metabelian, supersoluble, monomial
Aliases: C33⋊17SD16, C12.54S32, (C3×C6).39D12, C3⋊2(C24⋊2S3), C32⋊4Q8⋊7S3, (C3×C12).120D6, (C32×C6).37D4, C32⋊8(C24⋊C2), C6.14(C12⋊S3), C33⋊12D4.3C2, C2.6(C33⋊8D4), C3⋊1(C32⋊5SD16), C6.10(C3⋊D12), C32⋊11(Q8⋊2S3), (C32×C12).16C22, (C3×C3⋊C8)⋊3S3, C3⋊C8⋊3(C3⋊S3), C4.3(S3×C3⋊S3), (C32×C3⋊C8)⋊4C2, C12.14(C2×C3⋊S3), (C3×C32⋊4Q8)⋊3C2, (C3×C6).79(C3⋊D4), SmallGroup(432,444)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C33⋊17SD16
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=d3 >
Subgroups: 1616 in 184 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C3, C3 [×4], C3 [×4], C4, C4, C22, S3 [×13], C6, C6 [×4], C6 [×4], C8, D4, Q8, C32, C32 [×4], C32 [×4], Dic3 [×4], C12, C12 [×4], C12 [×5], D6 [×13], SD16, C3⋊S3 [×13], C3×C6, C3×C6 [×4], C3×C6 [×4], C3⋊C8, C24 [×4], Dic6 [×4], D12 [×9], C3×Q8, C33, C3×Dic3 [×4], C3⋊Dic3, C3×C12, C3×C12 [×4], C3×C12 [×4], C2×C3⋊S3 [×13], C24⋊C2 [×4], Q8⋊2S3, C33⋊C2, C32×C6, C3×C3⋊C8 [×4], C3×C24, C3×Dic6 [×4], C32⋊4Q8, C12⋊S3 [×9], C3×C3⋊Dic3, C32×C12, C2×C33⋊C2, C32⋊5SD16 [×4], C24⋊2S3, C32×C3⋊C8, C3×C32⋊4Q8, C33⋊12D4, C33⋊17SD16
Quotients: C1, C2 [×3], C22, S3 [×5], D4, D6 [×5], SD16, C3⋊S3, D12 [×4], C3⋊D4, S32 [×4], C2×C3⋊S3, C24⋊C2 [×4], Q8⋊2S3, C3⋊D12 [×4], C12⋊S3, S3×C3⋊S3, C32⋊5SD16 [×4], C24⋊2S3, C33⋊8D4, C33⋊17SD16
(1 27 44)(2 28 45)(3 29 46)(4 30 47)(5 31 48)(6 32 41)(7 25 42)(8 26 43)(9 63 72)(10 64 65)(11 57 66)(12 58 67)(13 59 68)(14 60 69)(15 61 70)(16 62 71)(17 53 39)(18 54 40)(19 55 33)(20 56 34)(21 49 35)(22 50 36)(23 51 37)(24 52 38)
(1 22 66)(2 23 67)(3 24 68)(4 17 69)(5 18 70)(6 19 71)(7 20 72)(8 21 65)(9 25 56)(10 26 49)(11 27 50)(12 28 51)(13 29 52)(14 30 53)(15 31 54)(16 32 55)(33 62 41)(34 63 42)(35 64 43)(36 57 44)(37 58 45)(38 59 46)(39 60 47)(40 61 48)
(1 36 11)(2 12 37)(3 38 13)(4 14 39)(5 40 15)(6 16 33)(7 34 9)(8 10 35)(17 30 60)(18 61 31)(19 32 62)(20 63 25)(21 26 64)(22 57 27)(23 28 58)(24 59 29)(41 71 55)(42 56 72)(43 65 49)(44 50 66)(45 67 51)(46 52 68)(47 69 53)(48 54 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 5)(2 8)(4 6)(9 34)(10 37)(11 40)(12 35)(13 38)(14 33)(15 36)(16 39)(17 71)(18 66)(19 69)(20 72)(21 67)(22 70)(23 65)(24 68)(25 42)(26 45)(27 48)(28 43)(29 46)(30 41)(31 44)(32 47)(49 58)(50 61)(51 64)(52 59)(53 62)(54 57)(55 60)(56 63)
G:=sub<Sym(72)| (1,27,44)(2,28,45)(3,29,46)(4,30,47)(5,31,48)(6,32,41)(7,25,42)(8,26,43)(9,63,72)(10,64,65)(11,57,66)(12,58,67)(13,59,68)(14,60,69)(15,61,70)(16,62,71)(17,53,39)(18,54,40)(19,55,33)(20,56,34)(21,49,35)(22,50,36)(23,51,37)(24,52,38), (1,22,66)(2,23,67)(3,24,68)(4,17,69)(5,18,70)(6,19,71)(7,20,72)(8,21,65)(9,25,56)(10,26,49)(11,27,50)(12,28,51)(13,29,52)(14,30,53)(15,31,54)(16,32,55)(33,62,41)(34,63,42)(35,64,43)(36,57,44)(37,58,45)(38,59,46)(39,60,47)(40,61,48), (1,36,11)(2,12,37)(3,38,13)(4,14,39)(5,40,15)(6,16,33)(7,34,9)(8,10,35)(17,30,60)(18,61,31)(19,32,62)(20,63,25)(21,26,64)(22,57,27)(23,28,58)(24,59,29)(41,71,55)(42,56,72)(43,65,49)(44,50,66)(45,67,51)(46,52,68)(47,69,53)(48,54,70), (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), (1,5)(2,8)(4,6)(9,34)(10,37)(11,40)(12,35)(13,38)(14,33)(15,36)(16,39)(17,71)(18,66)(19,69)(20,72)(21,67)(22,70)(23,65)(24,68)(25,42)(26,45)(27,48)(28,43)(29,46)(30,41)(31,44)(32,47)(49,58)(50,61)(51,64)(52,59)(53,62)(54,57)(55,60)(56,63)>;
G:=Group( (1,27,44)(2,28,45)(3,29,46)(4,30,47)(5,31,48)(6,32,41)(7,25,42)(8,26,43)(9,63,72)(10,64,65)(11,57,66)(12,58,67)(13,59,68)(14,60,69)(15,61,70)(16,62,71)(17,53,39)(18,54,40)(19,55,33)(20,56,34)(21,49,35)(22,50,36)(23,51,37)(24,52,38), (1,22,66)(2,23,67)(3,24,68)(4,17,69)(5,18,70)(6,19,71)(7,20,72)(8,21,65)(9,25,56)(10,26,49)(11,27,50)(12,28,51)(13,29,52)(14,30,53)(15,31,54)(16,32,55)(33,62,41)(34,63,42)(35,64,43)(36,57,44)(37,58,45)(38,59,46)(39,60,47)(40,61,48), (1,36,11)(2,12,37)(3,38,13)(4,14,39)(5,40,15)(6,16,33)(7,34,9)(8,10,35)(17,30,60)(18,61,31)(19,32,62)(20,63,25)(21,26,64)(22,57,27)(23,28,58)(24,59,29)(41,71,55)(42,56,72)(43,65,49)(44,50,66)(45,67,51)(46,52,68)(47,69,53)(48,54,70), (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), (1,5)(2,8)(4,6)(9,34)(10,37)(11,40)(12,35)(13,38)(14,33)(15,36)(16,39)(17,71)(18,66)(19,69)(20,72)(21,67)(22,70)(23,65)(24,68)(25,42)(26,45)(27,48)(28,43)(29,46)(30,41)(31,44)(32,47)(49,58)(50,61)(51,64)(52,59)(53,62)(54,57)(55,60)(56,63) );
G=PermutationGroup([(1,27,44),(2,28,45),(3,29,46),(4,30,47),(5,31,48),(6,32,41),(7,25,42),(8,26,43),(9,63,72),(10,64,65),(11,57,66),(12,58,67),(13,59,68),(14,60,69),(15,61,70),(16,62,71),(17,53,39),(18,54,40),(19,55,33),(20,56,34),(21,49,35),(22,50,36),(23,51,37),(24,52,38)], [(1,22,66),(2,23,67),(3,24,68),(4,17,69),(5,18,70),(6,19,71),(7,20,72),(8,21,65),(9,25,56),(10,26,49),(11,27,50),(12,28,51),(13,29,52),(14,30,53),(15,31,54),(16,32,55),(33,62,41),(34,63,42),(35,64,43),(36,57,44),(37,58,45),(38,59,46),(39,60,47),(40,61,48)], [(1,36,11),(2,12,37),(3,38,13),(4,14,39),(5,40,15),(6,16,33),(7,34,9),(8,10,35),(17,30,60),(18,61,31),(19,32,62),(20,63,25),(21,26,64),(22,57,27),(23,28,58),(24,59,29),(41,71,55),(42,56,72),(43,65,49),(44,50,66),(45,67,51),(46,52,68),(47,69,53),(48,54,70)], [(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)], [(1,5),(2,8),(4,6),(9,34),(10,37),(11,40),(12,35),(13,38),(14,33),(15,36),(16,39),(17,71),(18,66),(19,69),(20,72),(21,67),(22,70),(23,65),(24,68),(25,42),(26,45),(27,48),(28,43),(29,46),(30,41),(31,44),(32,47),(49,58),(50,61),(51,64),(52,59),(53,62),(54,57),(55,60),(56,63)])
60 conjugacy classes
class | 1 | 2A | 2B | 3A | ··· | 3E | 3F | 3G | 3H | 3I | 4A | 4B | 6A | ··· | 6E | 6F | 6G | 6H | 6I | 8A | 8B | 12A | ··· | 12H | 12I | ··· | 12Q | 12R | 12S | 24A | ··· | 24P |
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 |
size | 1 | 1 | 108 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | 36 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 36 | 36 | 6 | ··· | 6 |
60 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⋊17SD16 | C32×C3⋊C8 | C3×C32⋊4Q8 | C33⋊12D4 | C3×C3⋊C8 | C32⋊4Q8 | C32×C6 | C3×C12 | C33 | C3×C6 | C3×C6 | C32 | C12 | C32 | C6 | C3 |
# reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 5 | 2 | 8 | 2 | 16 | 4 | 1 | 4 | 8 |
Matrix representation of C33⋊17SD16 ►in GL8(𝔽73)
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 | 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 | 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 | 0 | 72 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 72 |
72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 55 | 0 | 0 | 0 | 0 |
0 | 0 | 4 | 61 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 72 | 0 | 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 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 48 | 1 | 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 | 0 | 1 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(73))| [72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,55,61,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,48,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
C33⋊17SD16 in GAP, Magma, Sage, TeX
C_3^3\rtimes_{17}{\rm SD}_{16}
% in TeX
G:=Group("C3^3:17SD16");
// GroupNames label
G:=SmallGroup(432,444);
// by ID
G=gap.SmallGroup(432,444);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,197,64,135,58,571,2028,14118]);
// 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^3>;
// generators/relations