C4^2:28D10 - GroupNames

G = C₄²:₂₈D₁₀ order 320 = 2⁶·5

28^th semidirect product of C₄² and D₁₀ acting via D₁₀/C₅=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄²:₂₈D₁₀, C₁₀.₈₀2₊¹⁺⁴, C₄:₁D₄:₉D₅, (C₂xD₄):₁₃D₁₀, (C₄xC₂₀):₃₇C₂², C₂³:D₁₀:₂₈C₂, (D₄xC₁₀):₃₄C₂², C₄²:₂D₅:₁₉C₂, Dic₅:D₄:₃₉C₂, (C₂xC₁₀).₂₆₄C₂⁴, (C₂xC₂₀).₆₃₈C₂³, C₂.₈₄(D₄:₆D₁₀), C₂³.D₅:₃₈C₂², C₂³.₇₀(C₂²xD₅), C₅:₅(C₂².₅₄C₂⁴), C₁₀.D₄:₃₇C₂², (C₂²xC₁₀).₇₈C₂³, (C₂³xD₅).₇₃C₂², C₂².₂₈₅(C₂³xD₅), D₁₀:C₄.₇₅C₂², C₂³.₁₈D₁₀:₂₈C₂, (C₂xDic₅).₁₃₈C₂³, (C₂²xDic₅):₃₀C₂², (C₂²xD₅).₁₁₈C₂³, (C₅xC₄:₁D₄):₁₅C₂, (C₂xC₄).₂₁₆(C₂²xD₅), (C₂xC₅:D₄).₈₀C₂², SmallGroup(320,1392)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₀ — C₄²:₂₈D₁₀

Chief series

C₁ — C₅ — C₁₀ — C₂xC₁₀ — C₂²xD₅ — C₂³xD₅ — C₂³:D₁₀ — C₄²:₂₈D₁₀

Lower central

C₅ — C₂xC₁₀ — C₄²:₂₈D₁₀

Upper central

C₁ — C₂² — C₄:₁D₄

Generators and relations for C₄²:₂₈D₁₀
G = < a,b,c,d | a⁴=b⁴=c¹⁰=d²=1, ab=ba, cac^-1=a^-1, dad=a^-1b², cbc^-1=b^-1, dbd=a²b, dcd=c^-1 >

Subgroups: 1062 in 252 conjugacy classes, 91 normal (12 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₂xC₄, C₂xC₄, D₄, C₂³, C₂³, C₂³, D₅, C₁₀, C₁₀, C₄², C₂²:C₄, C₄:C₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂⁴, Dic₅, C₂₀, D₁₀, C₂xC₁₀, C₂xC₁₀, C₂²wrC₂, C₄:D₄, C₂².D₄, C₄²:₂C₂, C₄:₁D₄, C₂xDic₅, C₂xDic₅, C₅:D₄, C₂xC₂₀, C₅xD₄, C₂²xD₅, C₂²xD₅, C₂²xC₁₀, C₂²xC₁₀, C₂².₅₄C₂⁴, C₁₀.D₄, D₁₀:C₄, C₂³.D₅, C₄xC₂₀, C₂²xDic₅, C₂xC₅:D₄, D₄xC₁₀, C₂³xD₅, C₄²:₂D₅, C₂³.₁₈D₁₀, C₂³:D₁₀, Dic₅:D₄, C₅xC₄:₁D₄, C₄²:₂₈D₁₀
Quotients: C₁, C₂, C₂², C₂³, D₅, C₂⁴, D₁₀, 2₊¹⁺⁴, C₂²xD₅, C₂².₅₄C₂⁴, C₂³xD₅, D₄:₆D₁₀, C₄²:₂₈D₁₀

Smallest permutation representation of C₄²:₂₈D₁₀
►On 80 points

Generators in S₈₀

(1 70 20 65)(2 66 16 61)(3 62 17 67)(4 68 18 63)(5 64 19 69)(6 77 15 72)(7 73 11 78)(8 79 12 74)(9 75 13 80)(10 71 14 76)(21 32 53 43)(22 44 54 33)(23 34 55 45)(24 46 56 35)(25 36 57 47)(26 48 58 37)(27 38 59 49)(28 50 60 39)(29 40 51 41)(30 42 52 31)

(1 28 9 23)(2 24 10 29)(3 30 6 25)(4 26 7 21)(5 22 8 27)(11 53 18 58)(12 59 19 54)(13 55 20 60)(14 51 16 56)(15 57 17 52)(31 72 47 67)(32 68 48 73)(33 74 49 69)(34 70 50 75)(35 76 41 61)(36 62 42 77)(37 78 43 63)(38 64 44 79)(39 80 45 65)(40 66 46 71)

(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 73 74 75 76 77 78 79 80)

(1 12)(2 11)(3 15)(4 14)(5 13)(6 17)(7 16)(8 20)(9 19)(10 18)(21 24)(22 23)(25 30)(26 29)(27 28)(31 42)(32 41)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(51 58)(52 57)(53 56)(54 55)(59 60)(61 63)(64 70)(65 69)(66 68)(71 73)(74 80)(75 79)(76 78)

G:=sub<Sym(80)| (1,70,20,65)(2,66,16,61)(3,62,17,67)(4,68,18,63)(5,64,19,69)(6,77,15,72)(7,73,11,78)(8,79,12,74)(9,75,13,80)(10,71,14,76)(21,32,53,43)(22,44,54,33)(23,34,55,45)(24,46,56,35)(25,36,57,47)(26,48,58,37)(27,38,59,49)(28,50,60,39)(29,40,51,41)(30,42,52,31), (1,28,9,23)(2,24,10,29)(3,30,6,25)(4,26,7,21)(5,22,8,27)(11,53,18,58)(12,59,19,54)(13,55,20,60)(14,51,16,56)(15,57,17,52)(31,72,47,67)(32,68,48,73)(33,74,49,69)(34,70,50,75)(35,76,41,61)(36,62,42,77)(37,78,43,63)(38,64,44,79)(39,80,45,65)(40,66,46,71), (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)(71,72,73,74,75,76,77,78,79,80), (1,12)(2,11)(3,15)(4,14)(5,13)(6,17)(7,16)(8,20)(9,19)(10,18)(21,24)(22,23)(25,30)(26,29)(27,28)(31,42)(32,41)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(51,58)(52,57)(53,56)(54,55)(59,60)(61,63)(64,70)(65,69)(66,68)(71,73)(74,80)(75,79)(76,78)>;

G:=Group( (1,70,20,65)(2,66,16,61)(3,62,17,67)(4,68,18,63)(5,64,19,69)(6,77,15,72)(7,73,11,78)(8,79,12,74)(9,75,13,80)(10,71,14,76)(21,32,53,43)(22,44,54,33)(23,34,55,45)(24,46,56,35)(25,36,57,47)(26,48,58,37)(27,38,59,49)(28,50,60,39)(29,40,51,41)(30,42,52,31), (1,28,9,23)(2,24,10,29)(3,30,6,25)(4,26,7,21)(5,22,8,27)(11,53,18,58)(12,59,19,54)(13,55,20,60)(14,51,16,56)(15,57,17,52)(31,72,47,67)(32,68,48,73)(33,74,49,69)(34,70,50,75)(35,76,41,61)(36,62,42,77)(37,78,43,63)(38,64,44,79)(39,80,45,65)(40,66,46,71), (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)(71,72,73,74,75,76,77,78,79,80), (1,12)(2,11)(3,15)(4,14)(5,13)(6,17)(7,16)(8,20)(9,19)(10,18)(21,24)(22,23)(25,30)(26,29)(27,28)(31,42)(32,41)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(51,58)(52,57)(53,56)(54,55)(59,60)(61,63)(64,70)(65,69)(66,68)(71,73)(74,80)(75,79)(76,78) );

G=PermutationGroup([[(1,70,20,65),(2,66,16,61),(3,62,17,67),(4,68,18,63),(5,64,19,69),(6,77,15,72),(7,73,11,78),(8,79,12,74),(9,75,13,80),(10,71,14,76),(21,32,53,43),(22,44,54,33),(23,34,55,45),(24,46,56,35),(25,36,57,47),(26,48,58,37),(27,38,59,49),(28,50,60,39),(29,40,51,41),(30,42,52,31)], [(1,28,9,23),(2,24,10,29),(3,30,6,25),(4,26,7,21),(5,22,8,27),(11,53,18,58),(12,59,19,54),(13,55,20,60),(14,51,16,56),(15,57,17,52),(31,72,47,67),(32,68,48,73),(33,74,49,69),(34,70,50,75),(35,76,41,61),(36,62,42,77),(37,78,43,63),(38,64,44,79),(39,80,45,65),(40,66,46,71)], [(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),(71,72,73,74,75,76,77,78,79,80)], [(1,12),(2,11),(3,15),(4,14),(5,13),(6,17),(7,16),(8,20),(9,19),(10,18),(21,24),(22,23),(25,30),(26,29),(27,28),(31,42),(32,41),(33,50),(34,49),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43),(51,58),(52,57),(53,56),(54,55),(59,60),(61,63),(64,70),(65,69),(66,68),(71,73),(74,80),(75,79),(76,78)]])

47 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	···	4I	5A	5B	10A	···	10F	10G	···	10N	20A	···	20L
order	1	2	2	2	2	2	2	2	2	2	4	4	4	4	···	4	5	5	10	···	10	10	···	10	20	···	20
size	1	1	1	1	4	4	4	4	20	20	4	4	4	20	···	20	2	2	2	···	2	8	···	8	4	···	4

47 irreducible representations

dim	1	1	1	1	1	1	2	2	2	4	4
type	+	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	D₅	D₁₀	D₁₀	2₊¹⁺⁴	D₄:₆D₁₀
kernel	C₄²:₂₈D₁₀	C₄²:₂D₅	C₂³.₁₈D₁₀	C₂³:D₁₀	Dic₅:D₄	C₅xC₄:₁D₄	C₄:₁D₄	C₄²	C₂xD₄	C₁₀	C₂
# reps	1	2	3	3	6	1	2	2	12	3	12

Matrix representation of C₄²:₂₈D₁₀ ►in GL₈(F₄₁)

18	6	0	0	0	0	0	0
35	23	0	0	0	0	0	0
0	0	18	6	0	0	0	0
0	0	35	23	0	0	0	0
0	0	0	0	26	0	7	7
0	0	0	0	39	30	36	12
0	0	0	0	13	12	13	2
0	0	0	0	25	29	2	13

0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
40	0	0	0	0	0	0	0
0	40	0	0	0	0	0	0
0	0	0	0	24	34	38	3
0	0	0	0	6	17	38	0
0	0	0	0	0	28	18	6
0	0	0	0	13	28	35	23

1	6	0	0	0	0	0	0
35	6	0	0	0	0	0	0
0	0	40	35	0	0	0	0
0	0	6	35	0	0	0	0
0	0	0	0	0	35	3	16
0	0	0	0	7	35	16	28
0	0	0	0	0	0	40	7
0	0	0	0	0	0	34	7

6	1	0	0	0	0	0	0
6	35	0	0	0	0	0	0
0	0	6	1	0	0	0	0
0	0	6	35	0	0	0	0
0	0	0	0	6	35	1	40
0	0	0	0	40	35	1	0
0	0	0	0	0	0	7	40
0	0	0	0	0	0	7	34

G:=sub<GL(8,GF(41))| [18,35,0,0,0,0,0,0,6,23,0,0,0,0,0,0,0,0,18,35,0,0,0,0,0,0,6,23,0,0,0,0,0,0,0,0,26,39,13,25,0,0,0,0,0,30,12,29,0,0,0,0,7,36,13,2,0,0,0,0,7,12,2,13],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,24,6,0,13,0,0,0,0,34,17,28,28,0,0,0,0,38,38,18,35,0,0,0,0,3,0,6,23],[1,35,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,40,6,0,0,0,0,0,0,35,35,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,35,35,0,0,0,0,0,0,3,16,40,34,0,0,0,0,16,28,7,7],[6,6,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,35,35,0,0,0,0,0,0,1,1,7,7,0,0,0,0,40,0,40,34] >;

C₄²:₂₈D₁₀ in GAP, Magma, Sage, TeX

C_4^2\rtimes_{28}D_{10}

% in TeX

G:=Group("C4^2:28D10");

// GroupNames label

G:=SmallGroup(320,1392);

// by ID

G=gap.SmallGroup(320,1392);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,1571,570,297,136,12550]);

// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^2,c*b*c^-1=b^-1,d*b*d=a^2*b,d*c*d=c^-1>;

// generators/relations

18	6	0	0	0	0	0	0
35	23	0	0	0	0	0	0
0	0	18	6	0	0	0	0
0	0	35	23	0	0	0	0
0	0	0	0	26	0	7	7
0	0	0	0	39	30	36	12
0	0	0	0	13	12	13	2
0	0	0	0	25	29	2	13

0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
40	0	0	0	0	0	0	0
0	40	0	0	0	0	0	0
0	0	0	0	24	34	38	3
0	0	0	0	6	17	38	0
0	0	0	0	0	28	18	6
0	0	0	0	13	28	35	23

1	6	0	0	0	0	0	0
35	6	0	0	0	0	0	0
0	0	40	35	0	0	0	0
0	0	6	35	0	0	0	0
0	0	0	0	0	35	3	16
0	0	0	0	7	35	16	28
0	0	0	0	0	0	40	7
0	0	0	0	0	0	34	7

6	1	0	0	0	0	0	0
6	35	0	0	0	0	0	0
0	0	6	1	0	0	0	0
0	0	6	35	0	0	0	0
0	0	0	0	6	35	1	40
0	0	0	0	40	35	1	0
0	0	0	0	0	0	7	40
0	0	0	0	0	0	7	34

18	6	0	0	0	0	0	0
35	23	0	0	0	0	0	0
0	0	18	6	0	0	0	0
0	0	35	23	0	0	0	0
0	0	0	0	26	0	7	7
0	0	0	0	39	30	36	12
0	0	0	0	13	12	13	2
0	0	0	0	25	29	2	13

0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
40	0	0	0	0	0	0	0
0	40	0	0	0	0	0	0
0	0	0	0	24	34	38	3
0	0	0	0	6	17	38	0
0	0	0	0	0	28	18	6
0	0	0	0	13	28	35	23

1	6	0	0	0	0	0	0
35	6	0	0	0	0	0	0
0	0	40	35	0	0	0	0
0	0	6	35	0	0	0	0
0	0	0	0	0	35	3	16
0	0	0	0	7	35	16	28
0	0	0	0	0	0	40	7
0	0	0	0	0	0	34	7

6	1	0	0	0	0	0	0
6	35	0	0	0	0	0	0
0	0	6	1	0	0	0	0
0	0	6	35	0	0	0	0
0	0	0	0	6	35	1	40
0	0	0	0	40	35	1	0
0	0	0	0	0	0	7	40
0	0	0	0	0	0	7	34

G = C42:28D10 order 320 = 26·5

28th semidirect product of C42 and D10 acting via D10/C5=C22

G = C₄²:₂₈D₁₀ order 320 = 2⁶·5

28^th semidirect product of C₄² and D₁₀ acting via D₁₀/C₅=C₂²

18	6	0	0	0	0	0	0
35	23	0	0	0	0	0	0
0	0	18	6	0	0	0	0
0	0	35	23	0	0	0	0
0	0	0	0	26	0	7	7
0	0	0	0	39	30	36	12
0	0	0	0	13	12	13	2
0	0	0	0	25	29	2	13

0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
40	0	0	0	0	0	0	0
0	40	0	0	0	0	0	0
0	0	0	0	24	34	38	3
0	0	0	0	6	17	38	0
0	0	0	0	0	28	18	6
0	0	0	0	13	28	35	23

1	6	0	0	0	0	0	0
35	6	0	0	0	0	0	0
0	0	40	35	0	0	0	0
0	0	6	35	0	0	0	0
0	0	0	0	0	35	3	16
0	0	0	0	7	35	16	28
0	0	0	0	0	0	40	7
0	0	0	0	0	0	34	7

6	1	0	0	0	0	0	0
6	35	0	0	0	0	0	0
0	0	6	1	0	0	0	0
0	0	6	35	0	0	0	0
0	0	0	0	6	35	1	40
0	0	0	0	40	35	1	0
0	0	0	0	0	0	7	40
0	0	0	0	0	0	7	34