D72:C2 - GroupNames

G = D₇₂⋊C₂ order 288 = 2⁵·3²

6^th semidirect product of D₇₂ and C₂ acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₈⋊₃D₁₈, D₇₂⋊₆C₂, Q₈⋊₃D₁₈, C₇₂⋊₃C₂², SD₁₆⋊₁D₉, D₁₈.₇D₄, D₄.₃D₁₈, C₂₄.₂₉D₆, D₃₆⋊₂C₂², C₃₆.₅C₂³, Dic₉.₉D₄, D₄⋊D₉⋊₃C₂, (D₄×D₉)⋊₃C₂, C₉⋊C₈⋊₂C₂², C₈⋊D₉⋊₁C₂, C₉⋊₃(C₈⋊C₂²), C₃.(Q₈⋊₃D₆), (C₃×D₄).₅D₆, C₆.₉₃(S₃×D₄), C₂.₁₉(D₄×D₉), Q₈⋊₂D₉⋊₂C₂, Q₈⋊₃D₉⋊₁C₂, (C₉×SD₁₆)⋊₁C₂, C₁₈.₃₁(C₂×D₄), (C₃×Q₈).₂₅D₆, (Q₈×C₉)⋊₂C₂², C₄.₅(C₂²×D₉), (C₃×SD₁₆).₁S₃, (C₄×D₉).₂C₂², (D₄×C₉).₃C₂², C₁₂.₄₄(C₂²×S₃), SmallGroup(288,124)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃₆ — D₇₂⋊C₂

Chief series

C₁ — C₃ — C₉ — C₁₈ — C₃₆ — C₄×D₉ — D₄×D₉ — D₇₂⋊C₂

Lower central

C₉ — C₁₈ — C₃₆ — D₇₂⋊C₂

Upper central

C₁ — C₂ — C₄ — SD₁₆

Generators and relations for D₇₂⋊C₂
G = < a,b,c | a⁷²=b²=c²=1, bab=a^-1, cac=a¹⁹, bc=cb >

Subgroups: 604 in 102 conjugacy classes, 34 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₈, C₈, C₂×C₄, D₄, D₄, Q₈, C₂³, C₉, Dic₃, C₁₂, C₁₂, D₆, C₂×C₆, M₄(2), D₈, SD₁₆, SD₁₆, C₂×D₄, C₄○D₄, D₉, C₁₈, C₁₈, C₃⋊C₈, C₂₄, C₄×S₃, D₁₂, C₃⋊D₄, C₃×D₄, C₃×Q₈, C₂²×S₃, C₈⋊C₂², Dic₉, C₃₆, C₃₆, D₁₈, D₁₈, C₂×C₁₈, C₈⋊S₃, D₂₄, D₄⋊S₃, Q₈⋊₂S₃, C₃×SD₁₆, S₃×D₄, Q₈⋊₃S₃, C₉⋊C₈, C₇₂, C₄×D₉, C₄×D₉, D₃₆, D₃₆, C₉⋊D₄, D₄×C₉, Q₈×C₉, C₂²×D₉, Q₈⋊₃D₆, C₈⋊D₉, D₇₂, D₄⋊D₉, Q₈⋊₂D₉, C₉×SD₁₆, D₄×D₉, Q₈⋊₃D₉, D₇₂⋊C₂
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, D₉, C₂²×S₃, C₈⋊C₂², D₁₈, S₃×D₄, C₂²×D₉, Q₈⋊₃D₆, D₄×D₉, D₇₂⋊C₂

Smallest permutation representation of D₇₂⋊C₂
►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 9)(2 8)(3 7)(4 6)(10 72)(11 71)(12 70)(13 69)(14 68)(15 67)(16 66)(17 65)(18 64)(19 63)(20 62)(21 61)(22 60)(23 59)(24 58)(25 57)(26 56)(27 55)(28 54)(29 53)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)

(2 20)(3 39)(4 58)(6 24)(7 43)(8 62)(10 28)(11 47)(12 66)(14 32)(15 51)(16 70)(18 36)(19 55)(22 40)(23 59)(26 44)(27 63)(30 48)(31 67)(34 52)(35 71)(38 56)(42 60)(46 64)(50 68)(54 72)

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

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

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

39 conjugacy classes

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C	6A	6B	8A	8B	9A	9B	9C	12A	12B	18A	18B	18C	18D	18E	18F	24A	24B	36A	36B	36C	36D	36E	36F	72A	···	72F
order	1	2	2	2	2	2	3	4	4	4	6	6	8	8	9	9	9	12	12	18	18	18	18	18	18	24	24	36	36	36	36	36	36	72	···	72
size	1	1	4	18	36	36	2	2	4	18	2	8	4	36	2	2	2	4	8	2	2	2	8	8	8	4	4	4	4	4	8	8	8	4	···	4

39 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

D₉

D₁₈

C₈⋊C₂²

S₃×D₄

Q₈⋊₃D₆

D₄×D₉

D₇₂⋊C₂

kernel

D₇₂⋊C₂

C₈⋊D₉

D₇₂

D₄⋊D₉

Q₈⋊₂D₉

C₉×SD₁₆

D₄×D₉

Q₈⋊₃D₉

C₃×SD₁₆

Dic₉

D₁₈

C₂₄

C₃×D₄

C₃×Q₈

SD₁₆

C₈

D₄

Q₈

C₉

C₆

C₃

C₂

C₁

# reps

Matrix representation of D₇₂⋊C₂ ►in GL₈(𝔽₇₃)

0	0	45	3	0	0	0	0
0	0	70	42	0	0	0	0
28	70	0	0	0	0	0	0
3	31	0	0	0	0	0	0
0	0	0	0	28	28	70	71
0	0	0	0	44	44	3	0
0	0	0	0	63	39	0	5
0	0	0	0	1	0	0	1

28	70	0	0	0	0	0	0
42	45	0	0	0	0	0	0
0	0	45	3	0	0	0	0
0	0	31	28	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	72	0	0
0	0	0	0	49	54	1	25
0	0	0	0	72	0	0	72

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	72	0	0	0	0	0
0	0	0	72	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	72	0	0
0	0	0	0	19	24	72	48
0	0	0	0	0	1	0	1

G:=sub<GL(8,GF(73))| [0,0,28,3,0,0,0,0,0,0,70,31,0,0,0,0,45,70,0,0,0,0,0,0,3,42,0,0,0,0,0,0,0,0,0,0,28,44,63,1,0,0,0,0,28,44,39,0,0,0,0,0,70,3,0,0,0,0,0,0,71,0,5,1],[28,42,0,0,0,0,0,0,70,45,0,0,0,0,0,0,0,0,45,31,0,0,0,0,0,0,3,28,0,0,0,0,0,0,0,0,1,0,49,72,0,0,0,0,0,72,54,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,25,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,19,0,0,0,0,0,0,72,24,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,48,1] >;

D₇₂⋊C₂ in GAP, Magma, Sage, TeX

D_{72}\rtimes C_2

% in TeX

G:=Group("D72:C2");

// GroupNames label

G:=SmallGroup(288,124);

// by ID

G=gap.SmallGroup(288,124);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,135,100,346,185,80,6725,292,9414]);

// Polycyclic

G:=Group<a,b,c|a^72=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^19,b*c=c*b>;

// generators/relations

0	0	45	3	0	0	0	0
0	0	70	42	0	0	0	0
28	70	0	0	0	0	0	0
3	31	0	0	0	0	0	0
0	0	0	0	28	28	70	71
0	0	0	0	44	44	3	0
0	0	0	0	63	39	0	5
0	0	0	0	1	0	0	1

28	70	0	0	0	0	0	0
42	45	0	0	0	0	0	0
0	0	45	3	0	0	0	0
0	0	31	28	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	72	0	0
0	0	0	0	49	54	1	25
0	0	0	0	72	0	0	72

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	72	0	0	0	0	0
0	0	0	72	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	72	0	0
0	0	0	0	19	24	72	48
0	0	0	0	0	1	0	1

0	0	45	3	0	0	0	0
0	0	70	42	0	0	0	0
28	70	0	0	0	0	0	0
3	31	0	0	0	0	0	0
0	0	0	0	28	28	70	71
0	0	0	0	44	44	3	0
0	0	0	0	63	39	0	5
0	0	0	0	1	0	0	1

28	70	0	0	0	0	0	0
42	45	0	0	0	0	0	0
0	0	45	3	0	0	0	0
0	0	31	28	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	72	0	0
0	0	0	0	49	54	1	25
0	0	0	0	72	0	0	72

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	72	0	0	0	0	0
0	0	0	72	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	72	0	0
0	0	0	0	19	24	72	48
0	0	0	0	0	1	0	1

G = D72⋊C2 order 288 = 25·32

6th semidirect product of D72 and C2 acting faithfully

G = D₇₂⋊C₂ order 288 = 2⁵·3²

6^th semidirect product of D₇₂ and C₂ acting faithfully

0	0	45	3	0	0	0	0
0	0	70	42	0	0	0	0
28	70	0	0	0	0	0	0
3	31	0	0	0	0	0	0
0	0	0	0	28	28	70	71
0	0	0	0	44	44	3	0
0	0	0	0	63	39	0	5
0	0	0	0	1	0	0	1

28	70	0	0	0	0	0	0
42	45	0	0	0	0	0	0
0	0	45	3	0	0	0	0
0	0	31	28	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	72	0	0
0	0	0	0	49	54	1	25
0	0	0	0	72	0	0	72

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	72	0	0	0	0	0
0	0	0	72	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	72	0	0
0	0	0	0	19	24	72	48
0	0	0	0	0	1	0	1