C10xC2^2wrC2 - GroupNames

G = C₁₀×C₂²≀C₂ order 320 = 2⁶·5

Direct product of C₁₀ and C₂²≀C₂

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

Aliases: C₁₀×C₂²≀C₂, C₂⁵⋊₂C₁₀, C₂³⋊₅(C₅×D₄), (C₂⁴×C₁₀)⋊₁C₂, C₂⁴⋊₇(C₂×C₁₀), C₂²⋊₃(D₄×C₁₀), (C₂×C₂₀)⋊₁₀C₂³, (C₂²×D₄)⋊₃C₁₀, (C₂²×C₁₀)⋊₁₇D₄, (D₄×C₁₀)⋊₆₀C₂², C₂³⋊₁(C₂²×C₁₀), (C₂²×C₁₀)⋊₃C₂³, (C₂³×C₁₀)⋊₂C₂², (C₂×C₁₀).₃₄₁C₂⁴, (C₂²×C₂₀)⋊₄₅C₂², C₁₀.₁₈₀(C₂²×D₄), C₂².₁₅(C₂³×C₁₀), C₂.₄(D₄×C₂×C₁₀), (D₄×C₂×C₁₀)⋊₁₈C₂, (C₂×D₄)⋊₈(C₂×C₁₀), (C₂×C₁₀)⋊₁₅(C₂×D₄), (C₂×C₂²⋊C₄)⋊₈C₁₀, (C₂²×C₄)⋊₅(C₂×C₁₀), (C₂×C₄)⋊₁(C₂²×C₁₀), (C₁₀×C₂²⋊C₄)⋊₂₈C₂, C₂²⋊C₄⋊₁₀(C₂×C₁₀), (C₅×C₂²⋊C₄)⋊₆₄C₂², SmallGroup(320,1523)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₁₀×C₂²≀C₂

Chief series

C₁ — C₂ — C₂² — C₂×C₁₀ — C₂²×C₁₀ — D₄×C₁₀ — C₅×C₂²≀C₂ — C₁₀×C₂²≀C₂

Lower central

C₁ — C₂² — C₁₀×C₂²≀C₂

Upper central

C₁ — C₂²×C₁₀ — C₁₀×C₂²≀C₂

Generators and relations for C₁₀×C₂²≀C₂
G = < a,b,c,d,e,f | a¹⁰=b²=c²=d²=e²=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, cd=dc, fcf=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 1138 in 662 conjugacy classes, 210 normal (12 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₁₀, C₁₀, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, C₂⁴, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₂²⋊C₄, C₂²≀C₂, C₂²×D₄, C₂⁵, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₂×C₂²≀C₂, C₅×C₂²⋊C₄, C₂²×C₂₀, D₄×C₁₀, D₄×C₁₀, C₂³×C₁₀, C₂³×C₁₀, C₂³×C₁₀, C₁₀×C₂²⋊C₄, C₅×C₂²≀C₂, D₄×C₂×C₁₀, C₂⁴×C₁₀, C₁₀×C₂²≀C₂
Quotients: C₁, C₂, C₂², C₅, D₄, C₂³, C₁₀, C₂×D₄, C₂⁴, C₂×C₁₀, C₂²≀C₂, C₂²×D₄, C₅×D₄, C₂²×C₁₀, C₂×C₂²≀C₂, D₄×C₁₀, C₂³×C₁₀, C₅×C₂²≀C₂, D₄×C₂×C₁₀, C₁₀×C₂²≀C₂

Smallest permutation representation of C₁₀×C₂²≀C₂
►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 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 21)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 71)(38 72)(39 73)(40 74)(41 54)(42 55)(43 56)(44 57)(45 58)(46 59)(47 60)(48 51)(49 52)(50 53)

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

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

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

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

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

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

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

140 conjugacy classes

class	1	2A	···	2G	2H	···	2S	2T	2U	4A	···	4F	5A	5B	5C	5D	10A	···	10AB	10AC	···	10BX	10BY	···	10CF	20A	···	20X
order	1	2	···	2	2	···	2	2	2	4	···	4	5	5	5	5	10	···	10	10	···	10	10	···	10	20	···	20
size	1	1	···	1	2	···	2	4	4	4	···	4	1	1	1	1	1	···	1	2	···	2	4	···	4	4	···	4

140 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2
type	+	+	+	+	+						+
image	C₁	C₂	C₂	C₂	C₂	C₅	C₁₀	C₁₀	C₁₀	C₁₀	D₄	C₅×D₄
kernel	C₁₀×C₂²≀C₂	C₁₀×C₂²⋊C₄	C₅×C₂²≀C₂	D₄×C₂×C₁₀	C₂⁴×C₁₀	C₂×C₂²≀C₂	C₂×C₂²⋊C₄	C₂²≀C₂	C₂²×D₄	C₂⁵	C₂²×C₁₀	C₂³
# reps	1	3	8	3	1	4	12	32	12	4	12	48

Matrix representation of C₁₀×C₂²≀C₂ ►in GL₅(𝔽₄₁)

40	0	0	0	0
0	18	0	0	0
0	0	18	0	0
0	0	0	23	0
0	0	0	0	23

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

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

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

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

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

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,23,0,0,0,0,0,23],[1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,1,0] >;

C₁₀×C₂²≀C₂ in GAP, Magma, Sage, TeX

C_{10}\times C_2^2\wr C_2

% in TeX

G:=Group("C10xC2^2wrC2");

// GroupNames label

G:=SmallGroup(320,1523);

// by ID

G=gap.SmallGroup(320,1523);

# by ID

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

// Polycyclic

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

// generators/relations

G = C10×C22≀C2 order 320 = 26·5

Direct product of C10 and C22≀C2

G = C₁₀×C₂²≀C₂ order 320 = 2⁶·5

Direct product of C₁₀ and C₂²≀C₂