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## 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 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6 [×2], C6 [×7], C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×2], C2×C6 [×2], C2×C6 [×3], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3×S3 [×2], C3×C6, C3×C6, C24 [×4], C24 [×2], Dic6 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×3], C3×D4 [×4], C3×Q8 [×2], C4○D8, C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×2], C62, C24⋊C2 [×2], D24, Dic12, C2×C24 [×2], C2×C24, C3×D8, C3×SD16 [×2], C3×Q16, C4○D12 [×2], C3×C4○D4 [×2], C3×C24 [×2], C3×Dic6 [×2], S3×C12 [×2], C3×D12 [×2], C3×C3⋊D4 [×2], C6×C12, C4○D24, C3×C4○D8, C3×C24⋊C2 [×2], C3×D24, C3×Dic12, C6×C24, C3×C4○D12 [×2], C3×C4○D24
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C3×S3, D12 [×2], C3×D4 [×2], C22×S3, C22×C6, C4○D8, S3×C6 [×3], C2×D12, C6×D4, C3×D12 [×2], 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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)

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

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

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

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

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