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G = C3×C4○D24  order 288 = 25·32

Direct product of C3 and C4○D24

direct product, metabelian, supersoluble, monomial

Aliases: C3×C4○D24, D24⋊7C6, C24.82D6, Dic12⋊7C6, C12.95D12, C62.87D4, (C6×C24)⋊9C2, (C2×C24)⋊6C6, (C2×C24)⋊9S3, C24⋊C2⋊7C6, C4○D12⋊1C6, C8.17(S3×C6), C6.11(C6×D4), (C3×D24)⋊15C2, C24.19(C2×C6), D12.7(C2×C6), C12.35(C3×D4), C6.99(C2×D12), (C2×C6).19D12, C2.13(C6×D12), C4.20(C3×D12), (C3×C12).140D4, (C2×C12).443D6, C32⋊16(C4○D8), Dic6.6(C2×C6), C22.1(C3×D12), (C3×Dic12)⋊15C2, C12.30(C22×C6), (C3×C24).57C22, (C3×C12).162C23, C12.217(C22×S3), (C6×C12).314C22, (C3×D12).46C22, (C3×Dic6).46C22, (C2×C8)⋊4(C3×S3), C3⋊1(C3×C4○D8), C4.28(S3×C2×C6), (C3×C4○D12)⋊5C2, (C2×C4).80(S3×C6), (C2×C6).22(C3×D4), (C3×C24⋊C2)⋊15C2, (C3×C6).181(C2×D4), (C2×C12).115(C2×C6), SmallGroup(288,675)

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 2A2B2C2D3A3B3C3D3E4A4B4C4D4E6A6B6C···6M6N6O6P6Q8A8B8C8D12A12B12C12D12E···12R12S12T12U12V24A···24AF
order122223333344444666···6666688881212121212···121212121224···24
size1121212112221121212112···212121212222211112···2121212122···2

90 irreducible representations

dim111111111111222222222222222222
type+++++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D4D4D6D6C3×S3D12C3×D4D12C3×D4C4○D8S3×C6S3×C6C3×D12C3×D12C4○D24C3×C4○D8C3×C4○D24
kernelC3×C4○D24C3×C24⋊C2C3×D24C3×Dic12C6×C24C3×C4○D12C4○D24C24⋊C2D24Dic12C2×C24C4○D12C2×C24C3×C12C62C24C2×C12C2×C8C12C12C2×C6C2×C6C32C8C2×C4C4C22C3C3C1
# reps1211122422241112122222442448816

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

640
064
,
460
046
,
430
017
,
017
430
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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