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G = C33⋊7D8  order 432 = 24·33

4th semidirect product of C33 and D8 acting via D8/C4=C22

metabelian, supersoluble, monomial

Aliases: C33⋊7D8, C32⋊10D24, C12.26S32, (C3×D12)⋊3S3, D12⋊1(C3⋊S3), C32⋊4C8⋊8S3, (C3×C6).78D12, C3⋊2(C3⋊D24), C32⋊6(D4⋊S3), (C3×C12).113D6, (C32×D12)⋊4C2, C33⋊12D4⋊2C2, C3⋊1(C32⋊7D8), (C32×C6).30D4, C6.1(C32⋊7D4), C2.4(C33⋊7D4), C6.22(C3⋊D12), (C32×C12).9C22, C4.8(S3×C3⋊S3), C12.30(C2×C3⋊S3), (C3×C32⋊4C8)⋊2C2, (C3×C6).56(C3⋊D4), SmallGroup(432,437)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C12 — C33⋊7D8
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C32×C12 — C32×D12 — C33⋊7D8
Lower central
C33 — C32×C6 — C32×C12 — C33⋊7D8
Upper central
C1 — C2 — C4

Generators and relations for C33⋊7D8
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, bc=cb, dbd-1=ebe=b-1, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 1672 in 196 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, C8, D4, C32, C32, C32, C12, C12, C12, D6, C2×C6, D8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, D12, D12, C3×D4, C33, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C62, D24, D4⋊S3, S3×C32, C33⋊C2, C32×C6, C3×C3⋊C8, C32⋊4C8, C3×D12, C12⋊S3, D4×C32, C32×C12, S3×C3×C6, C2×C33⋊C2, C3⋊D24, C32⋊7D8, C3×C32⋊4C8, C32×D12, C33⋊12D4, C33⋊7D8
Quotients: C1, C2, C22, S3, D4, D6, D8, C3⋊S3, D12, C3⋊D4, S32, C2×C3⋊S3, D24, D4⋊S3, C3⋊D12, C32⋊7D4, S3×C3⋊S3, C3⋊D24, C32⋊7D8, C33⋊7D4, C33⋊7D8

Smallest permutation representation of C33⋊7D8
►On 72 points
Generators in S72
(1 50 58)(2 59 51)(3 52 60)(4 61 53)(5 54 62)(6 63 55)(7 56 64)(8 57 49)(9 20 40)(10 33 21)(11 22 34)(12 35 23)(13 24 36)(14 37 17)(15 18 38)(16 39 19)(25 45 72)(26 65 46)(27 47 66)(28 67 48)(29 41 68)(30 69 42)(31 43 70)(32 71 44)

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

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

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

51 conjugacy classes

class 1 2A2B2C3A···3E3F3G3H3I 4 6A···6E6F6G6H6I6J···6Q8A8B12A12B12C···12N24A24B24C24D
order12223···3333346···666666···688121212···1224242424
size11121082···2444422···2444412···121818224···418181818

51 irreducible representations

dim1111222222224444
type+++++++++++++++
imageC1C2C2C2S3S3D4D6D8D12C3⋊D4D24S32D4⋊S3C3⋊D12C3⋊D24
kernelC33⋊7D8C3×C32⋊4C8C32×D12C33⋊12D4C32⋊4C8C3×D12C32×C6C3×C12C33C3×C6C3×C6C32C12C32C6C3
# reps1111141522844448

Matrix representation of C33⋊7D8 ►in GL8(𝔽73)

10000000
01000000
0071700000
00110000
000007200
000017200
00000010
00000001
,
10000000
01000000
0071700000
00110000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00001000
00000100
0000007272
00000010
,
1216000000
829000000
007200000
00110000
000007200
000072000
00000010
00000001
,
10000000
3372000000
00100000
0072720000
00000100
00001000
00000010
0000007272

G:=sub<GL(8,GF(73))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,71,1,0,0,0,0,0,0,70,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,71,1,0,0,0,0,0,0,70,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0],[12,8,0,0,0,0,0,0,16,29,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,33,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72] >;
 

C33⋊7D8 in GAP, Magma, Sage, TeX 

C_3^3\rtimes_7D_8
 
% in TeX
 
G:=Group("C3^3:7D8");
 
// GroupNames label
 
G:=SmallGroup(432,437);
 
// by ID
 
G=gap.SmallGroup(432,437);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,92,254,58,571,2028,14118]);
 
// 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,d*a*d^-1=e*a*e=a^-1,b*c=c*b,d*b*d^-1=e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
 
// generators/relations
 

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