Dic5.21C2^4 - GroupNames

G = Dic₅.₂₁C₂⁴ order 320 = 2⁶·5

21^st non-split extension by Dic₅ of C₂⁴ acting via C₂⁴/C₂³=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic₅.₂₁C₂⁴, D₅⋊(C₈○D₄), (D₄×D₅).₂C₄, D₄.F₅⋊₇C₂, C₅⋊C₈.₆C₂³, Q₈.F₅⋊₇C₂, C₄○D₄.₄F₅, (Q₈×D₅).₂C₄, D₅⋊C₈⋊₈C₂², C₄○D₂₀.₃C₄, D₄.₁₃(C₂×F₅), Q₈.₁₄(C₂×F₅), D₂₀.₁₃(C₂×C₄), D₅⋊M₄(2)⋊₉C₂, C₄.F₅⋊₁₁C₂², C₄.₃₂(C₂²×F₅), C₂.₁₆(C₂³×F₅), C₁₀.₁₅(C₂³×C₄), C₂₀.₃₂(C₂²×C₄), (C₄×D₅).₅₅C₂³, D₁₀.₆(C₂²×C₄), C₂².F₅⋊₅C₂², Dic₁₀.₁₄(C₂×C₄), C₂².₃(C₂²×F₅), Dic₅.₆(C₂²×C₄), D₄⋊₂D₅.₁₇C₂², Q₈⋊₂D₅.₁₇C₂², (C₂×Dic₅).₁₇₉C₂³, C₅⋊₃(C₂×C₈○D₄), (C₂×D₅⋊C₈)⋊₇C₂, (C₂×C₅⋊C₈)⋊₁₂C₂², (D₅×C₄○D₄).₉C₂, (C₅×C₄○D₄).₃C₄, C₅⋊D₄.₃(C₂×C₄), (C₂×C₄).₉₄(C₂×F₅), (C₂×C₂₀).₇₆(C₂×C₄), (C₅×D₄).₁₃(C₂×C₄), (C₄×D₅).₄₇(C₂×C₄), (C₅×Q₈).₁₄(C₂×C₄), (C₂×C₁₀).₄(C₂²×C₄), (C₂×C₄×D₅).₂₂₂C₂², (C₂²×D₅).₆₄(C₂×C₄), SmallGroup(320,1601)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — Dic₅.₂₁C₂⁴

Chief series

C₁ — C₅ — C₁₀ — Dic₅ — C₅⋊C₈ — C₂×C₅⋊C₈ — C₂×D₅⋊C₈ — Dic₅.₂₁C₂⁴

Lower central

C₅ — C₁₀ — Dic₅.₂₁C₂⁴

Upper central

C₁ — C₄ — C₄○D₄

Generators and relations for Dic₅.₂₁C₂⁴
G = < a,b,c,d,e,f | a¹⁰=d²=f²=1, b²=e²=a⁵, c²=b, bab^-1=a^-1, cac^-1=a³, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=a⁵d, ef=fe >

Subgroups: 778 in 266 conjugacy classes, 138 normal (18 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, D₅, D₅, C₁₀, C₁₀, C₂×C₈, M₄(2), C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, Dic₅, Dic₅, C₂₀, C₂₀, D₁₀, D₁₀, D₁₀, C₂×C₁₀, C₂²×C₈, C₂×M₄(2), C₈○D₄, C₂×C₄○D₄, C₅⋊C₈, Dic₁₀, C₄×D₅, C₄×D₅, D₂₀, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₅×D₄, C₅×Q₈, C₂²×D₅, C₂×C₈○D₄, D₅⋊C₈, D₅⋊C₈, C₄.F₅, C₂×C₅⋊C₈, C₂².F₅, C₂×C₄×D₅, C₄○D₂₀, D₄×D₅, D₄⋊₂D₅, Q₈×D₅, Q₈⋊₂D₅, C₅×C₄○D₄, C₂×D₅⋊C₈, D₅⋊M₄(2), D₄.F₅, Q₈.F₅, D₅×C₄○D₄, Dic₅.₂₁C₂⁴
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, C₂²×C₄, C₂⁴, F₅, C₈○D₄, C₂³×C₄, C₂×F₅, C₂×C₈○D₄, C₂²×F₅, C₂³×F₅, Dic₅.₂₁C₂⁴

Smallest permutation representation of Dic₅.₂₁C₂⁴
►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 16 6 11)(2 15 7 20)(3 14 8 19)(4 13 9 18)(5 12 10 17)(21 34 26 39)(22 33 27 38)(23 32 28 37)(24 31 29 36)(25 40 30 35)(41 54 46 59)(42 53 47 58)(43 52 48 57)(44 51 49 56)(45 60 50 55)(61 74 66 79)(62 73 67 78)(63 72 68 77)(64 71 69 76)(65 80 70 75)

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

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

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

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

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

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

50 conjugacy classes

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

50 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	4	4	4	4	8
type	+	+	+	+	+	+						+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	C₈○D₄	F₅	C₂×F₅	C₂×F₅	C₂×F₅	Dic₅.₂₁C₂⁴
kernel	Dic₅.₂₁C₂⁴	C₂×D₅⋊C₈	D₅⋊M₄(2)	D₄.F₅	Q₈.F₅	D₅×C₄○D₄	C₄○D₂₀	D₄×D₅	Q₈×D₅	C₅×C₄○D₄	D₅	C₄○D₄	C₂×C₄	D₄	Q₈	C₁
# reps	1	3	3	6	2	1	6	6	2	2	8	1	3	3	1	2

Matrix representation of Dic₅.₂₁C₂⁴ ►in GL₆(𝔽₄₁)

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

32	0	0	0	0	0
0	32	0	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0
0	0	0	1	0	0
0	0	1	0	0	0

27	0	0	0	0	0
0	27	0	0	0	0
0	0	9	32	32	0
0	0	0	32	0	9
0	0	9	0	32	0
0	0	0	32	32	9

0	1	0	0	0	0
1	0	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	1

9	0	0	0	0	0
0	9	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	1

1	0	0	0	0	0
0	40	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	1

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,40,40,40,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,9,0,9,0,0,0,32,32,0,32,0,0,32,0,32,32,0,0,0,9,0,9],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic₅.₂₁C₂⁴ in GAP, Magma, Sage, TeX

{\rm Dic}_5._{21}C_2^4

% in TeX

G:=Group("Dic5.21C2^4");

// GroupNames label

G:=SmallGroup(320,1601);

// by ID

G=gap.SmallGroup(320,1601);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,570,102,6278,818]);

// Polycyclic

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

// generators/relations

G = Dic5.21C24 order 320 = 26·5

21st non-split extension by Dic5 of C24 acting via C24/C23=C2

G = Dic₅.₂₁C₂⁴ order 320 = 2⁶·5

21^st non-split extension by Dic₅ of C₂⁴ acting via C₂⁴/C₂³=C₂