(D4xC10).29C4 - GroupNames

G = (D₄×C₁₀).₂₉C₄ order 320 = 2⁶·5

10^th non-split extension by D₄×C₁₀ of C₄ acting via C₄/C₂=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (D₄×C₁₀).₂₉C₄, C₂₀.₂₁₄(C₂×D₄), (C₂×C₂₀).₁₉₉D₄, (C₂×D₄).₉Dic₅, (C₂×D₄).₂₀₃D₁₀, C₂₀.D₄⋊₁₃C₂, (C₂²×C₂₀).₁₃C₄, (C₂×Q₈).₁₇₂D₁₀, C₂₀.₁₀D₄⋊₁₃C₂, C₂³.₃(C₂×Dic₅), (C₂²×C₄).₆Dic₅, (C₂×C₂₀).₄₈₃C₂³, (C₂²×C₄).₁₆₃D₁₀, C₄.₃₄(C₂³.D₅), C₂₀.₁₃₈(C₂²⋊C₄), (D₄×C₁₀).₂₄₄C₂², (Q₈×C₁₀).₂₀₇C₂², C₄.Dic₅.₄₇C₂², C₂².₇(C₂³.D₅), C₂².₈(C₂²×Dic₅), (C₂²×C₂₀).₂₀₉C₂², C₅⋊₆(M₄(2).₈C₂²), (C₂×C₄○D₄).₆D₅, C₄.₉₆(C₂×C₅⋊D₄), (C₁₀×C₄○D₄).₆C₂, (C₂×C₄).₅(C₂×Dic₅), (C₂×C₂₀).₁₈₉(C₂×C₄), (C₂×C₄).₉₁(C₅⋊D₄), (C₂×C₄.Dic₅)⋊₂₃C₂, C₂.₂₂(C₂×C₂³.D₅), C₁₀.₁₂₇(C₂×C₂²⋊C₄), (C₂×C₄).₁₃₁(C₂²×D₅), (C₂×C₁₀).₉₂(C₂²⋊C₄), (C₂²×C₁₀).₁₄₃(C₂×C₄), (C₂×C₁₀).₃₀₂(C₂²×C₄), SmallGroup(320,864)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — (D₄×C₁₀).₂₉C₄

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₂×C₂₀ — C₄.Dic₅ — C₂×C₄.Dic₅ — (D₄×C₁₀).₂₉C₄

Lower central

C₅ — C₁₀ — C₂×C₁₀ — (D₄×C₁₀).₂₉C₄

Upper central

C₁ — C₄ — C₂²×C₄ — C₂×C₄○D₄

Generators and relations for (D₄×C₁₀).₂₉C₄
G = < a,b,c,d | a¹⁰=b⁴=c²=1, d⁴=b², ab=ba, ac=ca, dad^-1=a^-1b², cbc=b^-1, dbd^-1=a⁵b, dcd^-1=b²c >

Subgroups: 350 in 150 conjugacy classes, 67 normal (25 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₁₀, C₁₀, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂₀, C₂₀, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₄.D₄, C₄.₁₀D₄, C₂×M₄(2), C₂×C₄○D₄, C₅⋊₂C₈, C₂×C₂₀, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₅×Q₈, C₂²×C₁₀, C₂²×C₁₀, M₄(2).₈C₂², C₂×C₅⋊₂C₈, C₄.Dic₅, C₄.Dic₅, C₂²×C₂₀, C₂²×C₂₀, D₄×C₁₀, D₄×C₁₀, Q₈×C₁₀, C₅×C₄○D₄, C₂₀.D₄, C₂₀.₁₀D₄, C₂×C₄.Dic₅, C₁₀×C₄○D₄, (D₄×C₁₀).₂₉C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, D₅, C₂²⋊C₄, C₂²×C₄, C₂×D₄, Dic₅, D₁₀, C₂×C₂²⋊C₄, C₂×Dic₅, C₅⋊D₄, C₂²×D₅, M₄(2).₈C₂², C₂³.D₅, C₂²×Dic₅, C₂×C₅⋊D₄, C₂×C₂³.D₅, (D₄×C₁₀).₂₉C₄

Smallest permutation representation of (D₄×C₁₀).₂₉C₄
►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 37 9 34)(2 38 10 35)(3 39 6 31)(4 40 7 32)(5 36 8 33)(11 24 18 30)(12 25 19 26)(13 21 20 27)(14 22 16 28)(15 23 17 29)(41 72 46 77)(42 73 47 78)(43 74 48 79)(44 75 49 80)(45 76 50 71)(51 62 56 67)(52 63 57 68)(53 64 58 69)(54 65 59 70)(55 66 60 61)

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

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

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

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

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

62 conjugacy classes

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

62 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+			+	+	-	+	-	+	+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	D₄	D₅	Dic₅	D₁₀	Dic₅	D₁₀	D₁₀	C₅⋊D₄	M₄(2).₈C₂²	(D₄×C₁₀).₂₉C₄
kernel	(D₄×C₁₀).₂₉C₄	C₂₀.D₄	C₂₀.₁₀D₄	C₂×C₄.Dic₅	C₁₀×C₄○D₄	C₂²×C₂₀	D₄×C₁₀	C₂×C₂₀	C₂×C₄○D₄	C₂²×C₄	C₂²×C₄	C₂×D₄	C₂×D₄	C₂×Q₈	C₂×C₄	C₅	C₁
# reps	1	2	2	2	1	4	4	4	2	4	2	4	2	2	16	2	8

Matrix representation of (D₄×C₁₀).₂₉C₄ ►in GL₆(𝔽₄₁)

31	0	0	0	0	0
0	4	0	0	0	0
0	0	40	0	5	37
0	0	0	40	4	13
0	0	0	0	1	0
0	0	0	0	0	1

40	0	0	0	0	0
0	1	0	0	0	0
0	0	32	0	0	5
0	0	0	9	5	0
0	0	0	0	32	0
0	0	0	0	0	9

40	0	0	0	0	0
0	40	0	0	0	0
0	0	0	32	36	40
0	0	9	0	1	36
0	0	0	0	0	9
0	0	0	0	32	0

0	1	0	0	0	0
40	0	0	0	0	0
0	0	37	36	40	4
0	0	13	37	37	31
0	0	0	39	4	13
0	0	2	0	36	4

G:=sub<GL(6,GF(41))| [31,0,0,0,0,0,0,4,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,5,4,1,0,0,0,37,13,0,1],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,5,32,0,0,0,5,0,0,9],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,0,32,0,0,0,0,0,36,1,0,32,0,0,40,36,9,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,37,13,0,2,0,0,36,37,39,0,0,0,40,37,4,36,0,0,4,31,13,4] >;

(D₄×C₁₀).₂₉C₄ in GAP, Magma, Sage, TeX

(D_4\times C_{10})._{29}C_4

% in TeX

G:=Group("(D4xC10).29C4");

// GroupNames label

G:=SmallGroup(320,864);

// by ID

G=gap.SmallGroup(320,864);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,297,136,1684,12550]);

// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=c^2=1,d^4=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c=b^-1,d*b*d^-1=a^5*b,d*c*d^-1=b^2*c>;

// generators/relations

G = (D4×C10).29C4 order 320 = 26·5

10th non-split extension by D4×C10 of C4 acting via C4/C2=C2

G = (D₄×C₁₀).₂₉C₄ order 320 = 2⁶·5

10^th non-split extension by D₄×C₁₀ of C₄ acting via C₄/C₂=C₂