Copied to
clipboard

G = He3⋊8SD16  order 432 = 24·33

1st semidirect product of He3 and SD16 acting via SD16/D4=C2

metabelian, supersoluble, monomial

Aliases: He3⋊8SD16, C12.3(S3×C6), (C3×C12).8D6, He3⋊3C8⋊9C2, C32⋊9SD16⋊C3, C32⋊4C8⋊1C6, He3⋊3Q8⋊3C2, D4.(C32⋊C6), (D4×He3).1C2, C32⋊4Q8⋊2C6, (C2×He3).27D4, (D4×C32).1S3, (D4×C32).1C6, C32⋊4(C3×SD16), C2.4(He3⋊6D4), C32⋊5(D4.S3), (C4×He3).8C22, (C3×C12).1(C2×C6), (C3×D4).3(C3×S3), (C3×C6).12(C3×D4), C6.20(C3×C3⋊D4), C4.1(C2×C32⋊C6), C3.2(C3×D4.S3), (C3×C6).23(C3⋊D4), SmallGroup(432,152)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — He3⋊8SD16
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C4×He3 — He3⋊3Q8 — He3⋊8SD16
Lower central
C32 — C3×C6 — C3×C12 — He3⋊8SD16
Upper central
C1 — C2 — C4 — D4

Generators and relations for He3⋊8SD16
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, cac-1=ab-1, dad-1=a-1, ae=ea, bc=cb, dbd-1=b-1, be=eb, cd=dc, ce=ec, ede=d3 >

Subgroups: 365 in 86 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C8, D4, Q8, C32, C32, Dic3, C12, C12, C2×C6, SD16, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C3×D4, C3×D4, C3×Q8, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C62, D4.S3, C3×SD16, C2×He3, C2×He3, C3×C3⋊C8, C32⋊4C8, C3×Dic6, C32⋊4Q8, D4×C32, D4×C32, C32⋊C12, C4×He3, C22×He3, C3×D4.S3, C32⋊9SD16, He3⋊3C8, He3⋊3Q8, D4×He3, He3⋊8SD16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, SD16, C3×S3, C3⋊D4, C3×D4, S3×C6, D4.S3, C3×SD16, C32⋊C6, C3×C3⋊D4, C2×C32⋊C6, C3×D4.S3, He3⋊6D4, He3⋊8SD16

Smallest permutation representation of He3⋊8SD16
►On 72 points
Generators in S72
(9 31 48)(10 41 32)(11 25 42)(12 43 26)(13 27 44)(14 45 28)(15 29 46)(16 47 30)(33 55 60)(34 61 56)(35 49 62)(36 63 50)(37 51 64)(38 57 52)(39 53 58)(40 59 54)

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

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

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

41 conjugacy classes

class 1 2A2B3A3B3C3D3E3F4A4B6A6B6C6D6E6F6G6H6I···6P8A8B12A12B12C12D12E12F12G12H24A24B24C24D
order12233333344666666666···688121212121212121224242424
size1142336662362334466612···121818466121212363618181818

41 irreducible representations

dim1111111112222222222244666
type++++-+++-++
imageC1C2C2C2C3C6C6C6He3⋊8SD16S3D4D6SD16C3×S3C3⋊D4C3×D4S3×C6C3×SD16C3×C3⋊D4D4.S3C3×D4.S3C32⋊C6C2×C32⋊C6He3⋊6D4
kernelHe3⋊8SD16He3⋊3C8He3⋊3Q8D4×He3C32⋊9SD16C32⋊4C8C32⋊4Q8D4×C32C1D4×C32C2×He3C3×C12He3C3×D4C3×C6C3×C6C12C32C6C32C3D4C4C2
# reps111122221111222224412112

Matrix representation of He3⋊8SD16 ►in GL10(𝔽73)

07200000000
17200000000
00072000000
00172000000
0000100000
000064640000
0000908000
0000000100
000016340080
00006161665864
,
1000000000
0100000000
0010000000
0001000000
00006400000
00000640000
00000064000
00005500800
000016340080
0000157157008
,
1000000000
0100000000
0010000000
0001000000
000064630000
0000091000
00000650000
0000000010
0000821665963
000001200064
,
00012000000
00120000000
067012000000
670120000000
00005800200
0000058042290
000013581147422
000033001500
0000683500150
000049381256062
,
1000000000
0100000000
10720000000
01072000000
0000100000
0000010000
0000001000
000015007200
0000116900720
00002426620072

G:=sub<GL(10,GF(73))| [0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,1,64,9,0,16,6,0,0,0,0,0,64,0,0,34,16,0,0,0,0,0,0,8,0,0,16,0,0,0,0,0,0,0,1,0,65,0,0,0,0,0,0,0,0,8,8,0,0,0,0,0,0,0,0,0,64],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,64,0,0,55,16,15,0,0,0,0,0,64,0,0,34,71,0,0,0,0,0,0,64,0,0,57,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,64,0,0,0,8,0,0,0,0,0,63,9,65,0,2,12,0,0,0,0,0,1,0,0,16,0,0,0,0,0,0,0,0,0,65,0,0,0,0,0,0,0,0,1,9,0,0,0,0,0,0,0,0,0,63,64],[0,0,0,67,0,0,0,0,0,0,0,0,67,0,0,0,0,0,0,0,0,12,0,12,0,0,0,0,0,0,12,0,12,0,0,0,0,0,0,0,0,0,0,0,58,0,13,33,68,49,0,0,0,0,0,58,58,0,35,38,0,0,0,0,0,0,11,0,0,12,0,0,0,0,2,42,47,15,0,56,0,0,0,0,0,29,42,0,15,0,0,0,0,0,0,0,2,0,0,62],[1,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,0,0,15,11,24,0,0,0,0,0,1,0,0,69,26,0,0,0,0,0,0,1,0,0,62,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72] >;
 

He3⋊8SD16 in GAP, Magma, Sage, TeX 

{\rm He}_3\rtimes_8{\rm SD}_{16}
 
% in TeX
 
G:=Group("He3:8SD16");
 
// GroupNames label
 
G:=SmallGroup(432,152);
 
// by ID
 
G=gap.SmallGroup(432,152);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,1011,514,80,4037,2035,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^3>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁