D8xES-(3,1) - GroupNames

G = D₈×3_-¹⁺² order 432 = 2⁴·3³

Direct product of D₈ and 3_-¹⁺²

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

Aliases: D₈×3_-¹⁺², C₇₂⋊₅C₆, C₁₂.₁₉C₆², (C₉×D₈)⋊C₃, C₉⋊₃(C₃×D₈), (D₄×C₉)⋊₄C₆, C₃².(C₃×D₈), (C₃×C₂₄).₂C₆, C₂₄.₄(C₃×C₆), (C₃²×D₈).C₃, C₃₆.₁₇(C₂×C₆), C₁₈.₁₄(C₃×D₄), C₃.₃(C₃²×D₈), (D₄×C₃²).₂C₆, (C₃×D₈).₃C₃², C₆.₃₀(D₄×C₃²), D₄⋊(C₂×3_-¹⁺²), C₈⋊₁(C₂×3_-¹⁺²), (D₄×3_-¹⁺²)⋊₄C₂, (C₈×3_-¹⁺²)⋊₅C₂, C₂.₃(D₄×3_-¹⁺²), (C₂×3_-¹⁺²).₁₄D₄, C₄.₁(C₂²×3_-¹⁺²), (C₄×3_-¹⁺²).₁₇C₂², (C₃×D₄).₄(C₃×C₆), (C₃×C₆).₃₃(C₃×D₄), (C₃×C₁₂).₁₈(C₂×C₆), SmallGroup(432,217)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — D₈×3_-¹⁺²

Chief series

C₁ — C₂ — C₆ — C₁₂ — C₃×C₁₂ — C₄×3_-¹⁺² — D₄×3_-¹⁺² — D₈×3_-¹⁺²

Lower central

C₁ — C₂ — C₁₂ — D₈×3_-¹⁺²

Upper central

C₁ — C₆ — C₄×3_-¹⁺² — D₈×3_-¹⁺²

Generators and relations for D₈×3_-¹⁺²
G = < a,b,c,d | a⁸=b²=c⁹=d³=1, bab=a^-1, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c⁴ >

Subgroups: 190 in 88 conjugacy classes, 49 normal (20 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₆, C₆, C₈, D₄, C₉, C₃², C₁₂, C₁₂, C₂×C₆, D₈, C₁₈, C₁₈, C₃×C₆, C₃×C₆, C₂₄, C₂₄, C₃×D₄, C₃×D₄, 3_-¹⁺², C₃₆, C₂×C₁₈, C₃×C₁₂, C₆², C₃×D₈, C₃×D₈, C₂×3_-¹⁺², C₂×3_-¹⁺², C₇₂, D₄×C₉, C₃×C₂₄, D₄×C₃², C₄×3_-¹⁺², C₂²×3_-¹⁺², C₉×D₈, C₃²×D₈, C₈×3_-¹⁺², D₄×3_-¹⁺², D₈×3_-¹⁺²
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, C₃², C₂×C₆, D₈, C₃×C₆, C₃×D₄, 3_-¹⁺², C₆², C₃×D₈, C₂×3_-¹⁺², D₄×C₃², C₂²×3_-¹⁺², C₃²×D₈, D₄×3_-¹⁺², D₈×3_-¹⁺²

Smallest permutation representation of D₈×3_-¹⁺²
►On 72 points

Generators in S₇₂

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

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

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

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

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

77 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	4	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	6K	6L	8A	8B	9A	···	9F	12A	12B	12C	12D	18A	···	18F	18G	···	18R	24A	24B	24C	24D	24E	24F	24G	24H	36A	···	36F	72A	···	72L
order	1	2	2	2	3	3	3	3	4	6	6	6	6	6	6	6	6	6	6	6	6	8	8	9	···	9	12	12	12	12	18	···	18	18	···	18	24	24	24	24	24	24	24	24	36	···	36	72	···	72
size	1	1	4	4	1	1	3	3	2	1	1	3	3	4	4	4	4	12	12	12	12	2	2	3	···	3	2	2	6	6	3	···	3	12	···	12	2	2	2	2	6	6	6	6	6	···	6	6	···	6

77 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	3	3	3	6	6
type	+	+	+							+	+
image	C₁	C₂	C₂	C₃	C₃	C₆	C₆	C₆	C₆	D₄	D₈	C₃×D₄	C₃×D₄	C₃×D₈	C₃×D₈	3_-¹⁺²	C₂×3_-¹⁺²	C₂×3_-¹⁺²	D₄×3_-¹⁺²	D₈×3_-¹⁺²
kernel	D₈×3_-¹⁺²	C₈×3_-¹⁺²	D₄×3_-¹⁺²	C₉×D₈	C₃²×D₈	C₇₂	D₄×C₉	C₃×C₂₄	D₄×C₃²	C₂×3_-¹⁺²	3_-¹⁺²	C₁₈	C₃×C₆	C₉	C₃²	D₈	C₈	D₄	C₂	C₁
# reps	1	1	2	6	2	6	12	2	4	1	2	6	2	12	4	2	2	4	2	4

Matrix representation of D₈×3_-¹⁺² ►in GL₅(𝔽₇₃)

57	16	0	0	0
57	57	0	0	0
0	0	72	0	0
0	0	0	72	0
0	0	0	0	72

57	16	0	0	0
16	16	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

8	0	0	0	0
0	8	0	0	0
0	0	8	63	0
0	0	64	65	64
0	0	0	9	0

64	0	0	0	0
0	64	0	0	0
0	0	1	0	0
0	0	8	64	0
0	0	72	0	8

G:=sub<GL(5,GF(73))| [57,57,0,0,0,16,57,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[57,16,0,0,0,16,16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[8,0,0,0,0,0,8,0,0,0,0,0,8,64,0,0,0,63,65,9,0,0,0,64,0],[64,0,0,0,0,0,64,0,0,0,0,0,1,8,72,0,0,0,64,0,0,0,0,0,8] >;

D₈×3_-¹⁺² in GAP, Magma, Sage, TeX

D_8\times 3_-^{1+2}

% in TeX

G:=Group("D8xES-(3,1)");

// GroupNames label

G:=SmallGroup(432,217);

// by ID

G=gap.SmallGroup(432,217);

# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-3,-2,533,394,605,8824,4421,242]);

// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^9=d^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;

// generators/relations

G = D8×3- 1+2 order 432 = 24·33

Direct product of D8 and 3- 1+2

G = D₈×3_-¹⁺² order 432 = 2⁴·3³

Direct product of D₈ and 3_-¹⁺²