matrix representation of C8×C32⋊C4 in GL4(𝔽73) generated by

 22 0 0 0 0 22 0 0 0 0 22 0 0 0 0 22
,
 0 72 0 0 1 72 0 0 0 0 0 72 0 0 1 72
,
 1 0 0 0 0 1 0 0 0 0 72 1 0 0 72 0
,
 0 0 72 0 0 0 0 72 1 72 0 0 0 72 0 0
G:=sub<GL(4,GF(73))| [22,0,0,0,0,22,0,0,0,0,22,0,0,0,0,22],[0,1,0,0,72,72,0,0,0,0,0,1,0,0,72,72],[1,0,0,0,0,1,0,0,0,0,72,72,0,0,1,0],[0,0,1,0,0,0,72,72,72,0,0,0,0,72,0,0] >;

C8×C32⋊C4 in GAP, Magma, Sage, TeX

C_8\times C_3^2\rtimes C_4
% in TeX

G:=Group("C8xC3^2:C4");
// GroupNames label

G:=SmallGroup(288,414);
// by ID

G=gap.SmallGroup(288,414);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,64,80,9413,691,12550,2372]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^3=c^3=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations

׿
×
𝔽