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G = He3⋊10SD16  order 432 = 24·33

1st semidirect product of He3 and SD16 acting via SD16/Q8=C2

metabelian, supersoluble, monomial

Aliases: He3⋊10SD16, (Q8×He3)⋊1C2, C12.10(S3×C6), C32⋊11SD16⋊C3, C32⋊4C8⋊3C6, (C3×C12).13D6, (Q8×C32)⋊1C6, (Q8×C32)⋊1S3, C12⋊S3.2C6, He3⋊3C8⋊10C2, (C2×He3).30D4, He3⋊4D4.3C2, Q8⋊3(C32⋊C6), C32⋊5(C3×SD16), C2.7(He3⋊6D4), C32⋊5(Q8⋊2S3), (C4×He3).11C22, (C3×C12).4(C2×C6), (C3×C6).15(C3×D4), C6.27(C3×C3⋊D4), C4.4(C2×C32⋊C6), (C3×Q8).24(C3×S3), C3.2(C3×Q8⋊2S3), (C3×C6).28(C3⋊D4), SmallGroup(432,161)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — He3⋊10SD16
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C4×He3 — He3⋊4D4 — He3⋊10SD16
Lower central
C32 — C3×C6 — C3×C12 — He3⋊10SD16
Upper central
C1 — C2 — C4 — Q8

Generators and relations for He3⋊10SD16
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, cac-1=ab-1, dad-1=eae=a-1, bc=cb, dbd-1=ebe=b-1, cd=dc, ce=ec, ede=d3 >

Subgroups: 429 in 82 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C32, C32, C12, C12, D6, C2×C6, SD16, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, D12, C3×D4, C3×Q8, C3×Q8, He3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, Q8⋊2S3, C3×SD16, C32⋊C6, C2×He3, C3×C3⋊C8, C32⋊4C8, C3×D12, C12⋊S3, Q8×C32, Q8×C32, C4×He3, C4×He3, C2×C32⋊C6, C3×Q8⋊2S3, C32⋊11SD16, He3⋊3C8, He3⋊4D4, Q8×He3, He3⋊10SD16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, SD16, C3×S3, C3⋊D4, C3×D4, S3×C6, Q8⋊2S3, C3×SD16, C32⋊C6, C3×C3⋊D4, C2×C32⋊C6, C3×Q8⋊2S3, He3⋊6D4, He3⋊10SD16

Smallest permutation representation of He3⋊10SD16
►On 72 points
Generators in S72
(9 29 24)(10 17 30)(11 31 18)(12 19 32)(13 25 20)(14 21 26)(15 27 22)(16 23 28)(33 41 56)(34 49 42)(35 43 50)(36 51 44)(37 45 52)(38 53 46)(39 47 54)(40 55 48)

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

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

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

41 conjugacy classes

class 1 2A2B3A3B3C3D3E3F4A4B6A6B6C6D6E6F6G6H8A8B12A12B12C12D12E12F···12P24A24B24C24D
order122333333446666666688121212121212···1224242424
size113623366624233666363618184446612···1218181818

41 irreducible representations

dim1111111112222222222244666
type+++++++++++
imageC1C2C2C2C3C6C6C6He3⋊10SD16S3D4D6SD16C3×S3C3⋊D4C3×D4S3×C6C3×SD16C3×C3⋊D4Q8⋊2S3C3×Q8⋊2S3C32⋊C6C2×C32⋊C6He3⋊6D4
kernelHe3⋊10SD16He3⋊3C8He3⋊4D4Q8×He3C32⋊11SD16C32⋊4C8C12⋊S3Q8×C32C1Q8×C32C2×He3C3×C12He3C3×Q8C3×C6C3×C6C12C32C6C32C3Q8C4C2
# reps111122221111222224412112

Matrix representation of He3⋊10SD16 ►in GL10(𝔽73)

07200000000
17200000000
00072000000
00172000000
0000100000
0000010000
00000072100
00000072000
000072720727272
0000001010
,
1000000000
0100000000
0010000000
0001000000
00007210000
00007200000
00000072100
00000072000
0000101001
00000720727272
,
1000000000
0100000000
0010000000
0001000000
0000001000
0000000100
00000000721
0000727272727172
0000000010
0000100010
,
676676000000
0606000000
667676000000
06706000000
0000010000
0000100000
0000000100
0000001000
0000000010
0000727272727272
,
17200000000
07200000000
00721000000
0001000000
0000010000
0000100000
0000000100
0000001000
0000000010
0000727272727272

G:=sub<GL(10,GF(73))| [0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,72,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,0,72,72,0,1,0,0,0,0,0,0,1,0,72,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,72,0,0,1,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,72,72,1,0,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,72,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,0,1,0,72,0,0,0,0,0,0,0,0,72,71,1,1,0,0,0,0,0,0,1,72,0,0],[67,0,6,0,0,0,0,0,0,0,6,6,67,67,0,0,0,0,0,0,67,0,67,0,0,0,0,0,0,0,6,6,6,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,72,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,72,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,72,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,72,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72] >;
 

He3⋊10SD16 in GAP, Magma, Sage, TeX 

{\rm He}_3\rtimes_{10}{\rm SD}_{16}
 
% in TeX
 
G:=Group("He3:10SD16");
 
// GroupNames label
 
G:=SmallGroup(432,161);
 
// by ID
 
G=gap.SmallGroup(432,161);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,176,1011,514,80,4037,2035,14118]);
 
// 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=e*a*e=a^-1,b*c=c*b,d*b*d^-1=e*b*e=b^-1,c*d=d*c,c*e=e*c,e*d*e=d^3>;
 
// generators/relations
 

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