(C2xC4).D20 - GroupNames

G = (C₂×C₄).D₂₀ order 320 = 2⁶·5

3^rd non-split extension by C₂×C₄ of D₂₀ acting via D₂₀/C₅=D₄

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂×C₄).₃D₂₀, (C₂×D₂₀).₄C₄, (C₂×C₂₀).₁₅D₄, (C₂×Q₈).₁D₁₀, C₄.₁₀D₄⋊₅D₅, (C₄×Dic₅).₁C₄, C₂₀.₁₀D₄⋊₁C₂, C₅⋊₃(C₄².C₄), (Q₈×C₁₀).₁C₂², C₁₀.₃₄(C₂³⋊C₄), C₂₀.₂₃D₄.₁C₂, C₂².₁₄(D₁₀⋊C₄), C₂.₁₄(C₂³.₁D₁₀), (C₂×C₄).₃(C₄×D₅), (C₂×C₂₀).₃(C₂×C₄), (C₂×C₄).₃(C₅⋊D₄), (C₅×C₄.₁₀D₄)⋊₁₁C₂, (C₂×C₁₀).₇₁(C₂²⋊C₄), SmallGroup(320,35)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₀ — (C₂×C₄).D₂₀

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂×C₂₀ — Q₈×C₁₀ — C₂₀.₂₃D₄ — (C₂×C₄).D₂₀

Lower central

C₅ — C₁₀ — C₂×C₁₀ — C₂×C₂₀ — (C₂×C₄).D₂₀

Upper central

C₁ — C₂ — C₂² — C₂×Q₈ — C₄.₁₀D₄

Generators and relations for (C₂×C₄).D₂₀
G = < a,b,c,d | a²=b²⁰=c²=1, d⁴=b¹⁰, ab=ba, ac=ca, dad^-1=ab¹⁰, cbc=b^-1, dbd^-1=ab¹¹, dcd^-1=b¹⁵c >

Subgroups: 334 in 64 conjugacy classes, 21 normal (all characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, D₅, C₁₀, C₁₀, C₄², C₂²⋊C₄, M₄(2), C₂×D₄, C₂×Q₈, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₄.₁₀D₄, C₄.₁₀D₄, C₄.₄D₄, C₅⋊₂C₈, C₄₀, D₂₀, C₂×Dic₅, C₂×C₂₀, C₅×Q₈, C₂²×D₅, C₄².C₄, C₄.Dic₅, C₄×Dic₅, D₁₀⋊C₄, C₅×M₄(2), C₂×D₂₀, Q₈×C₁₀, C₂₀.₁₀D₄, C₅×C₄.₁₀D₄, C₂₀.₂₃D₄, (C₂×C₄).D₂₀
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, D₅, C₂²⋊C₄, D₁₀, C₂³⋊C₄, C₄×D₅, D₂₀, C₅⋊D₄, C₄².C₄, D₁₀⋊C₄, C₂³.₁D₁₀, (C₂×C₄).D₂₀

Smallest permutation representation of (C₂×C₄).D₂₀
►On 80 points

Generators in S₈₀

(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)

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

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

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

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

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

35 conjugacy classes

class	1	2A	2B	2C	4A	4B	4C	4D	4E	5A	5B	8A	8B	8C	8D	10A	10B	10C	10D	20A	20B	20C	20D	20E	20F	20G	20H	40A	···	40H
order	1	2	2	2	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	40	4	4	4	20	20	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.C₄	C₂³.₁D₁₀	(C₂×C₄).D₂₀
kernel	(C₂×C₄).D₂₀	C₂₀.₁₀D₄	C₅×C₄.₁₀D₄	C₂₀.₂₃D₄	C₄×Dic₅	C₂×D₂₀	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×C₄).D₂₀ ►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	1	0
0	0	25	25	0	1

6	40	0	0	0	0
1	0	0	0	0	0
0	0	0	1	0	0
0	0	40	0	0	0
0	0	9	9	1	4
0	0	0	16	20	40

6	40	0	0	0	0
35	35	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	16	16	20	40

40	0	0	0	0	0
0	40	0	0	0	0
0	0	0	0	1	0
0	0	32	32	40	37
0	0	32	0	0	0
0	0	2	18	0	9

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,25,0,0,0,40,0,25,0,0,0,0,1,0,0,0,0,0,0,1],[6,1,0,0,0,0,40,0,0,0,0,0,0,0,0,40,9,0,0,0,1,0,9,16,0,0,0,0,1,20,0,0,0,0,4,40],[6,35,0,0,0,0,40,35,0,0,0,0,0,0,0,1,0,16,0,0,1,0,0,16,0,0,0,0,1,20,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,32,32,2,0,0,0,32,0,18,0,0,1,40,0,0,0,0,0,37,0,9] >;

(C₂×C₄).D₂₀ in GAP, Magma, Sage, TeX

(C_2\times C_4).D_{20}

% in TeX

G:=Group("(C2xC4).D20");

// GroupNames label

G:=SmallGroup(320,35);

// by ID

G=gap.SmallGroup(320,35);

# by ID

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

// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^2=1,d^4=b^10,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^10,c*b*c=b^-1,d*b*d^-1=a*b^11,d*c*d^-1=b^15*c>;

// generators/relations

G = (C2×C4).D20 order 320 = 26·5

3rd non-split extension by C2×C4 of D20 acting via D20/C5=D4

G = (C₂×C₄).D₂₀ order 320 = 2⁶·5

3^rd non-split extension by C₂×C₄ of D₂₀ acting via D₂₀/C₅=D₄