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G = C3×C3⋊D24  order 432 = 24·33

Direct product of C3 and C3⋊D24

direct product, metabelian, supersoluble, monomial

Aliases: C3×C3⋊D24, C33⋊5D8, C32⋊9D24, C12.74S32, C3⋊2(C3×D24), (C3×D12)⋊2C6, D12⋊1(C3×S3), C12⋊S3⋊8C6, C32⋊5(C3×D8), (C3×D12)⋊10S3, C12.26(S3×C6), C6.24(C3×D12), (C3×C6).71D12, C32⋊8(D4⋊S3), (C3×C12).172D6, (C32×D12)⋊2C2, (C32×C6).19D4, C6.42(C3⋊D12), (C32×C12).2C22, (C3×C3⋊C8)⋊1C6, (C3×C3⋊C8)⋊4S3, C3⋊C8⋊1(C3×S3), C4.1(C3×S32), C3⋊1(C3×D4⋊S3), (C32×C3⋊C8)⋊2C2, C6.1(C3×C3⋊D4), (C3×C6).18(C3×D4), (C3×C12⋊S3)⋊1C2, (C3×C12).36(C2×C6), C2.4(C3×C3⋊D12), (C3×C6).70(C3⋊D4), SmallGroup(432,419)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — C3×C3⋊D24
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C32×C12 — C32×D12 — C3×C3⋊D24
Lower central
C32 — C3×C6 — C3×C12 — C3×C3⋊D24
Upper central
C1 — C6 — C12

Generators and relations for C3×C3⋊D24
 G = < a,b,c,d | a3=b3=c24=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 616 in 134 conjugacy classes, 36 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, C32, C32, C12, C12, D6, C2×C6, D8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, D12, D12, C3×D4, C33, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C62, D24, D4⋊S3, C3×D8, S3×C32, C3×C3⋊S3, C32×C6, C3×C3⋊C8, C3×C3⋊C8, C3×C24, C3×D12, C3×D12, C12⋊S3, D4×C32, C32×C12, S3×C3×C6, C6×C3⋊S3, C3⋊D24, C3×D24, C3×D4⋊S3, C32×C3⋊C8, C32×D12, C3×C12⋊S3, C3×C3⋊D24
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, D8, C3×S3, D12, C3⋊D4, C3×D4, S32, S3×C6, D24, D4⋊S3, C3×D8, C3⋊D12, C3×D12, C3×C3⋊D4, C3×S32, C3⋊D24, C3×D24, C3×D4⋊S3, C3×C3⋊D12, C3×C3⋊D24

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

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

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

72 conjugacy classes

class 1 2A2B2C3A3B3C···3H3I3J3K 4 6A6B6C···6H6I6J6K6L···6S6T6U8A8B12A···12H12I···12Q24A···24P
order1222333···33334666···66666···6668812···1212···1224···24
size111236112···24442112···244412···123636662···24···46···6

72 irreducible representations

dim11111111222222222222222244444444
type+++++++++++++++
imageC1C2C2C2C3C6C6C6S3S3D4D6D8C3×S3C3×S3D12C3⋊D4C3×D4S3×C6D24C3×D8C3×D12C3×C3⋊D4C3×D24S32D4⋊S3C3⋊D12C3×S32C3⋊D24C3×D4⋊S3C3×C3⋊D12C3×C3⋊D24
kernelC3×C3⋊D24C32×C3⋊C8C32×D12C3×C12⋊S3C3⋊D24C3×C3⋊C8C3×D12C12⋊S3C3×C3⋊C8C3×D12C32×C6C3×C12C33C3⋊C8D12C3×C6C3×C6C3×C6C12C32C32C6C6C3C12C32C6C4C3C3C2C1
# reps11112222111222222244444811122224

Matrix representation of C3×C3⋊D24 ►in GL6(𝔽73)

100000
010000
008000
000800
000010
000001
,
100000
010000
001000
000100
0000072
0000172
,
0320000
57320000
0017200
001000
000001
000010
,
0320000
1600000
0072000
0072100
000001
000010

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[0,57,0,0,0,0,32,32,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,16,0,0,0,0,32,0,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

C3×C3⋊D24 in GAP, Magma, Sage, TeX 

C_3\times C_3\rtimes D_{24}
 
% in TeX
 
G:=Group("C3xC3:D24");
 
// GroupNames label
 
G:=SmallGroup(432,419);
 
// by ID
 
G=gap.SmallGroup(432,419);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,260,1011,80,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^24=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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