(D4xC10):18C4 - GroupNames

G = (D₄×C₁₀)⋊₁₈C₄ order 320 = 2⁶·5

2^nd semidirect product of D₄×C₁₀ and C₄ acting via C₄/C₂=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (D₄×C₁₀)⋊₁₈C₄, (C₂×D₄)⋊₄Dic₅, C₂₀.₂₀₃(C₂×D₄), (C₂×C₂₀).₁₈₉D₄, D₄.₅(C₂×Dic₅), (C₂²×D₄).₂D₅, (C₂×D₄).₁₉₆D₁₀, D₄⋊Dic₅⋊₃₇C₂, C₄⋊Dic₅⋊₆₈C₂², C₄.₈(C₂³.D₅), C₂₀.₇₈(C₂²⋊C₄), (C₂×C₂₀).₄₇₀C₂³, C₂₀.₁₃₉(C₂²×C₄), (C₂²×C₁₀).₁₉₅D₄, (C₂²×C₄).₁₄₇D₁₀, C₂³.₈₄(C₅⋊D₄), C₅⋊₅(C₂³.₃₇D₄), C₄.₁₀(C₂²×Dic₅), C₁₀.₁₀₁(C₈⋊C₂²), C₂.₅(D₄.D₁₀), (D₄×C₁₀).₂₃₈C₂², C₂³.₂₁D₁₀⋊₁₈C₂, (C₂²×C₂₀).₁₉₅C₂², C₂².₂₀(C₂³.D₅), (D₄×C₂×C₁₀).₂C₂, C₄.₈₉(C₂×C₅⋊D₄), (C₂×C₅⋊₂C₈)⋊₉C₂², (C₅×D₄).₃₆(C₂×C₄), (C₂×C₂₀).₂₈₉(C₂×C₄), C₂.₈(C₂×C₂³.D₅), (C₂×C₁₀).₅₅₂(C₂×D₄), (C₂×C₄.Dic₅)⋊₁₇C₂, (C₂×C₄).₂₃(C₂×Dic₅), C₂².₉₀(C₂×C₅⋊D₄), (C₂×C₄).₁₉₆(C₅⋊D₄), C₁₀.₁₁₃(C₂×C₂²⋊C₄), (C₂×C₄).₅₅₇(C₂²×D₅), (C₂×C₁₀).₁₇₅(C₂²⋊C₄), SmallGroup(320,842)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — (D₄×C₁₀)⋊₁₈C₄

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂×C₂₀ — C₄⋊Dic₅ — C₂³.₂₁D₁₀ — (D₄×C₁₀)⋊₁₈C₄

Lower central

C₅ — C₁₀ — C₂₀ — (D₄×C₁₀)⋊₁₈C₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²×D₄

Generators and relations for (D₄×C₁₀)⋊₁₈C₄
G = < a,b,c,d | a¹⁰=b⁴=c²=d⁴=1, ab=ba, ac=ca, dad^-1=a^-1b², cbc=dbd^-1=b^-1, dcd^-1=b^-1c >

Subgroups: 526 in 190 conjugacy classes, 71 normal (23 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, C₂³, C₂³, C₁₀, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, Dic₅, C₂₀, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, D₄⋊C₄, C₄²⋊C₂, C₂×M₄(2), C₂²×D₄, C₅⋊₂C₈, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₅×D₄, C₂²×C₁₀, C₂²×C₁₀, C₂³.₃₇D₄, C₂×C₅⋊₂C₈, C₄.Dic₅, C₄×Dic₅, C₄⋊Dic₅, C₂³.D₅, C₂²×C₂₀, D₄×C₁₀, D₄×C₁₀, C₂³×C₁₀, D₄⋊Dic₅, C₂×C₄.Dic₅, C₂³.₂₁D₁₀, D₄×C₂×C₁₀, (D₄×C₁₀)⋊₁₈C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, D₅, C₂²⋊C₄, C₂²×C₄, C₂×D₄, Dic₅, D₁₀, C₂×C₂²⋊C₄, C₈⋊C₂², C₂×Dic₅, C₅⋊D₄, C₂²×D₅, C₂³.₃₇D₄, C₂³.D₅, C₂²×Dic₅, C₂×C₅⋊D₄, D₄.D₁₀, C₂×C₂³.D₅, (D₄×C₁₀)⋊₁₈C₄

Smallest permutation representation of (D₄×C₁₀)⋊₁₈C₄
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

62 conjugacy classes

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

62 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+		+	+	+	+	-	+			+
image	C₁	C₂	C₂	C₂	C₂	C₄	D₄	D₄	D₅	D₁₀	Dic₅	D₁₀	C₅⋊D₄	C₅⋊D₄	C₈⋊C₂²	D₄.D₁₀
kernel	(D₄×C₁₀)⋊₁₈C₄	D₄⋊Dic₅	C₂×C₄.Dic₅	C₂³.₂₁D₁₀	D₄×C₂×C₁₀	D₄×C₁₀	C₂×C₂₀	C₂²×C₁₀	C₂²×D₄	C₂²×C₄	C₂×D₄	C₂×D₄	C₂×C₄	C₂³	C₁₀	C₂
# reps	1	4	1	1	1	8	3	1	2	2	8	4	12	4	2	8

Matrix representation of (D₄×C₁₀)⋊₁₈C₄ ►in GL₆(𝔽₄₁)

40	0	0	0	0	0
0	40	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	10	0
0	0	0	0	0	10

40	0	0	0	0	0
0	40	0	0	0	0
0	0	40	39	0	0
0	0	1	1	0	0
0	0	0	0	1	2
0	0	0	0	40	40

1	0	0	0	0	0
0	40	0	0	0	0
0	0	40	39	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	40	40

0	40	0	0	0	0
1	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	0	1	0	0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1,0,0,0,0,0,0,1,40,0,0,0,0,2,40],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,39,1,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

(D₄×C₁₀)⋊₁₈C₄ in GAP, Magma, Sage, TeX

(D_4\times C_{10})\rtimes_{18}C_4

% in TeX

G:=Group("(D4xC10):18C4");

// GroupNames label

G:=SmallGroup(320,842);

// by ID

G=gap.SmallGroup(320,842);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,1684,438,102,12550]);

// Polycyclic

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

// generators/relations

G = (D4×C10)⋊18C4 order 320 = 26·5

2nd semidirect product of D4×C10 and C4 acting via C4/C2=C2

G = (D₄×C₁₀)⋊₁₈C₄ order 320 = 2⁶·5

2^nd semidirect product of D₄×C₁₀ and C₄ acting via C₄/C₂=C₂