C4^2:16D10 - GroupNames

G = C₄²:₁₆D₁₀ order 320 = 2⁶·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄²:₁₆D₁₀, C₁₀.₁₈2₊¹⁺⁴, C₄:C₄:₄₉D₁₀, (C₄xD₄):₁₈D₅, (D₄xC₂₀):₂₀C₂, (C₂²xC₄):₅D₁₀, (C₄xC₂₀):₃₂C₂², D₁₀:Q₈:₈C₂, C₂²:C₄:₄₈D₁₀, C₄:Dic₅:₉C₂², (C₂xD₄).₂₁₇D₁₀, C₂³:D₁₀.₅C₂, C₄²:₂D₅:₁₆C₂, C₄²:D₅:₃₂C₂, D₁₀.₃₁(C₄oD₄), D₁₀.₁₂D₄:₇C₂, C₂₀.₄₈D₄:₁₁C₂, C₂³.D₅:₉C₂², (C₂xC₁₀).₁₀₀C₂⁴, (C₂xC₂₀).₆₉₉C₂³, (C₂²xC₂₀):₃₇C₂², Dic₅.₅D₄:₇C₂, (C₂xDic₁₀):₆C₂², (C₄xDic₅):₅₂C₂², C₂.₁₉(D₄:₆D₁₀), C₅:₃(C₂².₄₅C₂⁴), (D₄xC₁₀).₃₀₇C₂², C₂².₁₂(C₄oD₂₀), C₁₀.D₄:₄₂C₂², (C₂³xD₅).₄₁C₂², (C₂²xD₅).₃₅C₂³, C₂³.₁₇₄(C₂²xD₅), C₂².₁₂₅(C₂³xD₅), D₁₀:C₄.₈₅C₂², C₂³.₁₁D₁₀:₂₉C₂, C₂³.₂₃D₁₀:₁₆C₂, C₂³.₁₈D₁₀:₁₈C₂, (C₂²xC₁₀).₁₇₀C₂³, (C₂xDic₅).₂₁₇C₂³, (C₂²xDic₅).₉₈C₂², C₄:C₄:D₅:₇C₂, (C₄xC₅:D₄):₄₃C₂, C₂.₂₃(D₅xC₄oD₄), (C₅xC₄:C₄):₆₁C₂², (D₅xC₂²:C₄):₂₉C₂, C₂.₄₉(C₂xC₄oD₂₀), C₁₀.₁₄₀(C₂xC₄oD₄), (C₂xD₁₀:C₄):₂₂C₂, (C₂xC₄xD₅).₂₅₂C₂², (C₂xC₁₀).₁₆(C₄oD₄), (C₅xC₂²:C₄):₅₇C₂², (C₂xC₄).₂₈₄(C₂²xD₅), (C₂xC₅:D₄).₁₆C₂², SmallGroup(320,1228)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₀ — C₄²:₁₆D₁₀

Chief series

C₁ — C₅ — C₁₀ — C₂xC₁₀ — C₂²xD₅ — C₂³xD₅ — D₅xC₂²:C₄ — C₄²:₁₆D₁₀

Lower central

C₅ — C₂xC₁₀ — C₄²:₁₆D₁₀

Upper central

C₁ — C₂² — C₄xD₄

Generators and relations for C₄²:₁₆D₁₀
G = < a,b,c,d | a⁴=b⁴=c¹⁰=d²=1, ab=ba, cac^-1=dad=a^-1b², bc=cb, dbd=a²b, dcd=c^-1 >

Subgroups: 934 in 248 conjugacy classes, 97 normal (91 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₂³, D₅, C₁₀, C₁₀, C₄², C₄², C₂²:C₄, C₂²:C₄, C₄:C₄, C₄:C₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₂⁴, Dic₅, C₂₀, D₁₀, D₁₀, C₂xC₁₀, C₂xC₁₀, C₂xC₁₀, C₂xC₂²:C₄, C₄²:C₂, C₄xD₄, C₄xD₄, C₂²wrC₂, C₂²:Q₈, C₂².D₄, C₄.₄D₄, C₄²:₂C₂, Dic₁₀, C₄xD₅, C₂xDic₅, C₂xDic₅, C₅:D₄, C₂xC₂₀, C₂xC₂₀, C₅xD₄, C₂²xD₅, C₂²xD₅, C₂²xC₁₀, C₂².₄₅C₂⁴, C₄xDic₅, C₁₀.D₄, C₄:Dic₅, D₁₀:C₄, C₂³.D₅, C₄xC₂₀, C₅xC₂²:C₄, C₅xC₄:C₄, C₂xDic₁₀, C₂xC₄xD₅, C₂²xDic₅, C₂xC₅:D₄, C₂²xC₂₀, D₄xC₁₀, C₂³xD₅, C₄²:D₅, C₄²:₂D₅, C₂³.₁₁D₁₀, D₅xC₂²:C₄, D₁₀.₁₂D₄, Dic₅.₅D₄, D₁₀:Q₈, C₄:C₄:D₅, C₂₀.₄₈D₄, C₂xD₁₀:C₄, C₄xC₅:D₄, C₂³.₂₃D₁₀, C₂³.₁₈D₁₀, C₂³:D₁₀, D₄xC₂₀, C₄²:₁₆D₁₀
Quotients: C₁, C₂, C₂², C₂³, D₅, C₄oD₄, C₂⁴, D₁₀, C₂xC₄oD₄, 2₊¹⁺⁴, C₂²xD₅, C₂².₄₅C₂⁴, C₄oD₂₀, C₂³xD₅, C₂xC₄oD₂₀, D₄:₆D₁₀, D₅xC₄oD₄, C₄²:₁₆D₁₀

Smallest permutation representation of C₄²:₁₆D₁₀
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

65 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	···	4F	4G	4H	4I	4J	4K	···	4O	5A	5B	10A	···	10F	10G	···	10N	20A	···	20H	20I	···	20X
order	1	2	2	2	2	2	2	2	2	2	4	···	4	4	4	4	4	4	···	4	5	5	10	···	10	10	···	10	20	···	20	20	···	20
size	1	1	1	1	2	2	4	10	10	20	2	···	2	4	4	10	10	20	···	20	2	2	2	···	2	4	···	4	2	···	2	4	···	4

65 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+			+	+	+	+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₅	C₄oD₄	C₄oD₄	D₁₀	D₁₀	D₁₀	D₁₀	D₁₀	C₄oD₂₀	2₊¹⁺⁴	D₄:₆D₁₀	D₅xC₄oD₄
kernel	C₄²:₁₆D₁₀	C₄²:D₅	C₄²:₂D₅	C₂³.₁₁D₁₀	D₅xC₂²:C₄	D₁₀.₁₂D₄	Dic₅.₅D₄	D₁₀:Q₈	C₄:C₄:D₅	C₂₀.₄₈D₄	C₂xD₁₀:C₄	C₄xC₅:D₄	C₂³.₂₃D₁₀	C₂³.₁₈D₁₀	C₂³:D₁₀	D₄xC₂₀	C₄xD₄	D₁₀	C₂xC₁₀	C₄²	C₂²:C₄	C₄:C₄	C₂²xC₄	C₂xD₄	C₂²	C₁₀	C₂	C₂
# reps	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	4	4	2	4	2	4	2	16	1	4	4

Matrix representation of C₄²:₁₆D₁₀ ►in GL₆(F₄₁)

40	0	0	0	0	0
0	40	0	0	0	0
0	0	9	0	0	0
0	0	0	9	0	0
0	0	0	0	14	18
0	0	0	0	37	27

1	0	0	0	0	0
0	1	0	0	0	0
0	0	15	37	0	0
0	0	36	26	0	0
0	0	0	0	9	0
0	0	0	0	0	9

0	7	0	0	0	0
35	6	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	3	40

35	7	0	0	0	0
36	6	0	0	0	0
0	0	1	0	0	0
0	0	28	40	0	0
0	0	0	0	1	0
0	0	0	0	3	40

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,14,37,0,0,0,0,18,27],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,36,0,0,0,0,37,26,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[0,35,0,0,0,0,7,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,0,40],[35,36,0,0,0,0,7,6,0,0,0,0,0,0,1,28,0,0,0,0,0,40,0,0,0,0,0,0,1,3,0,0,0,0,0,40] >;

C₄²:₁₆D₁₀ in GAP, Magma, Sage, TeX

C_4^2\rtimes_{16}D_{10}

% in TeX

G:=Group("C4^2:16D10");

// GroupNames label

G:=SmallGroup(320,1228);

// by ID

G=gap.SmallGroup(320,1228);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,1571,136,12550]);

// Polycyclic

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

// generators/relations

G = C42:16D10 order 320 = 26·5

16th semidirect product of C42 and D10 acting via D10/C5=C22

G = C₄²:₁₆D₁₀ order 320 = 2⁶·5

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