C2^4.Dic5 - GroupNames

G = C₂⁴.Dic₅ order 320 = 2⁶·5

1^st non-split extension by C₂⁴ of Dic₅ acting via Dic₅/C₅=C₄

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴.₁Dic₅, C₅⋊₄(C₂³⋊C₈), C₂³⋊(C₅⋊₂C₈), (C₂²×C₁₀)⋊₂C₈, (C₂×C₂₀).₂₂₇D₄, (C₂³×C₁₀).₃C₄, (C₂²×C₂₀).₁C₄, (C₂²×C₄).₃D₁₀, C₂₀.₅₅D₄⋊₂₁C₂, (C₂²×C₄).₁Dic₅, C₁₀.₃₇(C₂³⋊C₄), C₁₀.₂₈(C₂²⋊C₈), C₂.₁(C₂₀.D₄), (C₂×C₁₀).₄₀M₄(2), C₂.₁(C₂³⋊Dic₅), C₂³.₂₀(C₂×Dic₅), C₁₀.₁₃(C₄.D₄), C₂.₃(C₂₀.₅₅D₄), C₂².₄(C₄.Dic₅), (C₂²×C₂₀).₃₂₄C₂², C₂².₂₃(C₂³.D₅), (C₂×C₁₀).₅₇(C₂×C₈), C₂².₂(C₂×C₅⋊₂C₈), (C₂×C₂²⋊C₄).₁D₅, (C₂×C₄).₁₅₉(C₅⋊D₄), (C₁₀×C₂²⋊C₄).₂₁C₂, (C₂²×C₁₀).₁₉₁(C₂×C₄), (C₂×C₁₀).₁₄₉(C₂²⋊C₄), SmallGroup(320,83)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — C₂⁴.Dic₅

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂×C₂₀ — C₂²×C₂₀ — C₂₀.₅₅D₄ — C₂⁴.Dic₅

Lower central

C₅ — C₁₀ — C₂×C₁₀ — C₂⁴.Dic₅

Upper central

C₁ — C₂² — C₂²×C₄ — C₂×C₂²⋊C₄

Generators and relations for C₂⁴.Dic₅
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e¹⁰=c, f²=e⁵, ab=ba, ac=ca, eae^-1=ad=da, faf^-1=abd, bc=cb, fbf^-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e⁹ >

Subgroups: 278 in 98 conjugacy classes, 35 normal (25 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₁₀, C₁₀, C₂²⋊C₄, C₂×C₈, C₂²×C₄, C₂⁴, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂²⋊C₈, C₂×C₂²⋊C₄, C₅⋊₂C₈, C₂×C₂₀, C₂×C₂₀, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₂³⋊C₈, C₂×C₅⋊₂C₈, C₅×C₂²⋊C₄, C₂²×C₂₀, C₂³×C₁₀, C₂₀.₅₅D₄, C₁₀×C₂²⋊C₄, C₂⁴.Dic₅
Quotients: C₁, C₂, C₄, C₂², C₈, C₂×C₄, D₄, D₅, C₂²⋊C₄, C₂×C₈, M₄(2), Dic₅, D₁₀, C₂²⋊C₈, C₂³⋊C₄, C₄.D₄, C₅⋊₂C₈, C₂×Dic₅, C₅⋊D₄, C₂³⋊C₈, C₂×C₅⋊₂C₈, C₄.Dic₅, C₂³.D₅, C₂₀.₅₅D₄, C₂₀.D₄, C₂³⋊Dic₅, C₂⁴.Dic₅

Smallest permutation representation of C₂⁴.Dic₅
►On 80 points

Generators in S₈₀

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

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

(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)(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)

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

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

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

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

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

62 conjugacy classes

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

62 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+				+	+		-	+	-				+	+
image	C₁	C₂	C₂	C₄	C₄	C₈	D₄	D₅	M₄(2)	Dic₅	D₁₀	Dic₅	C₅⋊D₄	C₅⋊₂C₈	C₄.Dic₅	C₂³⋊C₄	C₄.D₄	C₂₀.D₄	C₂³⋊Dic₅
kernel	C₂⁴.Dic₅	C₂₀.₅₅D₄	C₁₀×C₂²⋊C₄	C₂²×C₂₀	C₂³×C₁₀	C₂²×C₁₀	C₂×C₂₀	C₂×C₂²⋊C₄	C₂×C₁₀	C₂²×C₄	C₂²×C₄	C₂⁴	C₂×C₄	C₂³	C₂²	C₁₀	C₁₀	C₂	C₂
# reps	1	2	1	2	2	8	2	2	2	2	2	2	8	8	8	1	1	4	4

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

40	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	25	40	0	0
0	0	25	39	1	0
0	0	4	16	25	40

40	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	16	2	40	0
0	0	29	14	0	40

40	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

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	0
0	0	0	0	0	40

5	0	0	0	0	0
0	8	0	0	0	0
0	0	40	5	0	0
0	0	0	1	0	0
0	0	30	7	40	5
0	0	12	11	0	1

0	1	0	0	0	0
9	0	0	0	0	0
0	0	10	32	9	0
0	0	13	19	0	9
0	0	20	25	31	9
0	0	31	14	28	22

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,25,25,4,0,0,0,40,39,16,0,0,0,0,1,25,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,16,29,0,0,0,1,2,14,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,40,0,30,12,0,0,5,1,7,11,0,0,0,0,40,0,0,0,0,0,5,1],[0,9,0,0,0,0,1,0,0,0,0,0,0,0,10,13,20,31,0,0,32,19,25,14,0,0,9,0,31,28,0,0,0,9,9,22] >;

C₂⁴.Dic₅ in GAP, Magma, Sage, TeX

C_2^4.{\rm Dic}_5

% in TeX

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

// GroupNames label

G:=SmallGroup(320,83);

// by ID

G=gap.SmallGroup(320,83);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,387,100,1123,12550]);

// Polycyclic

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

// generators/relations

G = C24.Dic5 order 320 = 26·5

1st non-split extension by C24 of Dic5 acting via Dic5/C5=C4

G = C₂⁴.Dic₅ order 320 = 2⁶·5

1^st non-split extension by C₂⁴ of Dic₅ acting via Dic₅/C₅=C₄