C10.29C4wrC2 - GroupNames

G = C₁₀.₂₉C₄≀C₂ order 320 = 2⁶·5

5^th non-split extension by C₁₀ of C₄≀C₂ acting via C₄≀C₂/C₄○D₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄⋊C₄⋊₂Dic₅, C₁₀.₂₉C₄≀C₂, (Q₈×C₁₀)⋊₁₁C₄, (C₂×Q₈)⋊₁Dic₅, (C₂×C₁₀).₃Q₁₆, C₂²⋊Q₈.₁D₅, (C₂×C₂₀).₂₃₀D₄, (C₂×C₁₀).₁₁SD₁₆, (C₂²×C₁₀).₄₅D₄, (C₂²×C₄).₆₀D₁₀, C₂².₅(Q₈⋊D₅), C₁₀.₄₂(C₂³⋊C₄), C₂.₃(Q₈⋊Dic₅), C₅⋊₅(C₂³.₃₁D₄), C₂³.₄₉(C₅⋊D₄), C₂.₆(C₂³⋊Dic₅), C₂².₂(C₅⋊Q₁₆), C₁₀.₁₈(Q₈⋊C₄), C₂.₅(D₄⋊₂Dic₅), C₂₀.₅₅D₄.₁₆C₂, (C₂²×C₂₀).₃₇₂C₂², C₂².₃₈(C₂³.D₅), C₁₀.₁₀C₄².₃₅C₂, (C₅×C₄⋊C₄)⋊₉C₄, (C₂×C₄).₈(C₂×Dic₅), (C₂×C₂₀).₃₄₁(C₂×C₄), (C₅×C₂²⋊Q₈).₁₀C₂, (C₂×C₄).₁₆₄(C₅⋊D₄), (C₂×C₁₀).₁₆₀(C₂²⋊C₄), SmallGroup(320,96)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₀ — C₁₀.₂₉C₄≀C₂

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂²×C₁₀ — C₂²×C₂₀ — C₁₀.₁₀C₄² — C₁₀.₂₉C₄≀C₂

Lower central

C₅ — C₂×C₁₀ — C₂×C₂₀ — C₁₀.₂₉C₄≀C₂

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²⋊Q₈

Generators and relations for C₁₀.₂₉C₄≀C₂
G = < a,b,c,d | a¹⁰=b⁴=d⁴=1, c²=a⁵, ab=ba, ac=ca, dad^-1=a^-1, cbc^-1=b^-1, dbd^-1=a⁵b, dcd^-1=a⁵b^-1c >

Subgroups: 286 in 80 conjugacy classes, 31 normal (all characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, Q₈, C₂³, C₁₀, C₁₀, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×Q₈, Dic₅, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂.C₄², C₂²⋊C₈, C₂²⋊Q₈, C₅⋊₂C₈, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₅×Q₈, C₂²×C₁₀, C₂³.₃₁D₄, C₂×C₅⋊₂C₈, C₅×C₂²⋊C₄, C₅×C₄⋊C₄, C₅×C₄⋊C₄, C₂²×Dic₅, C₂²×C₂₀, Q₈×C₁₀, C₂₀.₅₅D₄, C₁₀.₁₀C₄², C₅×C₂²⋊Q₈, C₁₀.₂₉C₄≀C₂
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, D₅, C₂²⋊C₄, SD₁₆, Q₁₆, Dic₅, D₁₀, C₂³⋊C₄, Q₈⋊C₄, C₄≀C₂, C₂×Dic₅, C₅⋊D₄, C₂³.₃₁D₄, Q₈⋊D₅, C₅⋊Q₁₆, C₂³.D₅, C₂³⋊Dic₅, Q₈⋊Dic₅, D₄⋊₂Dic₅, C₁₀.₂₉C₄≀C₂

Smallest permutation representation of C₁₀.₂₉C₄≀C₂
►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 50 23 33)(2 41 24 34)(3 42 25 35)(4 43 26 36)(5 44 27 37)(6 45 28 38)(7 46 29 39)(8 47 30 40)(9 48 21 31)(10 49 22 32)(11 70 78 53)(12 61 79 54)(13 62 80 55)(14 63 71 56)(15 64 72 57)(16 65 73 58)(17 66 74 59)(18 67 75 60)(19 68 76 51)(20 69 77 52)

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

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

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

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

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

47 conjugacy classes

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

47 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4
type	+	+	+	+			+	+	+		-	-	+	-				+	+	-
image	C₁	C₂	C₂	C₂	C₄	C₄	D₄	D₄	D₅	SD₁₆	Q₁₆	Dic₅	D₁₀	Dic₅	C₄≀C₂	C₅⋊D₄	C₅⋊D₄	C₂³⋊C₄	Q₈⋊D₅	C₅⋊Q₁₆	C₂³⋊Dic₅	D₄⋊₂Dic₅
kernel	C₁₀.₂₉C₄≀C₂	C₂₀.₅₅D₄	C₁₀.₁₀C₄²	C₅×C₂²⋊Q₈	C₅×C₄⋊C₄	Q₈×C₁₀	C₂×C₂₀	C₂²×C₁₀	C₂²⋊Q₈	C₂×C₁₀	C₂×C₁₀	C₄⋊C₄	C₂²×C₄	C₂×Q₈	C₁₀	C₂×C₄	C₂³	C₁₀	C₂²	C₂²	C₂	C₂
# reps	1	1	1	1	2	2	1	1	2	2	2	2	2	2	4	4	4	1	2	2	4	4

Matrix representation of C₁₀.₂₉C₄≀C₂ ►in GL₆(𝔽₄₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	40	7
0	0	0	0	34	7

9	0	0	0	0	0
0	32	0	0	0	0
0	0	9	2	0	0
0	0	0	32	0	0
0	0	0	0	40	0
0	0	0	0	0	40

0	1	0	0	0	0
1	0	0	0	0	0
0	0	14	11	0	0
0	0	38	27	0	0
0	0	0	0	24	40
0	0	0	0	1	17

32	0	0	0	0	0
0	1	0	0	0	0
0	0	9	2	0	0
0	0	1	32	0	0
0	0	0	0	7	34
0	0	0	0	1	34

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,34,0,0,0,0,7,7],[9,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,2,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,14,38,0,0,0,0,11,27,0,0,0,0,0,0,24,1,0,0,0,0,40,17],[32,0,0,0,0,0,0,1,0,0,0,0,0,0,9,1,0,0,0,0,2,32,0,0,0,0,0,0,7,1,0,0,0,0,34,34] >;

C₁₀.₂₉C₄≀C₂ in GAP, Magma, Sage, TeX

C_{10}._{29}C_4\wr C_2

% in TeX

G:=Group("C10.29C4wrC2");

// GroupNames label

G:=SmallGroup(320,96);

// by ID

G=gap.SmallGroup(320,96);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,219,1571,570,136,12550]);

// Polycyclic

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

// generators/relations

G = C10.29C4≀C2 order 320 = 26·5

5th non-split extension by C10 of C4≀C2 acting via C4≀C2/C4○D4=C2

G = C₁₀.₂₉C₄≀C₂ order 320 = 2⁶·5

5^th non-split extension by C₁₀ of C₄≀C₂ acting via C₄≀C₂/C₄○D₄=C₂