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G = C338D8order 432 = 24·33

5th semidirect product of C33 and D8 acting via D8/C4=C22

metabelian, supersoluble, monomial

Aliases: C338D8, C325D24, C12.52S32, C12⋊S37S3, (C3×C6).37D12, C32(C325D8), C31(C3⋊D24), C329(D4⋊S3), (C3×C12).114D6, C3312D43C2, (C32×C6).31D4, C6.8(C3⋊D12), C6.12(C12⋊S3), C2.4(C338D4), (C32×C12).10C22, (C3×C3⋊C8)⋊1S3, C3⋊C81(C3⋊S3), C4.1(S3×C3⋊S3), (C32×C3⋊C8)⋊1C2, C12.10(C2×C3⋊S3), (C3×C12⋊S3)⋊3C2, (C3×C6).77(C3⋊D4), SmallGroup(432,438)

Series: Derived Chief Lower central Upper central

C1C32×C12 — C338D8
C1C3C32C33C32×C6C32×C12C32×C3⋊C8 — C338D8
C33C32×C6C32×C12 — C338D8
C1C2C4

Generators and relations for C338D8
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 1792 in 196 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C3 [×4], C4, C22 [×2], S3 [×17], C6, C6 [×4], C6 [×5], C8, D4 [×2], C32, C32 [×4], C32 [×4], C12, C12 [×4], C12 [×4], D6 [×17], C2×C6, D8, C3×S3 [×4], C3⋊S3 [×14], C3×C6, C3×C6 [×4], C3×C6 [×4], C3⋊C8, C24 [×4], D12 [×13], C3×D4, C33, C3×C12, C3×C12 [×4], C3×C12 [×4], S3×C6 [×4], C2×C3⋊S3 [×14], D24 [×4], D4⋊S3, C3×C3⋊S3, C33⋊C2, C32×C6, C3×C3⋊C8 [×4], C3×C24, C3×D12 [×4], C12⋊S3, C12⋊S3 [×9], C32×C12, C6×C3⋊S3, C2×C33⋊C2, C3⋊D24 [×4], C325D8, C32×C3⋊C8, C3×C12⋊S3, C3312D4, C338D8
Quotients: C1, C2 [×3], C22, S3 [×5], D4, D6 [×5], D8, C3⋊S3, D12 [×4], C3⋊D4, S32 [×4], C2×C3⋊S3, D24 [×4], D4⋊S3, C3⋊D12 [×4], C12⋊S3, S3×C3⋊S3, C3⋊D24 [×4], C325D8, C338D4, C338D8

Smallest permutation representation of C338D8
On 72 points
Generators in S72
(1 38 43)(2 39 44)(3 40 45)(4 33 46)(5 34 47)(6 35 48)(7 36 41)(8 37 42)(9 51 19)(10 52 20)(11 53 21)(12 54 22)(13 55 23)(14 56 24)(15 49 17)(16 50 18)(25 57 68)(26 58 69)(27 59 70)(28 60 71)(29 61 72)(30 62 65)(31 63 66)(32 64 67)
(1 54 57)(2 55 58)(3 56 59)(4 49 60)(5 50 61)(6 51 62)(7 52 63)(8 53 64)(9 30 48)(10 31 41)(11 32 42)(12 25 43)(13 26 44)(14 27 45)(15 28 46)(16 29 47)(17 71 33)(18 72 34)(19 65 35)(20 66 36)(21 67 37)(22 68 38)(23 69 39)(24 70 40)
(1 25 22)(2 23 26)(3 27 24)(4 17 28)(5 29 18)(6 19 30)(7 31 20)(8 21 32)(9 62 35)(10 36 63)(11 64 37)(12 38 57)(13 58 39)(14 40 59)(15 60 33)(16 34 61)(41 66 52)(42 53 67)(43 68 54)(44 55 69)(45 70 56)(46 49 71)(47 72 50)(48 51 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)
(2 8)(3 7)(4 6)(9 71)(10 70)(11 69)(12 68)(13 67)(14 66)(15 65)(16 72)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 32)(24 31)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 64)(56 63)

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

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

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

60 conjugacy classes

class 1 2A2B2C3A···3E3F3G3H3I 4 6A···6E6F6G6H6I6J6K8A8B12A···12H12I···12Q24A···24P
order12223···3333346···66666668812···1212···1224···24
size11361082···2444422···244443636662···24···46···6

60 irreducible representations

dim1111222222224444
type+++++++++++++++
imageC1C2C2C2S3S3D4D6D8D12C3⋊D4D24S32D4⋊S3C3⋊D12C3⋊D24
kernelC338D8C32×C3⋊C8C3×C12⋊S3C3312D4C3×C3⋊C8C12⋊S3C32×C6C3×C12C33C3×C6C3×C6C32C12C32C6C3
# reps11114115282164148

Matrix representation of C338D8 in GL6(𝔽73)

1700000
1710000
0007200
0017200
000010
000001
,
7130000
7210000
001000
000100
000010
000001
,
100000
010000
001000
000100
00007272
000010
,
13150000
68280000
0072000
0007200
000010
00007272
,
7230000
010000
000100
001000
000010
00007272

G:=sub<GL(6,GF(73))| [1,1,0,0,0,0,70,71,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[71,72,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[13,68,0,0,0,0,15,28,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[72,0,0,0,0,0,3,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;

C338D8 in GAP, Magma, Sage, TeX

C_3^3\rtimes_8D_8
% in TeX

G:=Group("C3^3:8D8");
// GroupNames label

G:=SmallGroup(432,438);
// by ID

G=gap.SmallGroup(432,438);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,85,135,58,571,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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