(C2xQ8).D10 - GroupNames

G = (C₂×Q₈).D₁₀ order 320 = 2⁶·5

2^nd non-split extension by C₂×Q₈ of D₁₀ acting via D₁₀/C₅=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂×C₄).₄D₂₀, (C₂×C₂₀).₁₆D₄, C₄.₁₀D₄.D₅, (C₂×Q₈).₂D₁₀, (C₄×Dic₅).₂C₄, C₅⋊₃(C₄².₃C₄), (C₂×Dic₁₀).₄C₄, (Q₈×C₁₀).₂C₂², C₁₀.₃₅(C₂³⋊C₄), Dic₅⋊Q₈.₁C₂, C₂₀.₁₀D₄.₁C₂, C₂².₁₅(D₁₀⋊C₄), C₂.₁₅(C₂³.₁D₁₀), (C₂×C₄).₄(C₄×D₅), (C₂×C₂₀).₄(C₂×C₄), (C₂×C₄).₄(C₅⋊D₄), (C₅×C₄.₁₀D₄).₁C₂, (C₂×C₁₀).₇₂(C₂²⋊C₄), SmallGroup(320,36)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₀ — (C₂×Q₈).D₁₀

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂×C₂₀ — Q₈×C₁₀ — Dic₅⋊Q₈ — (C₂×Q₈).D₁₀

Lower central

C₅ — C₁₀ — C₂×C₁₀ — C₂×C₂₀ — (C₂×Q₈).D₁₀

Upper central

C₁ — C₂ — C₂² — C₂×Q₈ — C₄.₁₀D₄

Generators and relations for (C₂×Q₈).D₁₀
G = < a,b,c,d | a²=b²⁰=1, c⁴=d²=b¹⁰, ab=ba, cac^-1=ab¹⁰, ad=da, cbc^-1=dbd^-1=ab⁹, dcd^-1=b⁵c³ >

Subgroups: 238 in 60 conjugacy classes, 21 normal (all characteristic)
C₁, C₂, C₂, C₄, C₂², C₅, C₈, C₂×C₄, C₂×C₄, Q₈, C₁₀, C₁₀, C₄², C₄⋊C₄, M₄(2), C₂×Q₈, C₂×Q₈, Dic₅, C₂₀, C₂×C₁₀, C₄.₁₀D₄, C₄.₁₀D₄, C₄⋊Q₈, C₅⋊₂C₈, C₄₀, Dic₁₀, C₂×Dic₅, C₂×C₂₀, C₅×Q₈, C₄².₃C₄, C₄.Dic₅, C₄×Dic₅, C₁₀.D₄, C₅×M₄(2), C₂×Dic₁₀, Q₈×C₁₀, C₂₀.₁₀D₄, C₅×C₄.₁₀D₄, Dic₅⋊Q₈, (C₂×Q₈).D₁₀
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, D₅, C₂²⋊C₄, D₁₀, C₂³⋊C₄, C₄×D₅, D₂₀, C₅⋊D₄, C₄².₃C₄, D₁₀⋊C₄, C₂³.₁D₁₀, (C₂×Q₈).D₁₀

Smallest permutation representation of (C₂×Q₈).D₁₀
►On 80 points

Generators in S₈₀

(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)

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

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

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

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

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

35 conjugacy classes

class	1	2A	2B	4A	4B	4C	4D	4E	4F	5A	5B	8A	8B	8C	8D	10A	10B	10C	10D	20A	20B	20C	20D	20E	20F	20G	20H	40A	···	40H
order	1	2	2	4	4	4	4	4	4	5	5	8	8	8	8	10	10	10	10	20	20	20	20	20	20	20	20	40	···	40
size	1	1	2	4	4	4	20	20	40	2	2	8	8	40	40	2	2	4	4	4	4	4	4	8	8	8	8	8	···	8

35 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	4	4	4	8
type	+	+	+	+			+	+	+		+		+	-		-
image	C₁	C₂	C₂	C₂	C₄	C₄	D₄	D₅	D₁₀	C₄×D₅	D₂₀	C₅⋊D₄	C₂³⋊C₄	C₄².₃C₄	C₂³.₁D₁₀	(C₂×Q₈).D₁₀
kernel	(C₂×Q₈).D₁₀	C₂₀.₁₀D₄	C₅×C₄.₁₀D₄	Dic₅⋊Q₈	C₄×Dic₅	C₂×Dic₁₀	C₂×C₂₀	C₄.₁₀D₄	C₂×Q₈	C₂×C₄	C₂×C₄	C₂×C₄	C₁₀	C₅	C₂	C₁
# reps	1	1	1	1	2	2	2	2	2	4	4	4	1	2	4	2

Matrix representation of (C₂×Q₈).D₁₀ ►in GL₆(𝔽₄₁)

1	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	0	40	0
0	0	21	21	0	40

6	40	0	0	0	0
1	0	0	0	0	0
0	0	20	3	0	0
0	0	3	21	0	0
0	0	3	3	17	29
0	0	10	0	31	24

1	0	0	0	0	0
6	40	0	0	0	0
0	0	20	20	18	2
0	0	3	3	17	29
0	0	40	0	0	0
0	0	6	6	19	18

1	0	0	0	0	0
6	40	0	0	0	0
0	0	20	3	0	0
0	0	3	21	0	0
0	0	20	20	18	2
0	0	0	31	22	23

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,21,0,0,0,1,0,21,0,0,0,0,40,0,0,0,0,0,0,40],[6,1,0,0,0,0,40,0,0,0,0,0,0,0,20,3,3,10,0,0,3,21,3,0,0,0,0,0,17,31,0,0,0,0,29,24],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,20,3,40,6,0,0,20,3,0,6,0,0,18,17,0,19,0,0,2,29,0,18],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,20,3,20,0,0,0,3,21,20,31,0,0,0,0,18,22,0,0,0,0,2,23] >;

(C₂×Q₈).D₁₀ in GAP, Magma, Sage, TeX

(C_2\times Q_8).D_{10}

% in TeX

G:=Group("(C2xQ8).D10");

// GroupNames label

G:=SmallGroup(320,36);

// by ID

G=gap.SmallGroup(320,36);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,141,36,422,184,1123,794,297,136,851,12550]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=1,c^4=d^2=b^10,a*b=b*a,c*a*c^-1=a*b^10,a*d=d*a,c*b*c^-1=d*b*d^-1=a*b^9,d*c*d^-1=b^5*c^3>;

// generators/relations

G = (C2×Q8).D10 order 320 = 26·5

2nd non-split extension by C2×Q8 of D10 acting via D10/C5=C22

G = (C₂×Q₈).D₁₀ order 320 = 2⁶·5

2^nd non-split extension by C₂×Q₈ of D₁₀ acting via D₁₀/C₅=C₂²