D8xHe3 - GroupNames

G = D₈xHe₃ order 432 = 2⁴·3³

Direct product of D₈ and He₃

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

Aliases: D₈xHe₃, C₁₂.₁₈C₆², D₄:(C₂xHe₃), (C₃xC₂₄):₃C₆, C₈:₁(C₂xHe₃), (C₃²xD₈):C₃, C₂₄.₃(C₃xC₆), (C₈xHe₃):₅C₂, (D₄xHe₃):₇C₂, C₃²:₆(C₃xD₈), C₂.₃(D₄xHe₃), (D₄xC₃²):₄C₆, C₃.₂(C₃²xD₈), (C₂xHe₃).₄₁D₄, (C₃xD₈).₂C₃², C₆.₂₉(D₄xC₃²), C₄.₁(C₂²xHe₃), (C₄xHe₃).₅₀C₂², (C₃xD₄).₃(C₃xC₆), (C₃xC₆).₃₂(C₃xD₄), (C₃xC₁₂).₁₇(C₂xC₆), SmallGroup(432,216)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — D₈xHe₃

Chief series

C₁ — C₂ — C₆ — C₁₂ — C₃xC₁₂ — C₄xHe₃ — D₄xHe₃ — D₈xHe₃

Lower central

C₁ — C₂ — C₁₂ — D₈xHe₃

Upper central

C₁ — C₆ — C₄xHe₃ — D₈xHe₃

Generators and relations for D₈xHe₃
G = < a,b,c,d,e | a⁸=b²=c³=d³=e³=1, bab=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece^-1=cd^-1, de=ed >

Subgroups: 361 in 121 conjugacy classes, 49 normal (15 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₆, C₆, C₈, D₄, C₃², C₁₂, C₁₂, C₂xC₆, D₈, C₃xC₆, C₃xC₆, C₂₄, C₂₄, C₃xD₄, C₃xD₄, He₃, C₃xC₁₂, C₆², C₃xD₈, C₃xD₈, C₂xHe₃, C₂xHe₃, C₃xC₂₄, D₄xC₃², C₄xHe₃, C₂²xHe₃, C₃²xD₈, C₈xHe₃, D₄xHe₃, D₈xHe₃
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, C₃², C₂xC₆, D₈, C₃xC₆, C₃xD₄, He₃, C₆², C₃xD₈, C₂xHe₃, D₄xC₃², C₂²xHe₃, C₃²xD₈, D₄xHe₃, D₈xHe₃

Smallest permutation representation of D₈xHe₃
►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)

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

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

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

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

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

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

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

77 conjugacy classes

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

77 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	3	3	3	6	6
type	+	+	+				+	+
image	C₁	C₂	C₂	C₃	C₆	C₆	D₄	D₈	C₃xD₄	C₃xD₈	He₃	C₂xHe₃	C₂xHe₃	D₄xHe₃	D₈xHe₃
kernel	D₈xHe₃	C₈xHe₃	D₄xHe₃	C₃²xD₈	C₃xC₂₄	D₄xC₃²	C₂xHe₃	He₃	C₃xC₆	C₃²	D₈	C₈	D₄	C₂	C₁
# reps	1	1	2	8	8	16	1	2	8	16	2	2	4	2	4

Matrix representation of D₈xHe₃ ►in GL₅(F₇₃)

0	41	0	0	0
16	41	0	0	0
0	0	72	0	0
0	0	0	72	0
0	0	0	0	72

0	41	0	0	0
57	0	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

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

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	0	64	0
0	0	0	0	64
0	0	64	0	0

G:=sub<GL(5,GF(73))| [0,16,0,0,0,41,41,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[0,57,0,0,0,41,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[64,0,0,0,0,0,64,0,0,0,0,0,1,0,0,0,0,0,64,0,0,0,0,0,8],[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],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,64,0,0,64,0,0,0,0,0,64,0] >;

D₈xHe₃ in GAP, Magma, Sage, TeX

D_8\times {\rm He}_3

% in TeX

G:=Group("D8xHe3");

// GroupNames label

G:=SmallGroup(432,216);

// by ID

G=gap.SmallGroup(432,216);

# by ID

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

// Polycyclic

G:=Group<a,b,c,d,e|a^8=b^2=c^3=d^3=e^3=1,b*a*b=a^-1,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 = D8xHe3 order 432 = 24·33

Direct product of D8 and He3

G = D₈xHe₃ order 432 = 2⁴·3³

Direct product of D₈ and He₃