Copied to
clipboard

## G = C3×C4○D24order 288 = 25·32

### Direct product of C3 and C4○D24

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×C4○D24
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C3×D12 — C3×C4○D12 — C3×C4○D24
 Lower central C3 — C6 — C12 — C3×C4○D24
 Upper central C1 — C12 — C2×C12 — C2×C24

Generators and relations for C3×C4○D24
G = < a,b,c,d | a3=b4=d2=1, c12=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c11 >

Subgroups: 346 in 135 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C3×S3, C3×C6, C3×C6, C24, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C4○D8, C3×Dic3, C3×C12, S3×C6, C62, C24⋊C2, D24, Dic12, C2×C24, C2×C24, C3×D8, C3×SD16, C3×Q16, C4○D12, C3×C4○D4, C3×C24, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, C4○D24, C3×C4○D8, C3×C24⋊C2, C3×D24, C3×Dic12, C6×C24, C3×C4○D12, C3×C4○D24
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, D12, C3×D4, C22×S3, C22×C6, C4○D8, S3×C6, C2×D12, C6×D4, C3×D12, S3×C2×C6, C4○D24, C3×C4○D8, C6×D12, C3×C4○D24

Smallest permutation representation of C3×C4○D24
On 48 points
Generators in S48
(1 9 17)(2 10 18)(3 11 19)(4 12 20)(5 13 21)(6 14 22)(7 15 23)(8 16 24)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)
(1 19 13 7)(2 20 14 8)(3 21 15 9)(4 22 16 10)(5 23 17 11)(6 24 18 12)(25 31 37 43)(26 32 38 44)(27 33 39 45)(28 34 40 46)(29 35 41 47)(30 36 42 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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 48)(11 47)(12 46)(13 45)(14 44)(15 43)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 36)(23 35)(24 34)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 6A 6B 6C ··· 6M 6N 6O 6P 6Q 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12R 12S 12T 12U 12V 24A ··· 24AF order 1 2 2 2 2 3 3 3 3 3 4 4 4 4 4 6 6 6 ··· 6 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 2 12 12 1 1 2 2 2 1 1 2 12 12 1 1 2 ··· 2 12 12 12 12 2 2 2 2 1 1 1 1 2 ··· 2 12 12 12 12 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D4 D4 D6 D6 C3×S3 D12 C3×D4 D12 C3×D4 C4○D8 S3×C6 S3×C6 C3×D12 C3×D12 C4○D24 C3×C4○D8 C3×C4○D24 kernel C3×C4○D24 C3×C24⋊C2 C3×D24 C3×Dic12 C6×C24 C3×C4○D12 C4○D24 C24⋊C2 D24 Dic12 C2×C24 C4○D12 C2×C24 C3×C12 C62 C24 C2×C12 C2×C8 C12 C12 C2×C6 C2×C6 C32 C8 C2×C4 C4 C22 C3 C3 C1 # reps 1 2 1 1 1 2 2 4 2 2 2 4 1 1 1 2 1 2 2 2 2 2 4 4 2 4 4 8 8 16

Matrix representation of C3×C4○D24 in GL2(𝔽73) generated by

 64 0 0 64
,
 46 0 0 46
,
 43 0 0 17
,
 0 17 43 0
G:=sub<GL(2,GF(73))| [64,0,0,64],[46,0,0,46],[43,0,0,17],[0,43,17,0] >;

C3×C4○D24 in GAP, Magma, Sage, TeX

C_3\times C_4\circ D_{24}
% in TeX

G:=Group("C3xC4oD24");
// GroupNames label

G:=SmallGroup(288,675);
// by ID

G=gap.SmallGroup(288,675);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,590,142,2524,102,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=d^2=1,c^12=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^11>;
// generators/relations

׿
×
𝔽