C2xC2^4:2D5 - GroupNames

G = C₂×C₂⁴⋊₂D₅ order 320 = 2⁶·5

Direct product of C₂ and C₂⁴⋊₂D₅

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂×C₂⁴⋊₂D₅, C₂⁵⋊₂D₅, C₂⁴⋊₁₂D₁₀, C₁₀⋊₃C₂²≀C₂, (C₂⁴×C₁₀)⋊₃C₂, (C₂²×C₁₀)⋊₁₈D₄, C₂³⋊₇(C₅⋊D₄), (C₂×Dic₅)⋊₄C₂³, (C₂²×D₅)⋊₃C₂³, (C₂×C₁₀).₃₂₂C₂⁴, (C₂³×C₁₀)⋊₁₈C₂², C₁₀.₁₇₆(C₂²×D₄), (C₂³×D₅)⋊₁₆C₂², C₂³.D₅⋊₆₆C₂², C₂².₃₃₀(C₂³×D₅), C₂³.₂₄₁(C₂²×D₅), (C₂²×C₁₀).₄₂₉C₂³, (C₂²×Dic₅)⋊₃₇C₂², C₅⋊₄(C₂×C₂²≀C₂), (C₂×C₁₀)⋊₁₆(C₂×D₄), C₂²⋊₄(C₂×C₅⋊D₄), (C₂²×C₅⋊D₄)⋊₁₈C₂, (C₂×C₅⋊D₄)⋊₄₉C₂², (C₂×C₂³.D₅)⋊₃₂C₂, C₂.₄₈(C₂²×C₅⋊D₄), SmallGroup(320,1512)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — C₂×C₂⁴⋊₂D₅

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂²×D₅ — C₂³×D₅ — C₂²×C₅⋊D₄ — C₂×C₂⁴⋊₂D₅

Lower central

C₅ — C₂×C₁₀ — C₂×C₂⁴⋊₂D₅

Upper central

C₁ — C₂³ — C₂⁵

Generators and relations for C₂×C₂⁴⋊₂D₅
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=f⁵=g²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, gbg=be=eb, bf=fb, gcg=cd=dc, ce=ec, cf=fc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, gfg=f^-1 >

Subgroups: 1918 in 662 conjugacy classes, 159 normal (10 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂×C₄, D₄, C₂³, C₂³, C₂³, D₅, C₁₀, C₁₀, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, Dic₅, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₂²⋊C₄, C₂²≀C₂, C₂²×D₄, C₂⁵, C₂×Dic₅, C₂×Dic₅, C₅⋊D₄, C₂²×D₅, C₂²×D₅, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₂×C₂²≀C₂, C₂³.D₅, C₂²×Dic₅, C₂×C₅⋊D₄, C₂×C₅⋊D₄, C₂³×D₅, C₂³×C₁₀, C₂³×C₁₀, C₂×C₂³.D₅, C₂⁴⋊₂D₅, C₂²×C₅⋊D₄, C₂⁴×C₁₀, C₂×C₂⁴⋊₂D₅
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂×D₄, C₂⁴, D₁₀, C₂²≀C₂, C₂²×D₄, C₅⋊D₄, C₂²×D₅, C₂×C₂²≀C₂, C₂×C₅⋊D₄, C₂³×D₅, C₂⁴⋊₂D₅, C₂²×C₅⋊D₄, C₂×C₂⁴⋊₂D₅

Smallest permutation representation of C₂×C₂⁴⋊₂D₅
►On 80 points

Generators in S₈₀

(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(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 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 76)(42 77)(43 78)(44 79)(45 80)(46 71)(47 72)(48 73)(49 74)(50 75)(51 66)(52 67)(53 68)(54 69)(55 70)(56 61)(57 62)(58 63)(59 64)(60 65)

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

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

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

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

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

92 conjugacy classes

class	1	2A	···	2G	2H	···	2S	2T	2U	4A	···	4F	5A	5B	10A	···	10BJ
order	1	2	···	2	2	···	2	2	2	4	···	4	5	5	10	···	10
size	1	1	···	1	2	···	2	20	20	20	···	20	2	2	2	···	2

92 irreducible representations

dim	1	1	1	1	1	2	2	2	2
type	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	D₄	D₅	D₁₀	C₅⋊D₄
kernel	C₂×C₂⁴⋊₂D₅	C₂×C₂³.D₅	C₂⁴⋊₂D₅	C₂²×C₅⋊D₄	C₂⁴×C₁₀	C₂²×C₁₀	C₂⁵	C₂⁴	C₂³
# reps	1	3	8	3	1	12	2	14	48

Matrix representation of C₂×C₂⁴⋊₂D₅ ►in GL₅(𝔽₄₁)

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

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

1	0	0	0	0
0	10	0	0	0
0	0	37	0	0
0	0	0	10	0
0	0	0	0	37

1	0	0	0	0
0	0	4	0	0
0	31	0	0	0
0	0	0	0	4
0	0	0	31	0

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1],[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],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,10,0,0,0,0,0,37,0,0,0,0,0,10,0,0,0,0,0,37],[1,0,0,0,0,0,0,31,0,0,0,4,0,0,0,0,0,0,0,31,0,0,0,4,0] >;

C₂×C₂⁴⋊₂D₅ in GAP, Magma, Sage, TeX

C_2\times C_2^4\rtimes_2D_5

% in TeX

G:=Group("C2xC2^4:2D5");

// GroupNames label

G:=SmallGroup(320,1512);

// by ID

G=gap.SmallGroup(320,1512);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,675,12550]);

// Polycyclic

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

// generators/relations

G = C2×C24⋊2D5 order 320 = 26·5

Direct product of C2 and C24⋊2D5

G = C₂×C₂⁴⋊₂D₅ order 320 = 2⁶·5

Direct product of C₂ and C₂⁴⋊₂D₅