He3:9SD16 - GroupNames

G = He₃⋊₉SD₁₆ order 432 = 2⁴·3³

2^nd semidirect product of He₃ and SD₁₆ acting via SD₁₆/D₄=C₂

Aliases: He₃⋊₉SD₁₆, He₃⋊₄C₈⋊₆C₂, He₃⋊₄Q₈⋊₂C₂, (C₃×C₁₂).₁₇D₆, (D₄×He₃).₂C₂, D₄.(He₃⋊C₂), (C₂×He₃).₃₅D₄, (D₄×C₃²).₄S₃, C₂.₅(He₃⋊₇D₄), C₃²⋊₆(D₄.S₃), C₆.₄₁(C₃²⋊₇D₄), (C₄×He₃).₁₃C₂², C₃.₂(C₃²⋊₉SD₁₆), C₁₂.₄₅(C₂×C₃⋊S₃), (C₃×D₄).₇(C₃⋊S₃), C₄.₂(C₂×He₃⋊C₂), (C₃×C₆).₃₆(C₃⋊D₄), SmallGroup(432,193)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — C₄×He₃ — He₃⋊₉SD₁₆

Chief series

C₁ — C₃ — C₃² — He₃ — C₂×He₃ — C₄×He₃ — He₃⋊₄Q₈ — He₃⋊₉SD₁₆

Lower central

He₃ — C₂×He₃ — C₄×He₃ — He₃⋊₉SD₁₆

Upper central

C₁ — C₆ — C₁₂ — C₃×D₄

Generators and relations for He₃⋊₉SD₁₆
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, bd=db, be=eb, dcd^-1=c^-1, ce=ec, ede=d³ >

Subgroups: 405 in 110 conjugacy classes, 31 normal (15 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₆, C₆, C₈, D₄, Q₈, 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×C₁₂, C₆², D₄.S₃, C₃×SD₁₆, C₂×He₃, C₂×He₃, C₃×C₃⋊C₈, C₃×Dic₆, D₄×C₃², He₃⋊₃C₄, C₄×He₃, C₂²×He₃, C₃×D₄.S₃, He₃⋊₄C₈, He₃⋊₄Q₈, D₄×He₃, He₃⋊₉SD₁₆
Quotients: C₁, C₂, C₂², S₃, D₄, D₆, SD₁₆, C₃⋊S₃, C₃⋊D₄, C₂×C₃⋊S₃, D₄.S₃, He₃⋊C₂, C₃²⋊₇D₄, C₂×He₃⋊C₂, C₃²⋊₉SD₁₆, He₃⋊₇D₄, He₃⋊₉SD₁₆

Smallest permutation representation of He₃⋊₉SD₁₆
►On 72 points

Generators in S₇₂

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

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

(9 22 64)(10 57 23)(11 24 58)(12 59 17)(13 18 60)(14 61 19)(15 20 62)(16 63 21)(33 66 53)(34 54 67)(35 68 55)(36 56 69)(37 70 49)(38 50 71)(39 72 51)(40 52 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)

(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 19)(18 22)(21 23)(25 29)(26 32)(28 30)(33 39)(35 37)(36 40)(42 44)(43 47)(46 48)(49 55)(51 53)(52 56)(57 63)(59 61)(60 64)(65 69)(66 72)(68 70)

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

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

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

41 conjugacy classes

class	1	2A	2B	3A	3B	3C	3D	3E	3F	4A	4B	6A	6B	6C	6D	6E	6F	6G	6H	6I	···	6P	8A	8B	12A	12B	12C	12D	12E	12F	12G	12H	24A	24B	24C	24D
order	1	2	2	3	3	3	3	3	3	4	4	6	6	6	6	6	6	6	6	6	···	6	8	8	12	12	12	12	12	12	12	12	24	24	24	24
size	1	1	4	1	1	6	6	6	6	2	36	1	1	4	4	6	6	6	6	12	···	12	18	18	2	2	12	12	12	12	36	36	18	18	18	18

41 irreducible representations

dim	1	1	1	1	2	2	2	2	2	3	3	4	6	6
type	+	+	+	+	+	+	+					-
image	C₁	C₂	C₂	C₂	S₃	D₄	D₆	SD₁₆	C₃⋊D₄	He₃⋊C₂	C₂×He₃⋊C₂	D₄.S₃	He₃⋊₇D₄	He₃⋊₉SD₁₆
kernel	He₃⋊₉SD₁₆	He₃⋊₄C₈	He₃⋊₄Q₈	D₄×He₃	D₄×C₃²	C₂×He₃	C₃×C₁₂	He₃	C₃×C₆	D₄	C₄	C₃²	C₂	C₁
# reps	1	1	1	1	4	1	4	2	8	4	4	4	2	4

Matrix representation of He₃⋊₉SD₁₆ ►in GL₅(𝔽₇₃)

1	0	0	0	0
0	1	0	0	0
0	0	0	1	0
0	0	0	0	1
0	0	1	0	0

1	0	0	0	0
0	1	0	0	0
0	0	64	0	0
0	0	0	64	0
0	0	0	0	64

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	8	0
0	0	0	0	64

0	55	0	0	0
4	12	0	0	0
0	0	1	0	0
0	0	0	0	1
0	0	0	1	0

1	0	0	0	0
48	72	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,64],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,64],[0,4,0,0,0,55,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0],[1,48,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

He₃⋊₉SD₁₆ in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_9{\rm SD}_{16}

% in TeX

G:=Group("He3:9SD16");

// GroupNames label

G:=SmallGroup(432,193);

// by ID

G=gap.SmallGroup(432,193);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,254,135,58,1124,4037,537]);

// 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,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e=d^3>;

// generators/relations

G = He3⋊9SD16 order 432 = 24·33

2nd semidirect product of He3 and SD16 acting via SD16/D4=C2

G = He₃⋊₉SD₁₆ order 432 = 2⁴·3³

2^nd semidirect product of He₃ and SD₁₆ acting via SD₁₆/D₄=C₂