D4.12D20 - GroupNames

G = D₄.₁₂D₂₀ order 320 = 2⁶·5

2^nd non-split extension by D₄ of D₂₀ acting through Inn(D₄)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₄.₁₂D₂₀, Q₈.₁₂D₂₀, D₄₀⋊₁₂C₂², C₄₀.₁₁C₂³, C₂₀.₆₂C₂⁴, M₄(2)⋊₂₁D₁₀, D₂₀.₂₅C₂³, Dic₂₀⋊₂₁C₂², Dic₁₀.₂₅C₂³, C₈○D₄⋊₄D₅, (C₂×C₈)⋊₇D₁₀, C₅⋊₁(D₄○D₈), (C₂×D₄₀)⋊₁₅C₂, (C₅×D₄).₂₄D₄, C₂₀.₇₄(C₂×D₄), C₄.₂₈(C₂×D₂₀), (C₅×Q₈).₂₄D₄, D₄⋊₈D₁₀⋊₄C₂, C₈⋊D₁₀⋊₁₂C₂, (C₂×C₄₀)⋊₁₀C₂², C₄○D₄.₃₉D₁₀, C₄○D₂₀⋊₂C₂², D₄₀⋊₇C₂⋊₁₂C₂, C₂².₄(C₂×D₂₀), C₈.₅₃(C₂²×D₅), C₄.₅₉(C₂³×D₅), (C₂×D₂₀)⋊₃₁C₂², C₄₀⋊C₂⋊₁₂C₂², C₁₀.₂₉(C₂²×D₄), C₂.₃₁(C₂²×D₂₀), (C₂×C₂₀).₅₁₆C₂³, (C₅×M₄(2))⋊₂₃C₂², (C₅×C₈○D₄)⋊₄C₂, (C₂×C₁₀).₉(C₂×D₄), (C₅×C₄○D₄).₄₆C₂², (C₂×C₄).₂₂₇(C₂²×D₅), SmallGroup(320,1424)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — D₄.₁₂D₂₀

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — D₂₀ — C₂×D₂₀ — D₄⋊₈D₁₀ — D₄.₁₂D₂₀

Lower central

C₅ — C₁₀ — C₂₀ — D₄.₁₂D₂₀

Upper central

C₁ — C₂ — C₄○D₄ — C₈○D₄

Generators and relations for D₄.₁₂D₂₀
G = < a,b,c,d | a⁴=b²=d²=1, c²⁰=a², bab=a^-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a²c¹⁹ >

Subgroups: 1286 in 268 conjugacy classes, 107 normal (16 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₅, C₈, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, D₅, C₁₀, C₁₀, C₂×C₈, M₄(2), D₈, SD₁₆, Q₁₆, C₂×D₄, C₄○D₄, C₄○D₄, Dic₅, C₂₀, C₂₀, D₁₀, C₂×C₁₀, C₈○D₄, C₂×D₈, C₄○D₈, C₈⋊C₂², 2₊¹⁺⁴, C₄₀, C₄₀, Dic₁₀, C₄×D₅, D₂₀, D₂₀, C₅⋊D₄, C₂×C₂₀, C₅×D₄, C₅×Q₈, C₂²×D₅, D₄○D₈, C₄₀⋊C₂, D₄₀, Dic₂₀, C₂×C₄₀, C₅×M₄(2), C₂×D₂₀, C₄○D₂₀, D₄×D₅, Q₈⋊₂D₅, C₅×C₄○D₄, C₂×D₄₀, D₄₀⋊₇C₂, C₈⋊D₁₀, C₅×C₈○D₄, D₄⋊₈D₁₀, D₄.₁₂D₂₀
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂×D₄, C₂⁴, D₁₀, C₂²×D₄, D₂₀, C₂²×D₅, D₄○D₈, C₂×D₂₀, C₂³×D₅, C₂²×D₂₀, D₄.₁₂D₂₀

Smallest permutation representation of D₄.₁₂D₂₀
►On 80 points

Generators in S₈₀

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

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

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

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

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

62 conjugacy classes

class	1	2A	2B	2C	2D	2E	···	2J	4A	4B	4C	4D	4E	4F	5A	5B	8A	8B	8C	8D	8E	10A	10B	10C	···	10H	20A	20B	20C	20D	20E	···	20J	40A	···	40H	40I	···	40T
order	1	2	2	2	2	2	···	2	4	4	4	4	4	4	5	5	8	8	8	8	8	10	10	10	···	10	20	20	20	20	20	···	20	40	···	40	40	···	40
size	1	1	2	2	2	20	···	20	2	2	2	2	20	20	2	2	2	2	4	4	4	2	2	4	···	4	2	2	2	2	4	···	4	2	···	2	4	···	4

62 irreducible representations

dim

type

image

C₁

C₂

D₄

D₅

D₁₀

D₂₀

D₄○D₈

D₄.₁₂D₂₀

kernel

D₄.₁₂D₂₀

C₂×D₄₀

D₄₀⋊₇C₂

C₈⋊D₁₀

C₅×C₈○D₄

D₄⋊₈D₁₀

C₅×D₄

C₅×Q₈

C₈○D₄

C₂×C₈

M₄(2)

C₄○D₄

D₄

Q₈

C₅

C₁

# reps

Matrix representation of D₄.₁₂D₂₀ ►in GL₆(𝔽₄₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	20	0
0	0	38	0	27	40
0	0	4	0	40	0
0	0	23	1	36	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	4	0	40	0
0	0	20	0	0	40

0	40	0	0	0	0
1	35	0	0	0	0
0	0	40	13	0	0
0	0	8	18	0	0
0	0	37	26	35	26
0	0	23	7	12	23

0	1	0	0	0	0
1	0	0	0	0	0
0	0	40	13	0	0
0	0	0	1	0	0
0	0	37	26	35	26
0	0	21	7	16	6

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,38,4,23,0,0,0,0,0,1,0,0,20,27,40,36,0,0,0,40,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,4,20,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,40,35,0,0,0,0,0,0,40,8,37,23,0,0,13,18,26,7,0,0,0,0,35,12,0,0,0,0,26,23],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,37,21,0,0,13,1,26,7,0,0,0,0,35,16,0,0,0,0,26,6] >;

D₄.₁₂D₂₀ in GAP, Magma, Sage, TeX

D_4._{12}D_{20}

% in TeX

G:=Group("D4.12D20");

// GroupNames label

G:=SmallGroup(320,1424);

// by ID

G=gap.SmallGroup(320,1424);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,675,192,1684,102,12550]);

// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^20=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^2*c^19>;

// generators/relations

G = D4.12D20 order 320 = 26·5

2nd non-split extension by D4 of D20 acting through Inn(D4)

G = D₄.₁₂D₂₀ order 320 = 2⁶·5

2^nd non-split extension by D₄ of D₂₀ acting through Inn(D₄)