C4^2:D10 - GroupNames

G = C₄²⋊D₁₀ order 320 = 2⁶·5

1^st semidirect product of C₄² and D₁₀ acting via D₁₀/C₅=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄²⋊₁D₁₀, M₄(2)⋊₁₆D₁₀, C₄≀C₂⋊₅D₅, (D₄×D₅)⋊₄C₄, (Q₈×D₅)⋊₄C₄, D₄⋊₂D₅⋊₄C₄, Q₈⋊₂D₅⋊₄C₄, D₄.₁₁(C₄×D₅), C₄.₂₀₁(D₄×D₅), Q₈.₁₁(C₄×D₅), D₂₀⋊₄C₄⋊₄C₂, D₂₀⋊₇C₄⋊₆C₂, (C₄×C₂₀)⋊₁₀C₂², C₄○D₄.₁₉D₁₀, (C₄×D₅).₁₀₄D₄, D₂₀.₁₉(C₂×C₄), C₂₀.₃₆₀(C₂×D₄), C₄²⋊D₅⋊₉C₂, (D₅×M₄(2))⋊₉C₂, C₂².₂₈(D₄×D₅), D₄⋊₂Dic₅⋊₂C₂, C₂₀.₅₃(C₂²×C₄), (C₂×Dic₅).₄₅D₄, (C₄×Dic₅)⋊₃C₂², (C₂²×D₅).₃₁D₄, C₄.Dic₅⋊₃C₂², (C₂×C₂₀).₂₆₁C₂³, Dic₁₀.₂₀(C₂×C₄), C₅⋊₄(C₄²⋊C₂²), C₄○D₂₀.₁₀C₂², D₁₀.₂₇(C₂²⋊C₄), (C₅×M₄(2))⋊₁₄C₂², Dic₅.₃₉(C₂²⋊C₄), (C₅×C₄≀C₂)⋊₆C₂, C₄.₁₈(C₂×C₄×D₅), (D₅×C₄○D₄).₂C₂, (C₄×D₅).₅(C₂×C₄), (C₅×D₄).₁₉(C₂×C₄), (C₂×C₁₀).₂₅(C₂×D₄), (C₅×Q₈).₂₀(C₂×C₄), C₂.₂₆(D₅×C₂²⋊C₄), (C₂×C₄×D₅).₃₃C₂², C₁₀.₆₆(C₂×C₂²⋊C₄), (C₅×C₄○D₄).₂C₂², (C₂×C₄).₃₆₈(C₂²×D₅), SmallGroup(320,448)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — C₄²⋊D₁₀

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₂×C₂₀ — C₂×C₄×D₅ — D₅×C₄○D₄ — C₄²⋊D₁₀

Lower central

C₅ — C₁₀ — C₂₀ — C₄²⋊D₁₀

Upper central

C₁ — C₄ — C₂×C₄ — C₄≀C₂

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

Subgroups: 638 in 154 conjugacy classes, 51 normal (all characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, D₅, C₁₀, C₁₀, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, Dic₅, Dic₅, C₂₀, C₂₀, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₄≀C₂, C₄≀C₂, C₄²⋊C₂, C₂×M₄(2), C₂×C₄○D₄, C₅⋊₂C₈, C₄₀, Dic₁₀, Dic₁₀, C₄×D₅, C₄×D₅, D₂₀, D₂₀, C₂×Dic₅, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₅×D₄, C₅×Q₈, C₂²×D₅, C₂²×D₅, C₄²⋊C₂², C₈×D₅, C₈⋊D₅, C₄.Dic₅, C₄×Dic₅, C₁₀.D₄, D₁₀⋊C₄, C₄×C₂₀, C₅×M₄(2), C₂×C₄×D₅, C₂×C₄×D₅, C₄○D₂₀, C₄○D₂₀, D₄×D₅, D₄×D₅, D₄⋊₂D₅, D₄⋊₂D₅, Q₈×D₅, Q₈⋊₂D₅, C₅×C₄○D₄, D₂₀⋊₄C₄, D₂₀⋊₇C₄, D₄⋊₂Dic₅, C₅×C₄≀C₂, C₄²⋊D₅, D₅×M₄(2), D₅×C₄○D₄, C₄²⋊D₁₀
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, D₅, C₂²⋊C₄, C₂²×C₄, C₂×D₄, D₁₀, C₂×C₂²⋊C₄, C₄×D₅, C₂²×D₅, C₄²⋊C₂², C₂×C₄×D₅, D₄×D₅, D₅×C₂²⋊C₄, C₄²⋊D₁₀

Smallest permutation representation of C₄²⋊D₁₀
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

50 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	5A	5B	8A	8B	8C	8D	10A	10B	10C	10D	10E	10F	20A	20B	20C	20D	20E	···	20N	20O	20P	40A	40B	40C	40D
order	1	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	4	4	5	5	8	8	8	8	10	10	10	10	10	10	20	20	20	20	20	···	20	20	20	40	40	40	40
size	1	1	2	4	10	10	20	1	1	2	4	4	4	10	10	20	20	20	2	2	4	4	20	20	2	2	4	4	8	8	2	2	2	2	4	···	4	8	8	8	8	8	8

50 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+	+	+	+	+					+	+	+	+	+	+	+				+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	D₄	D₄	D₄	D₅	D₁₀	D₁₀	D₁₀	C₄×D₅	C₄×D₅	C₄²⋊C₂²	D₄×D₅	D₄×D₅	C₄²⋊D₁₀
kernel	C₄²⋊D₁₀	D₂₀⋊₄C₄	D₂₀⋊₇C₄	D₄⋊₂Dic₅	C₅×C₄≀C₂	C₄²⋊D₅	D₅×M₄(2)	D₅×C₄○D₄	D₄×D₅	D₄⋊₂D₅	Q₈×D₅	Q₈⋊₂D₅	C₄×D₅	C₂×Dic₅	C₂²×D₅	C₄≀C₂	C₄²	M₄(2)	C₄○D₄	D₄	Q₈	C₅	C₄	C₂²	C₁
# reps	1	1	1	1	1	1	1	1	2	2	2	2	2	1	1	2	2	2	2	4	4	2	2	2	8

Matrix representation of C₄²⋊D₁₀ ►in GL₆(𝔽₄₁)

9	0	0	0	0	0
0	9	0	0	0	0
0	0	40	40	40	37
0	0	0	0	32	0
0	0	0	9	0	0
0	0	21	39	23	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	9	0	0	0
0	0	0	32	0	0
0	0	0	0	32	0
0	0	0	25	25	9

35	35	0	0	0	0
6	40	0	0	0	0
0	0	0	0	1	0
0	0	40	40	40	37
0	0	1	0	0	0
0	0	0	0	0	1

35	35	0	0	0	0
40	6	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	40	40	40	37
0	0	0	0	0	1

G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,21,0,0,40,0,9,39,0,0,40,32,0,23,0,0,37,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,25,0,0,0,0,32,25,0,0,0,0,0,9],[35,6,0,0,0,0,35,40,0,0,0,0,0,0,0,40,1,0,0,0,0,40,0,0,0,0,1,40,0,0,0,0,0,37,0,1],[35,40,0,0,0,0,35,6,0,0,0,0,0,0,0,1,40,0,0,0,1,0,40,0,0,0,0,0,40,0,0,0,0,0,37,1] >;

C₄²⋊D₁₀ in GAP, Magma, Sage, TeX

C_4^2\rtimes D_{10}

% in TeX

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

// GroupNames label

G:=SmallGroup(320,448);

// by ID

G=gap.SmallGroup(320,448);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,58,136,851,438,102,12550]);

// Polycyclic

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

// generators/relations

G = C42⋊D10 order 320 = 26·5

1st semidirect product of C42 and D10 acting via D10/C5=C22

G = C₄²⋊D₁₀ order 320 = 2⁶·5

1^st semidirect product of C₄² and D₁₀ acting via D₁₀/C₅=C₂²