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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, (C3×C9)⋊3D8, C12.2S32, C4.1(S3×D9), C3⋊1(D4⋊D9), (C9×D12)⋊2C2, (C3×C18).6D4, C36⋊S3⋊5C2, (C3×D12).2S3, (C3×C12).74D6, C6.1(C9⋊D4), (C3×C36).9C22, C2.4(C9⋊D12), C3.2(C3⋊D24), C32.3(D4⋊S3), C6.14(C3⋊D12), (C3×C9⋊C8)⋊1C2, (C3×C6).42(C3⋊D4), SmallGroup(432,69)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C36 — C9⋊D24
Chief series
C1 — C3 — C32 — C3×C9 — C3×C18 — C3×C36 — C9×D12 — C9⋊D24
Lower central
C3×C9 — C3×C18 — C3×C36 — 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, C2×C6, D8, D9, C18, C18, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24, D12, D12, C3×D4, C3×C9, C36, C36, D18, C2×C18, C3×C12, S3×C6, C2×C3⋊S3, D24, D4⋊S3, S3×C9, C9⋊S3, C3×C18, C9⋊C8, D36, D4×C9, C3×C3⋊C8, C3×D12, C12⋊S3, C3×C36, S3×C18, C2×C9⋊S3, D4⋊D9, C3⋊D24, C3×C9⋊C8, C9×D12, 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, S3×D9, 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⋊D12S3×D9D4⋊D9C3⋊D24C9⋊D12C9⋊D24
kernelC9⋊D24C3×C9⋊C8C9×D12C36⋊S3C9⋊C8C3×D12C3×C18C36C3×C12C3×C9D12C18C3×C6C12C9C6C12C32C6C4C3C3C2C1
# 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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