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## G = C33⋊15SD16order 432 = 24·33

### 7th semidirect product of C33 and SD16 acting via SD16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C12 — C33⋊15SD16
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×C12 — C32×Dic6 — C33⋊15SD16
 Lower central C33 — C32×C6 — C32×C12 — C33⋊15SD16
 Upper central C1 — C2 — C4

Generators and relations for C3315SD16
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=d3 >

Subgroups: 1576 in 184 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C8, D4, Q8, C32, C32, C32, Dic3, C12, C12, C12, D6, SD16, C3⋊S3, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, Dic6, D12, C3×Q8, C33, C3×Dic3, C3×C12, C3×C12, C3×C12, C2×C3⋊S3, C24⋊C2, Q82S3, C33⋊C2, C32×C6, C3×C3⋊C8, C324C8, C3×Dic6, C12⋊S3, Q8×C32, C32×Dic3, C32×C12, C2×C33⋊C2, C325SD16, C3211SD16, C3×C324C8, C32×Dic6, C3312D4, C3315SD16
Quotients: C1, C2, C22, S3, D4, D6, SD16, C3⋊S3, D12, C3⋊D4, S32, C2×C3⋊S3, C24⋊C2, Q82S3, C3⋊D12, C327D4, S3×C3⋊S3, C325SD16, C3211SD16, C337D4, C3315SD16

Smallest permutation representation of C3315SD16
On 72 points
Generators in S72
(1 58 43)(2 44 59)(3 60 45)(4 46 61)(5 62 47)(6 48 63)(7 64 41)(8 42 57)(9 52 29)(10 30 53)(11 54 31)(12 32 55)(13 56 25)(14 26 49)(15 50 27)(16 28 51)(17 65 40)(18 33 66)(19 67 34)(20 35 68)(21 69 36)(22 37 70)(23 71 38)(24 39 72)
(1 68 13)(2 14 69)(3 70 15)(4 16 71)(5 72 9)(6 10 65)(7 66 11)(8 12 67)(17 63 53)(18 54 64)(19 57 55)(20 56 58)(21 59 49)(22 50 60)(23 61 51)(24 52 62)(25 43 35)(26 36 44)(27 45 37)(28 38 46)(29 47 39)(30 40 48)(31 41 33)(32 34 42)
(1 25 20)(2 26 21)(3 27 22)(4 28 23)(5 29 24)(6 30 17)(7 31 18)(8 32 19)(9 39 62)(10 40 63)(11 33 64)(12 34 57)(13 35 58)(14 36 59)(15 37 60)(16 38 61)(41 54 66)(42 55 67)(43 56 68)(44 49 69)(45 50 70)(46 51 71)(47 52 72)(48 53 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 3)(2 6)(5 7)(9 66)(10 69)(11 72)(12 67)(13 70)(14 65)(15 68)(16 71)(17 26)(18 29)(19 32)(20 27)(21 30)(22 25)(23 28)(24 31)(33 52)(34 55)(35 50)(36 53)(37 56)(38 51)(39 54)(40 49)(41 62)(42 57)(43 60)(44 63)(45 58)(46 61)(47 64)(48 59)

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

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

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

51 conjugacy classes

 class 1 2A 2B 3A ··· 3E 3F 3G 3H 3I 4A 4B 6A ··· 6E 6F 6G 6H 6I 8A 8B 12A 12B 12C ··· 12N 12O ··· 12V 24A 24B 24C 24D order 1 2 2 3 ··· 3 3 3 3 3 4 4 6 ··· 6 6 6 6 6 8 8 12 12 12 ··· 12 12 ··· 12 24 24 24 24 size 1 1 108 2 ··· 2 4 4 4 4 2 12 2 ··· 2 4 4 4 4 18 18 2 2 4 ··· 4 12 ··· 12 18 18 18 18

51 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 S3 S3 D4 D6 SD16 D12 C3⋊D4 C24⋊C2 S32 Q8⋊2S3 C3⋊D12 C32⋊5SD16 kernel C33⋊15SD16 C3×C32⋊4C8 C32×Dic6 C33⋊12D4 C32⋊4C8 C3×Dic6 C32×C6 C3×C12 C33 C3×C6 C3×C6 C32 C12 C32 C6 C3 # reps 1 1 1 1 1 4 1 5 2 2 8 4 4 4 4 8

Matrix representation of C3315SD16 in GL8(𝔽73)

 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 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 72 72 0 0 0 0 0 0 1 0 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 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 72 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 61 12 0 0 0 0 0 0 67 0 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 0 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 1 71 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 0 0 0 0 0 0 0 72 72

G:=sub<GL(8,GF(73))| [0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[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],[1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,61,67,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,71,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72] >;

C3315SD16 in GAP, Magma, Sage, TeX

C_3^3\rtimes_{15}{\rm SD}_{16}
% in TeX

G:=Group("C3^3:15SD16");
// GroupNames label

G:=SmallGroup(432,442);
// by ID

G=gap.SmallGroup(432,442);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,36,254,58,571,2028,14118]);
// 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^3>;
// generators/relations

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