C5xC2^3:3D4 - GroupNames

G = C₅×C₂³⋊₃D₄ order 320 = 2⁶·5

Direct product of C₅ and C₂³⋊₃D₄

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C₅×C₂³⋊₃D₄, C₁₀.₁₅₁2₊¹⁺⁴, C₂³⋊₃(C₅×D₄), C₄⋊D₄⋊₅C₁₀, C₂⁴⋊₃(C₂×C₁₀), (C₂²×C₁₀)⋊₆D₄, C₂²≀C₂⋊₂C₁₀, (C₂²×D₄)⋊₆C₁₀, C₂².₂(D₄×C₁₀), (D₄×C₁₀)⋊₃₅C₂², (C₂³×C₁₀)⋊₃C₂², (C₂×C₂₀).₆₆₃C₂³, (C₂×C₁₀).₃₅₄C₂⁴, (C₂²×C₂₀)⋊₄₇C₂², C₂².D₄⋊₂C₁₀, C₁₀.₁₈₉(C₂²×D₄), C₂.₃(C₅×2₊¹⁺⁴), C₂².₂₈(C₂³×C₁₀), (C₂²×C₁₀).₉₀C₂³, C₂³.₃₆(C₂²×C₁₀), C₄⋊C₄⋊₃(C₂×C₁₀), (D₄×C₂×C₁₀)⋊₂₁C₂, C₂.₁₃(D₄×C₂×C₁₀), (C₂×D₄)⋊₃(C₂×C₁₀), (C₅×C₄⋊D₄)⋊₃₂C₂, C₂²⋊C₄⋊₃(C₂×C₁₀), (C₅×C₄⋊C₄)⋊₃₇C₂², (C₂²×C₄)⋊₇(C₂×C₁₀), (C₂×C₁₀).₉₀(C₂×D₄), (C₅×C₂²≀C₂)⋊₁₂C₂, (C₂×C₂²⋊C₄)⋊₁₂C₁₀, (C₁₀×C₂²⋊C₄)⋊₃₂C₂, (C₅×C₂²⋊C₄)⋊₃₈C₂², (C₂×C₄).₂₁(C₂²×C₁₀), (C₅×C₂².D₄)⋊₂₁C₂, SmallGroup(320,1536)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₅×C₂³⋊₃D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₁₀ — C₂²×C₁₀ — D₄×C₁₀ — C₅×C₄⋊D₄ — C₅×C₂³⋊₃D₄

Lower central

C₁ — C₂² — C₅×C₂³⋊₃D₄

Upper central

C₁ — C₂×C₁₀ — C₅×C₂³⋊₃D₄

Generators and relations for C₅×C₂³⋊₃D₄
G = < a,b,c,d,e,f | a⁵=b²=c²=d²=e⁴=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, ece^-1=fcf=cd=dc, de=ed, df=fd, fef=e^-1 >

Subgroups: 642 in 346 conjugacy classes, 162 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₁₀, C₁₀, C₁₀, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₂²⋊C₄, C₂²≀C₂, C₄⋊D₄, C₂².D₄, C₂²×D₄, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₂³⋊₃D₄, C₅×C₂²⋊C₄, C₅×C₄⋊C₄, C₂²×C₂₀, D₄×C₁₀, D₄×C₁₀, C₂³×C₁₀, C₂³×C₁₀, C₁₀×C₂²⋊C₄, C₅×C₂²≀C₂, C₅×C₄⋊D₄, C₅×C₂².D₄, D₄×C₂×C₁₀, C₅×C₂³⋊₃D₄
Quotients: C₁, C₂, C₂², C₅, D₄, C₂³, C₁₀, C₂×D₄, C₂⁴, C₂×C₁₀, C₂²×D₄, 2₊¹⁺⁴, C₅×D₄, C₂²×C₁₀, C₂³⋊₃D₄, D₄×C₁₀, C₂³×C₁₀, D₄×C₂×C₁₀, C₅×2₊¹⁺⁴, C₅×C₂³⋊₃D₄

Smallest permutation representation of C₅×C₂³⋊₃D₄
►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 36)(2 37)(3 38)(4 39)(5 40)(6 60)(7 56)(8 57)(9 58)(10 59)(11 68)(12 69)(13 70)(14 66)(15 67)(16 61)(17 62)(18 63)(19 64)(20 65)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(71 77)(72 78)(73 79)(74 80)(75 76)

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

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

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

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

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

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

110 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	2K	2L	2M	4A	···	4H	5A	5B	5C	5D	10A	···	10L	10M	···	10AJ	10AK	···	10AZ	20A	···	20AF
order	1	2	2	2	2	···	2	2	2	2	2	4	···	4	5	5	5	5	10	···	10	10	···	10	10	···	10	20	···	20
size	1	1	1	1	2	···	2	4	4	4	4	4	···	4	1	1	1	1	1	···	1	2	···	2	4	···	4	4	···	4

110 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	4	4
type	+	+	+	+	+	+							+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₅	C₁₀	C₁₀	C₁₀	C₁₀	C₁₀	D₄	C₅×D₄	2₊¹⁺⁴	C₅×2₊¹⁺⁴
kernel	C₅×C₂³⋊₃D₄	C₁₀×C₂²⋊C₄	C₅×C₂²≀C₂	C₅×C₄⋊D₄	C₅×C₂².D₄	D₄×C₂×C₁₀	C₂³⋊₃D₄	C₂×C₂²⋊C₄	C₂²≀C₂	C₄⋊D₄	C₂².D₄	C₂²×D₄	C₂²×C₁₀	C₂³	C₁₀	C₂
# reps	1	1	4	4	4	2	4	4	16	16	16	8	4	16	2	8

Matrix representation of C₅×C₂³⋊₃D₄ ►in GL₆(𝔽₄₁)

18	0	0	0	0	0
0	18	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

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

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	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

0	40	0	0	0	0
1	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	2	40
0	0	40	0	0	0
0	0	39	1	0	0

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

G:=sub<GL(6,GF(41))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,40,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,39,0,0,0,0,0,1,0,0,1,2,0,0,0,0,0,40,0,0],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₅×C₂³⋊₃D₄ in GAP, Magma, Sage, TeX

C_5\times C_2^3\rtimes_3D_4

% in TeX

G:=Group("C5xC2^3:3D4");

// GroupNames label

G:=SmallGroup(320,1536);

// by ID

G=gap.SmallGroup(320,1536);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,3446,891,2467]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f=b*d=d*b,b*e=e*b,e*c*e^-1=f*c*f=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;

// generators/relations

G = C5×C23⋊3D4 order 320 = 26·5

Direct product of C5 and C23⋊3D4

G = C₅×C₂³⋊₃D₄ order 320 = 2⁶·5

Direct product of C₅ and C₂³⋊₃D₄