C20.76C2^4 - GroupNames

G = C₂₀.₇₆C₂⁴ order 320 = 2⁶·5

23^rd non-split extension by C₂₀ of C₂⁴ acting via C₂⁴/C₂³=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂₀.₇₆C₂⁴, C₄oD₄.₄₅D₁₀, (D₄xC₁₀).₂₆C₄, C₅:₆(Q₈oM₄(2)), D₄.Dic₅:₉C₂, (Q₈xC₁₀).₂₃C₄, C₄oD₄.₂Dic₅, (C₂xQ₈).₈Dic₅, D₄.₉(C₂xDic₅), C₄.₇₅(C₂³xD₅), C₁₀.₇₀(C₂³xC₄), C₅:₂C₈.₃₄C₂³, (C₂xD₄).₁₀Dic₅, Q₈.₁₀(C₂xDic₅), (C₂xC₂₀).₅₅₄C₂³, C₂₀.₁₅₇(C₂²xC₄), (C₂²xC₄).₂₈₀D₁₀, C₄.Dic₅:₃₅C₂², C₄.₂₀(C₂²xDic₅), C₂.₁₁(C₂³xDic₅), C₂³.₁₁(C₂xDic₅), C₂².₂(C₂²xDic₅), (C₂²xC₂₀).₂₈₉C₂², (C₂xC₄oD₄).₉D₅, (C₅xC₄oD₄).₈C₄, (C₅xD₄).₄₀(C₂xC₄), (C₅xQ₈).₄₃(C₂xC₄), (C₂xC₂₀).₃₀₉(C₂xC₄), (C₂xC₅:₂C₈):₂₁C₂², (C₁₀xC₄oD₄).₁₀C₂, (C₂xC₄.Dic₅):₂₉C₂, (C₂xC₄).₃₁(C₂xDic₅), (C₅xC₄oD₄).₄₉C₂², (C₂xC₄).₆₃₅(C₂²xD₅), (C₂²xC₁₀).₁₄₉(C₂xC₄), (C₂xC₁₀).₁₃₀(C₂²xC₄), SmallGroup(320,1491)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₂₀.₇₆C₂⁴

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₅:₂C₈ — C₂xC₅:₂C₈ — D₄.Dic₅ — C₂₀.₇₆C₂⁴

Lower central

C₅ — C₁₀ — C₂₀.₇₆C₂⁴

Upper central

C₁ — C₄ — C₂xC₄oD₄

Generators and relations for C₂₀.₇₆C₂⁴
G = < a,b,c,d,e | a²⁰=c²=d²=e²=1, b²=a⁵, bab^-1=a⁹, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=a¹⁰b, dcd=a¹⁰c, ce=ec, de=ed >

Subgroups: 494 in 258 conjugacy classes, 187 normal (17 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₅, C₈, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₁₀, C₁₀, C₂xC₈, M₄(2), C₂²xC₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₂₀, C₂₀, C₂xC₁₀, C₂xC₁₀, C₂xC₁₀, C₂xM₄(2), C₈oD₄, C₂xC₄oD₄, C₅:₂C₈, C₂xC₂₀, C₂xC₂₀, C₅xD₄, C₅xQ₈, C₂²xC₁₀, Q₈oM₄(2), C₂xC₅:₂C₈, C₄.Dic₅, C₂²xC₂₀, D₄xC₁₀, Q₈xC₁₀, C₅xC₄oD₄, C₂xC₄.Dic₅, D₄.Dic₅, C₁₀xC₄oD₄, C₂₀.₇₆C₂⁴
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, C₂³, D₅, C₂²xC₄, C₂⁴, Dic₅, D₁₀, C₂³xC₄, C₂xDic₅, C₂²xD₅, Q₈oM₄(2), C₂²xDic₅, C₂³xD₅, C₂³xDic₅, C₂₀.₇₆C₂⁴

Smallest permutation representation of C₂₀.₇₆C₂⁴
►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 55 6 60 11 45 16 50)(2 44 7 49 12 54 17 59)(3 53 8 58 13 43 18 48)(4 42 9 47 14 52 19 57)(5 51 10 56 15 41 20 46)(21 66 26 71 31 76 36 61)(22 75 27 80 32 65 37 70)(23 64 28 69 33 74 38 79)(24 73 29 78 34 63 39 68)(25 62 30 67 35 72 40 77)

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

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

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

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

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

74 conjugacy classes

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

74 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	4	4
type	+	+	+	+				+	+	-	-	-	+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	D₅	D₁₀	Dic₅	Dic₅	Dic₅	D₁₀	Q₈oM₄(2)	C₂₀.₇₆C₂⁴
kernel	C₂₀.₇₆C₂⁴	C₂xC₄.Dic₅	D₄.Dic₅	C₁₀xC₄oD₄	D₄xC₁₀	Q₈xC₁₀	C₅xC₄oD₄	C₂xC₄oD₄	C₂²xC₄	C₂xD₄	C₂xQ₈	C₄oD₄	C₄oD₄	C₅	C₁
# reps	1	6	8	1	6	2	8	2	6	6	2	8	8	2	8

Matrix representation of C₂₀.₇₆C₂⁴ ►in GL₆(F₄₁)

23	0	0	0	0	0
38	25	0	0	0	0
0	0	9	0	0	0
0	0	0	9	0	0
0	0	0	0	9	0
0	0	0	0	0	9

5	24	0	0	0	0
16	36	0	0	0	0
0	0	0	15	1	23
0	0	0	17	0	26
0	0	9	40	0	1
0	0	0	5	0	24

40	0	0	0	0	0
0	40	0	0	0	0
0	0	0	9	26	32
0	0	32	0	24	40
0	0	0	0	1	0
0	0	0	0	36	40

40	0	0	0	0	0
0	40	0	0	0	0
0	0	0	1	29	27
0	0	1	0	11	32
0	0	0	0	9	20
0	0	0	0	37	32

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	35
0	0	0	1	0	26
0	0	0	0	40	0
0	0	0	0	0	40

G:=sub<GL(6,GF(41))| [23,38,0,0,0,0,0,25,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[5,16,0,0,0,0,24,36,0,0,0,0,0,0,0,0,9,0,0,0,15,17,40,5,0,0,1,0,0,0,0,0,23,26,1,24],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,32,0,0,0,0,9,0,0,0,0,0,26,24,1,36,0,0,32,40,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,29,11,9,37,0,0,27,32,20,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,35,26,0,40] >;

C₂₀.₇₆C₂⁴ in GAP, Magma, Sage, TeX

C_{20}._{76}C_2^4

% in TeX

G:=Group("C20.76C2^4");

// GroupNames label

G:=SmallGroup(320,1491);

// by ID

G=gap.SmallGroup(320,1491);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,387,1123,102,12550]);

// Polycyclic

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

// generators/relations

G = C20.76C24 order 320 = 26·5

23rd non-split extension by C20 of C24 acting via C24/C23=C2

G = C₂₀.₇₆C₂⁴ order 320 = 2⁶·5

23^rd non-split extension by C₂₀ of C₂⁴ acting via C₂⁴/C₂³=C₂