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G = C9:D24  order 432 = 24·33

The semidirect product of C9 and D24 acting via D24/D12=C2

metabelian, supersoluble, monomial

Aliases: C9:2D24, D12:1D9, C36.10D6, C12.30D18, C18.12D12, C9:C8:1S3, (C3xC9):3D8, C12.2S32, C4.1(S3xD9), C3:1(D4:D9), (C9xD12):2C2, (C3xC18).6D4, C36:S3:5C2, (C3xD12).2S3, (C3xC12).74D6, C6.1(C9:D4), (C3xC36).9C22, C2.4(C9:D12), C3.2(C3:D24), C32.3(D4:S3), C6.14(C3:D12), (C3xC9:C8):1C2, (C3xC6).42(C3:D4), SmallGroup(432,69)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC36 — C9:D24
Chief series
C1 — C3 — C32 — C3xC9 — C3xC18 — C3xC36 — C9xD12 — C9:D24
Lower central
C3xC9 — C3xC18 — C3xC36 — C9:D24
Upper central
C1 — C2 — C4

Generators and relations for C9:D24
 G = < a,b,c | a9=b24=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 724 in 82 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, C9, C9, C32, C12, C12, D6, C2xC6, D8, D9, C18, C18, C3xS3, C3:S3, C3xC6, C3:C8, C24, D12, D12, C3xD4, C3xC9, C36, C36, D18, C2xC18, C3xC12, S3xC6, C2xC3:S3, D24, D4:S3, S3xC9, C9:S3, C3xC18, C9:C8, D36, D4xC9, C3xC3:C8, C3xD12, C12:S3, C3xC36, S3xC18, C2xC9:S3, D4:D9, C3:D24, C3xC9:C8, C9xD12, C36:S3, C9:D24
Quotients: C1, C2, C22, S3, D4, D6, D8, D9, D12, C3:D4, D18, S32, D24, D4:S3, C9:D4, C3:D12, S3xD9, D4:D9, C3:D24, C9:D12, C9:D24

Smallest permutation representation of C9:D24
►On 72 points
Generators in S72
(1 58 34 17 50 26 9 66 42)(2 43 67 10 27 51 18 35 59)(3 60 36 19 52 28 11 68 44)(4 45 69 12 29 53 20 37 61)(5 62 38 21 54 30 13 70 46)(6 47 71 14 31 55 22 39 63)(7 64 40 23 56 32 15 72 48)(8 25 49 16 33 57 24 41 65)

(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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)

(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 24)(17 23)(18 22)(19 21)(25 65)(26 64)(27 63)(28 62)(29 61)(30 60)(31 59)(32 58)(33 57)(34 56)(35 55)(36 54)(37 53)(38 52)(39 51)(40 50)(41 49)(42 72)(43 71)(44 70)(45 69)(46 68)(47 67)(48 66)
 
G:=sub<Sym(72)| (1,58,34,17,50,26,9,66,42)(2,43,67,10,27,51,18,35,59)(3,60,36,19,52,28,11,68,44)(4,45,69,12,29,53,20,37,61)(5,62,38,21,54,30,13,70,46)(6,47,71,14,31,55,22,39,63)(7,64,40,23,56,32,15,72,48)(8,25,49,16,33,57,24,41,65), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,24)(17,23)(18,22)(19,21)(25,65)(26,64)(27,63)(28,62)(29,61)(30,60)(31,59)(32,58)(33,57)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,50)(41,49)(42,72)(43,71)(44,70)(45,69)(46,68)(47,67)(48,66)>;
 
G:=Group( (1,58,34,17,50,26,9,66,42)(2,43,67,10,27,51,18,35,59)(3,60,36,19,52,28,11,68,44)(4,45,69,12,29,53,20,37,61)(5,62,38,21,54,30,13,70,46)(6,47,71,14,31,55,22,39,63)(7,64,40,23,56,32,15,72,48)(8,25,49,16,33,57,24,41,65), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,24)(17,23)(18,22)(19,21)(25,65)(26,64)(27,63)(28,62)(29,61)(30,60)(31,59)(32,58)(33,57)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,50)(41,49)(42,72)(43,71)(44,70)(45,69)(46,68)(47,67)(48,66) );
 
G=PermutationGroup([[(1,58,34,17,50,26,9,66,42),(2,43,67,10,27,51,18,35,59),(3,60,36,19,52,28,11,68,44),(4,45,69,12,29,53,20,37,61),(5,62,38,21,54,30,13,70,46),(6,47,71,14,31,55,22,39,63),(7,64,40,23,56,32,15,72,48),(8,25,49,16,33,57,24,41,65)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,24),(17,23),(18,22),(19,21),(25,65),(26,64),(27,63),(28,62),(29,61),(30,60),(31,59),(32,58),(33,57),(34,56),(35,55),(36,54),(37,53),(38,52),(39,51),(40,50),(41,49),(42,72),(43,71),(44,70),(45,69),(46,68),(47,67),(48,66)]])
 

51 conjugacy classes

class 1 2A2B2C3A3B3C 4 6A6B6C6D6E8A8B9A9B9C9D9E9F12A12B12C12D12E18A18B18C18D18E18F18G···18L24A24B24C24D36A···36I
order122233346666688999999121212121218181818181818···182424242436···36
size11121082242224121218182224442244422244412···12181818184···4

51 irreducible representations

dim111122222222222244444444
type++++++++++++++++++++++
imageC1C2C2C2S3S3D4D6D6D8D9D12C3:D4D18D24C9:D4S32D4:S3C3:D12S3xD9D4:D9C3:D24C9:D12C9:D24
kernelC9:D24C3xC9:C8C9xD12C36:S3C9:C8C3xD12C3xC18C36C3xC12C3xC9D12C18C3xC6C12C9C6C12C32C6C4C3C3C2C1
# reps111111111232234611133236

Matrix representation of C9:D24 ►in GL4(F73) generated by

1000
0100
00283
007031
,
51800
552300
00046
00460
,
7700
146600
00597
006614
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,28,70,0,0,3,31],[5,55,0,0,18,23,0,0,0,0,0,46,0,0,46,0],[7,14,0,0,7,66,0,0,0,0,59,66,0,0,7,14] >;
 

C9:D24 in GAP, Magma, Sage, TeX 

C_9\rtimes D_{24}
 
% in TeX
 
G:=Group("C9:D24");
 
// GroupNames label
 
G:=SmallGroup(432,69);
 
// by ID
 
G=gap.SmallGroup(432,69);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,92,254,58,3091,662,4037,7069]);
 
// Polycyclic
 
G:=Group<a,b,c|a^9=b^24=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
 
// generators/relations
 

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