SD16xHe3 - GroupNames

G = SD₁₆×He₃ order 432 = 2⁴·3³

Direct product of SD₁₆ and He₃

direct product, metabelian, nilpotent (class 3), monomial

Aliases: SD₁₆×He₃, C₁₂.₂₁C₆², (C₃×C₂₄)⋊₆C₆, C₈⋊₂(C₂×He₃), D₄.(C₂×He₃), Q₈⋊₂(C₂×He₃), (Q₈×He₃)⋊₇C₂, C₂.₄(D₄×He₃), C₂₄.₁₁(C₃×C₆), (C₈×He₃)⋊₁₀C₂, (D₄×He₃).₃C₂, (C₃²×SD₁₆)⋊C₃, (Q₈×C₃²)⋊₇C₆, (C₂×He₃).₄₂D₄, (D₄×C₃²).₃C₆, C₃²⋊₉(C₃×SD₁₆), C₆.₃₁(D₄×C₃²), C₄.₂(C₂²×He₃), C₃.₂(C₃²×SD₁₆), (C₄×He₃).₅₁C₂², (C₃×SD₁₆).₂C₃², (C₃×D₄).₆(C₃×C₆), (C₃×C₆).₃₄(C₃×D₄), (C₃×C₁₂).₁₉(C₂×C₆), (C₃×Q₈).₁₅(C₃×C₆), SmallGroup(432,219)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — SD₁₆×He₃

Chief series

C₁ — C₂ — C₆ — C₁₂ — C₃×C₁₂ — C₄×He₃ — Q₈×He₃ — SD₁₆×He₃

Lower central

C₁ — C₂ — C₁₂ — SD₁₆×He₃

Upper central

C₁ — C₆ — C₄×He₃ — SD₁₆×He₃

Generators and relations for SD₁₆×He₃
G = < a,b,c,d,e | a⁸=b²=c³=d³=e³=1, bab=a³, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece^-1=cd^-1, de=ed >

Subgroups: 285 in 110 conjugacy classes, 49 normal (21 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₆, C₆, C₈, D₄, Q₈, C₃², C₁₂, C₁₂, C₂×C₆, SD₁₆, C₃×C₆, C₃×C₆, C₂₄, C₂₄, C₃×D₄, C₃×D₄, C₃×Q₈, C₃×Q₈, He₃, C₃×C₁₂, C₃×C₁₂, C₆², C₃×SD₁₆, C₃×SD₁₆, C₂×He₃, C₂×He₃, C₃×C₂₄, D₄×C₃², Q₈×C₃², C₄×He₃, C₄×He₃, C₂²×He₃, C₃²×SD₁₆, C₈×He₃, D₄×He₃, Q₈×He₃, SD₁₆×He₃
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, C₃², C₂×C₆, SD₁₆, C₃×C₆, C₃×D₄, He₃, C₆², C₃×SD₁₆, C₂×He₃, D₄×C₃², C₂²×He₃, C₃²×SD₁₆, D₄×He₃, SD₁₆×He₃

Smallest permutation representation of SD₁₆×He₃
►On 72 points

Generators in S₇₂

(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)(10 12)(11 15)(14 16)(18 20)(19 23)(22 24)(25 27)(26 30)(29 31)(33 37)(34 40)(36 38)(41 47)(43 45)(44 48)(49 55)(51 53)(52 56)(58 60)(59 63)(62 64)(65 69)(66 72)(68 70)

(9 67 17)(10 68 18)(11 69 19)(12 70 20)(13 71 21)(14 72 22)(15 65 23)(16 66 24)(33 44 52)(34 45 53)(35 46 54)(36 47 55)(37 48 56)(38 41 49)(39 42 50)(40 43 51)

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

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

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

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

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

77 conjugacy classes

class	1	2A	2B	3A	3B	3C	···	3J	4A	4B	6A	6B	6C	···	6J	6K	6L	6M	···	6T	8A	8B	12A	12B	12C	12D	12E	···	12L	12M	···	12T	24A	24B	24C	24D	24E	···	24T
order	1	2	2	3	3	3	···	3	4	4	6	6	6	···	6	6	6	6	···	6	8	8	12	12	12	12	12	···	12	12	···	12	24	24	24	24	24	···	24
size	1	1	4	1	1	3	···	3	2	4	1	1	3	···	3	4	4	12	···	12	2	2	2	2	4	4	6	···	6	12	···	12	2	2	2	2	6	···	6

77 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	3	3	3	3	6	6
type	+	+	+	+					+
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	D₄	SD₁₆	C₃×D₄	C₃×SD₁₆	He₃	C₂×He₃	C₂×He₃	C₂×He₃	D₄×He₃	SD₁₆×He₃
kernel	SD₁₆×He₃	C₈×He₃	D₄×He₃	Q₈×He₃	C₃²×SD₁₆	C₃×C₂₄	D₄×C₃²	Q₈×C₃²	C₂×He₃	He₃	C₃×C₆	C₃²	SD₁₆	C₈	D₄	Q₈	C₂	C₁
# reps	1	1	1	1	8	8	8	8	1	2	8	16	2	2	2	2	2	4

Matrix representation of SD₁₆×He₃ ►in GL₅(𝔽₇₃)

0	61	0	0	0
6	61	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

1	0	0	0	0
1	72	0	0	0
0	0	72	0	0
0	0	0	72	0
0	0	0	0	72

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

1	0	0	0	0
0	1	0	0	0
0	0	8	0	0
0	0	0	8	0
0	0	0	0	8

8	0	0	0	0
0	8	0	0	0
0	0	0	64	0
0	0	0	0	64
0	0	64	0	0

G:=sub<GL(5,GF(73))| [0,6,0,0,0,61,61,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,1,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[8,0,0,0,0,0,8,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,64],[1,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,8],[8,0,0,0,0,0,8,0,0,0,0,0,0,0,64,0,0,64,0,0,0,0,0,64,0] >;

SD₁₆×He₃ in GAP, Magma, Sage, TeX

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

% in TeX

G:=Group("SD16xHe3");

// GroupNames label

G:=SmallGroup(432,219);

// by ID

G=gap.SmallGroup(432,219);

# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-3,-2,1512,533,605,8824,4421,242]);

// Polycyclic

G:=Group<a,b,c,d,e|a^8=b^2=c^3=d^3=e^3=1,b*a*b=a^3,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;

// generators/relations

G = SD16×He3 order 432 = 24·33

Direct product of SD16 and He3

G = SD₁₆×He₃ order 432 = 2⁴·3³

Direct product of SD₁₆ and He₃