metabelian, supersoluble, monomial
Aliases: C33⋊15SD16, C12.30S32, (C3×Dic6)⋊3S3, (C3×C6).80D12, Dic6⋊1(C3⋊S3), C32⋊4C8⋊10S3, (C3×C12).118D6, (C32×C6).35D4, (C32×Dic6)⋊4C2, C33⋊12D4.2C2, C6.3(C32⋊7D4), C2.6(C33⋊7D4), C32⋊13(C24⋊C2), C3⋊2(C32⋊5SD16), C6.24(C3⋊D12), C3⋊1(C32⋊11SD16), C32⋊8(Q8⋊2S3), (C32×C12).14C22, C4.10(S3×C3⋊S3), C12.32(C2×C3⋊S3), (C3×C32⋊4C8)⋊5C2, (C3×C6).58(C3⋊D4), SmallGroup(432,442)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C33⋊15SD16
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 [×4], C3 [×4], C4, C4, C22, S3 [×13], C6, C6 [×4], C6 [×4], C8, D4, Q8, C32, C32 [×4], C32 [×4], Dic3, C12, C12 [×4], C12 [×8], D6 [×13], SD16, C3⋊S3 [×13], C3×C6, C3×C6 [×4], C3×C6 [×4], C3⋊C8 [×4], C24, Dic6, D12 [×9], C3×Q8 [×4], C33, C3×Dic3 [×4], C3×C12, C3×C12 [×4], C3×C12 [×5], C2×C3⋊S3 [×13], C24⋊C2, Q8⋊2S3 [×4], C33⋊C2, C32×C6, C3×C3⋊C8 [×4], C32⋊4C8, C3×Dic6 [×4], C12⋊S3 [×9], Q8×C32, C32×Dic3, C32×C12, C2×C33⋊C2, C32⋊5SD16 [×4], C32⋊11SD16, C3×C32⋊4C8, C32×Dic6, C33⋊12D4, C33⋊15SD16
Quotients: C1, C2 [×3], C22, S3 [×5], D4, D6 [×5], SD16, C3⋊S3, D12, C3⋊D4 [×4], S32 [×4], C2×C3⋊S3, C24⋊C2, Q8⋊2S3 [×4], C3⋊D12 [×4], C32⋊7D4, S3×C3⋊S3, C32⋊5SD16 [×4], C32⋊11SD16, C33⋊7D4, C33⋊15SD16
(1 39 43)(2 44 40)(3 33 45)(4 46 34)(5 35 47)(6 48 36)(7 37 41)(8 42 38)(9 21 69)(10 70 22)(11 23 71)(12 72 24)(13 17 65)(14 66 18)(15 19 67)(16 68 20)(25 50 61)(26 62 51)(27 52 63)(28 64 53)(29 54 57)(30 58 55)(31 56 59)(32 60 49)
(1 68 52)(2 53 69)(3 70 54)(4 55 71)(5 72 56)(6 49 65)(7 66 50)(8 51 67)(9 44 28)(10 29 45)(11 46 30)(12 31 47)(13 48 32)(14 25 41)(15 42 26)(16 27 43)(17 36 60)(18 61 37)(19 38 62)(20 63 39)(21 40 64)(22 57 33)(23 34 58)(24 59 35)
(1 27 20)(2 28 21)(3 29 22)(4 30 23)(5 31 24)(6 32 17)(7 25 18)(8 26 19)(9 40 53)(10 33 54)(11 34 55)(12 35 56)(13 36 49)(14 37 50)(15 38 51)(16 39 52)(41 61 66)(42 62 67)(43 63 68)(44 64 69)(45 57 70)(46 58 71)(47 59 72)(48 60 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 60)(10 63)(11 58)(12 61)(13 64)(14 59)(15 62)(16 57)(17 28)(18 31)(19 26)(20 29)(21 32)(22 27)(23 30)(24 25)(33 43)(34 46)(35 41)(36 44)(37 47)(38 42)(39 45)(40 48)(49 69)(50 72)(51 67)(52 70)(53 65)(54 68)(55 71)(56 66)
G:=sub<Sym(72)| (1,39,43)(2,44,40)(3,33,45)(4,46,34)(5,35,47)(6,48,36)(7,37,41)(8,42,38)(9,21,69)(10,70,22)(11,23,71)(12,72,24)(13,17,65)(14,66,18)(15,19,67)(16,68,20)(25,50,61)(26,62,51)(27,52,63)(28,64,53)(29,54,57)(30,58,55)(31,56,59)(32,60,49), (1,68,52)(2,53,69)(3,70,54)(4,55,71)(5,72,56)(6,49,65)(7,66,50)(8,51,67)(9,44,28)(10,29,45)(11,46,30)(12,31,47)(13,48,32)(14,25,41)(15,42,26)(16,27,43)(17,36,60)(18,61,37)(19,38,62)(20,63,39)(21,40,64)(22,57,33)(23,34,58)(24,59,35), (1,27,20)(2,28,21)(3,29,22)(4,30,23)(5,31,24)(6,32,17)(7,25,18)(8,26,19)(9,40,53)(10,33,54)(11,34,55)(12,35,56)(13,36,49)(14,37,50)(15,38,51)(16,39,52)(41,61,66)(42,62,67)(43,63,68)(44,64,69)(45,57,70)(46,58,71)(47,59,72)(48,60,65), (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)(9,60)(10,63)(11,58)(12,61)(13,64)(14,59)(15,62)(16,57)(17,28)(18,31)(19,26)(20,29)(21,32)(22,27)(23,30)(24,25)(33,43)(34,46)(35,41)(36,44)(37,47)(38,42)(39,45)(40,48)(49,69)(50,72)(51,67)(52,70)(53,65)(54,68)(55,71)(56,66)>;
G:=Group( (1,39,43)(2,44,40)(3,33,45)(4,46,34)(5,35,47)(6,48,36)(7,37,41)(8,42,38)(9,21,69)(10,70,22)(11,23,71)(12,72,24)(13,17,65)(14,66,18)(15,19,67)(16,68,20)(25,50,61)(26,62,51)(27,52,63)(28,64,53)(29,54,57)(30,58,55)(31,56,59)(32,60,49), (1,68,52)(2,53,69)(3,70,54)(4,55,71)(5,72,56)(6,49,65)(7,66,50)(8,51,67)(9,44,28)(10,29,45)(11,46,30)(12,31,47)(13,48,32)(14,25,41)(15,42,26)(16,27,43)(17,36,60)(18,61,37)(19,38,62)(20,63,39)(21,40,64)(22,57,33)(23,34,58)(24,59,35), (1,27,20)(2,28,21)(3,29,22)(4,30,23)(5,31,24)(6,32,17)(7,25,18)(8,26,19)(9,40,53)(10,33,54)(11,34,55)(12,35,56)(13,36,49)(14,37,50)(15,38,51)(16,39,52)(41,61,66)(42,62,67)(43,63,68)(44,64,69)(45,57,70)(46,58,71)(47,59,72)(48,60,65), (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)(9,60)(10,63)(11,58)(12,61)(13,64)(14,59)(15,62)(16,57)(17,28)(18,31)(19,26)(20,29)(21,32)(22,27)(23,30)(24,25)(33,43)(34,46)(35,41)(36,44)(37,47)(38,42)(39,45)(40,48)(49,69)(50,72)(51,67)(52,70)(53,65)(54,68)(55,71)(56,66) );
G=PermutationGroup([(1,39,43),(2,44,40),(3,33,45),(4,46,34),(5,35,47),(6,48,36),(7,37,41),(8,42,38),(9,21,69),(10,70,22),(11,23,71),(12,72,24),(13,17,65),(14,66,18),(15,19,67),(16,68,20),(25,50,61),(26,62,51),(27,52,63),(28,64,53),(29,54,57),(30,58,55),(31,56,59),(32,60,49)], [(1,68,52),(2,53,69),(3,70,54),(4,55,71),(5,72,56),(6,49,65),(7,66,50),(8,51,67),(9,44,28),(10,29,45),(11,46,30),(12,31,47),(13,48,32),(14,25,41),(15,42,26),(16,27,43),(17,36,60),(18,61,37),(19,38,62),(20,63,39),(21,40,64),(22,57,33),(23,34,58),(24,59,35)], [(1,27,20),(2,28,21),(3,29,22),(4,30,23),(5,31,24),(6,32,17),(7,25,18),(8,26,19),(9,40,53),(10,33,54),(11,34,55),(12,35,56),(13,36,49),(14,37,50),(15,38,51),(16,39,52),(41,61,66),(42,62,67),(43,63,68),(44,64,69),(45,57,70),(46,58,71),(47,59,72),(48,60,65)], [(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),(9,60),(10,63),(11,58),(12,61),(13,64),(14,59),(15,62),(16,57),(17,28),(18,31),(19,26),(20,29),(21,32),(22,27),(23,30),(24,25),(33,43),(34,46),(35,41),(36,44),(37,47),(38,42),(39,45),(40,48),(49,69),(50,72),(51,67),(52,70),(53,65),(54,68),(55,71),(56,66)])
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 C33⋊15SD16 ►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] >;
C33⋊15SD16 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