C2^4.38D10 - GroupNames

G = C₂⁴.₃₈D₁₀ order 320 = 2⁶·5

38^th non-split extension by C₂⁴ of D₁₀ acting via D₁₀/C₅=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴.₃₈D₁₀, C₁₀.₈₇2₊¹⁺⁴, (D₄×C₁₀)⋊₂₄C₄, D₄⋊₆(C₂×Dic₅), (C₂×D₄)⋊₁₁Dic₅, (D₄×Dic₅)⋊₃₇C₂, C₂³⋊₃(C₂×Dic₅), (C₂×D₄).₂₅₁D₁₀, C₁₀.₆₆(C₂³×C₄), C₄⋊Dic₅⋊₇₆C₂², (C₂²×D₄).₁₂D₅, C₂.₅(D₄⋊₆D₁₀), C₂.₇(C₂³×Dic₅), C₂₀.₁₅₃(C₂²×C₄), (C₂×C₁₀).₂₉₃C₂⁴, (C₂×C₂₀).₅₄₁C₂³, C₅⋊₅(C₂².₁₁C₂⁴), (C₄×Dic₅)⋊₄₀C₂², (C₂²×C₄).₂₇₀D₁₀, C₂³.D₅⋊₅₉C₂², C₄.₁₇(C₂²×Dic₅), C₂².₄₅(C₂³×D₅), (D₄×C₁₀).₂₇₀C₂², (C₂³×C₁₀).₇₅C₂², C₂³.₂₀₄(C₂²×D₅), C₂³.₂₁D₁₀⋊₃₂C₂, C₂².₁(C₂²×Dic₅), (C₂²×C₁₀).₂₂₉C₂³, (C₂²×C₂₀).₂₇₄C₂², (C₂×Dic₅).₂₉₃C₂³, (C₂²×Dic₅)⋊₃₁C₂², (D₄×C₂×C₁₀).₉C₂, (C₂×C₂₀)⋊₂₈(C₂×C₄), (C₅×D₄)⋊₃₀(C₂×C₄), (C₂×C₄)⋊₄(C₂×Dic₅), (C₂²×C₁₀)⋊₁₉(C₂×C₄), (C₂×C₂³.D₅)⋊₂₆C₂, (C₂×C₄).₆₂₄(C₂²×D₅), (C₂×C₁₀).₁₂₈(C₂²×C₄), SmallGroup(320,1470)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₂⁴.₃₈D₁₀

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂×Dic₅ — C₂²×Dic₅ — D₄×Dic₅ — C₂⁴.₃₈D₁₀

Lower central

C₅ — C₁₀ — C₂⁴.₃₈D₁₀

Upper central

C₁ — C₂² — C₂²×D₄

Generators and relations for C₂⁴.₃₈D₁₀
G = < a,b,c,d,e,f | a²=b²=c²=d²=e¹⁰=1, f²=c, ab=ba, ac=ca, eae^-1=faf^-1=ad=da, bc=cb, fbf^-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e^-1 >

Subgroups: 910 in 338 conjugacy classes, 191 normal (13 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₅, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₁₀, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, Dic₅, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₂²⋊C₄, C₄²⋊C₂, C₄×D₄, C₂²×D₄, C₂×Dic₅, C₂×Dic₅, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₂².₁₁C₂⁴, C₄×Dic₅, C₄⋊Dic₅, C₂³.D₅, C₂²×Dic₅, C₂²×C₂₀, D₄×C₁₀, C₂³×C₁₀, C₂³.₂₁D₁₀, D₄×Dic₅, C₂×C₂³.D₅, D₄×C₂×C₁₀, C₂⁴.₃₈D₁₀
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, D₅, C₂²×C₄, C₂⁴, Dic₅, D₁₀, C₂³×C₄, 2₊¹⁺⁴, C₂×Dic₅, C₂²×D₅, C₂².₁₁C₂⁴, C₂²×Dic₅, C₂³×D₅, D₄⋊₆D₁₀, C₂³×Dic₅, C₂⁴.₃₈D₁₀

Smallest permutation representation of C₂⁴.₃₈D₁₀
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

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

74 conjugacy classes

class	1	2A	2B	2C	2D	···	2M	4A	4B	4C	4D	4E	···	4T	5A	5B	10A	···	10N	10O	···	10AD	20A	···	20H
order	1	2	2	2	2	···	2	4	4	4	4	4	···	4	5	5	10	···	10	10	···	10	20	···	20
size	1	1	1	1	2	···	2	2	2	2	2	10	···	10	2	2	2	···	2	4	···	4	4	···	4

74 irreducible representations

dim

type

image

C₁

C₂

C₄

D₅

D₁₀

Dic₅

D₁₀

2₊¹⁺⁴

D₄⋊₆D₁₀

kernel

C₂⁴.₃₈D₁₀

C₂³.₂₁D₁₀

D₄×Dic₅

C₂×C₂³.D₅

D₄×C₂×C₁₀

D₄×C₁₀

C₂²×D₄

C₂²×C₄

C₂×D₄

C₂⁴

C₁₀

C₂

# reps

Matrix representation of C₂⁴.₃₈D₁₀ ►in GL₆(𝔽₄₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

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	17	14	40	0
0	0	14	17	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

23	0	0	0	0	0
0	25	0	0	0	0
0	0	1	0	0	0
0	0	0	40	0	0
0	0	0	27	1	0
0	0	14	0	0	40

0	21	0	0	0	0
39	0	0	0	0	0
0	0	25	23	26	0
0	0	18	16	0	15
0	0	20	0	16	23
0	0	0	21	18	25

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,17,14,0,0,0,1,14,17,0,0,0,0,40,0,0,0,0,0,0,40],[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],[23,0,0,0,0,0,0,25,0,0,0,0,0,0,1,0,0,14,0,0,0,40,27,0,0,0,0,0,1,0,0,0,0,0,0,40],[0,39,0,0,0,0,21,0,0,0,0,0,0,0,25,18,20,0,0,0,23,16,0,21,0,0,26,0,16,18,0,0,0,15,23,25] >;

C₂⁴.₃₈D₁₀ in GAP, Magma, Sage, TeX

C_2^4._{38}D_{10}

% in TeX

G:=Group("C2^4.38D10");

// GroupNames label

G:=SmallGroup(320,1470);

// by ID

G=gap.SmallGroup(320,1470);

# by ID

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

// Polycyclic

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

// generators/relations

G = C24.38D10 order 320 = 26·5

38th non-split extension by C24 of D10 acting via D10/C5=C22

G = C₂⁴.₃₈D₁₀ order 320 = 2⁶·5

38^th non-split extension by C₂⁴ of D₁₀ acting via D₁₀/C₅=C₂²