C20.72C2^4 - GroupNames

G = C₂₀.₇₂C₂⁴ order 320 = 2⁶·5

19^th non-split extension by C₂₀ of C₂⁴ acting via C₂⁴/C₂³=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂₀.₇₂C₂⁴, C₄₀.₄₉C₂³, M₄(2)⋊₂₈D₁₀, C₈○D₄⋊₈D₅, (C₂×C₈)⋊₂₃D₁₀, (D₄×D₅).₁C₄, (Q₈×D₅).₁C₄, D₄.₁₃(C₄×D₅), Q₈.₁₄(C₄×D₅), (C₂×C₄₀)⋊₂₆C₂², C₄○D₄.₄₃D₁₀, D₂₀.₃₅(C₂×C₄), D₄⋊₂D₅.₁C₄, (C₈×D₅)⋊₁₂C₂², C₅⋊₅(Q₈○M₄(2)), D₄.Dic₅⋊₈C₂, Q₈⋊₂D₅.₁C₄, C₈.₅₆(C₂²×D₅), C₄.₇₁(C₂³×D₅), C₈⋊D₅⋊₂₂C₂², (D₅×M₄(2))⋊₁₁C₂, C₂₀.₇₄(C₂²×C₄), C₁₀.₅₆(C₂³×C₄), C₅⋊₂C₈.₃₃C₂³, (C₄×D₅).₇₃C₂³, D₂₀.₂C₄⋊₁₃C₂, D₂₀.₃C₄⋊₁₇C₂, (C₂×C₂₀).₅₁₄C₂³, Dic₁₀.₃₇(C₂×C₄), C₄○D₂₀.₅₂C₂², D₁₀.₂₅(C₂²×C₄), C₄.Dic₅⋊₂₇C₂², (C₅×M₄(2))⋊₂₈C₂², Dic₅.₂₄(C₂²×C₄), C₄.₃₉(C₂×C₄×D₅), (C₅×C₈○D₄)⋊₉C₂, C₂².₅(C₂×C₄×D₅), (D₅×C₄○D₄).₃C₂, C₅⋊D₄.₆(C₂×C₄), (C₂×C₈⋊D₅)⋊₂₈C₂, C₂.₃₆(D₅×C₂²×C₄), (C₄×D₅).₁₁(C₂×C₄), (C₅×D₄).₃₁(C₂×C₄), (C₅×Q₈).₃₃(C₂×C₄), (C₂×C₅⋊₂C₈)⋊₁₃C₂², (C₂×C₄×D₅).₁₆₃C₂², (C₂×C₁₀).₁₂(C₂²×C₄), (C₂×Dic₅).₄₀(C₂×C₄), (C₅×C₄○D₄).₄₄C₂², (C₂²×D₅).₃₃(C₂×C₄), (C₂×C₄).₆₀₇(C₂²×D₅), SmallGroup(320,1422)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₂₀.₇₂C₂⁴

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₄×D₅ — C₂×C₄×D₅ — D₅×C₄○D₄ — C₂₀.₇₂C₂⁴

Lower central

C₅ — C₁₀ — C₂₀.₇₂C₂⁴

Upper central

C₁ — C₄ — C₈○D₄

Generators and relations for C₂₀.₇₂C₂⁴
G = < a,b,c,d | a⁴⁰=b²=c²=d²=1, bab=a²⁹, cac=a²¹, ad=da, bc=cb, bd=db, dcd=a²⁰c >

Subgroups: 734 in 258 conjugacy classes, 147 normal (24 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₅, C₈, C₈, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, D₅, C₁₀, C₁₀, C₂×C₈, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, Dic₅, Dic₅, C₂₀, C₂₀, D₁₀, D₁₀, D₁₀, C₂×C₁₀, C₂×M₄(2), C₈○D₄, C₈○D₄, C₂×C₄○D₄, C₅⋊₂C₈, C₅⋊₂C₈, C₄₀, C₄₀, Dic₁₀, C₄×D₅, C₄×D₅, D₂₀, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₅×D₄, C₅×Q₈, C₂²×D₅, Q₈○M₄(2), C₈×D₅, C₈⋊D₅, C₈⋊D₅, C₂×C₅⋊₂C₈, C₄.Dic₅, C₂×C₄₀, C₅×M₄(2), C₂×C₄×D₅, C₄○D₂₀, D₄×D₅, D₄⋊₂D₅, Q₈×D₅, Q₈⋊₂D₅, C₅×C₄○D₄, C₂×C₈⋊D₅, D₂₀.₃C₄, D₅×M₄(2), D₂₀.₂C₄, D₄.Dic₅, C₅×C₈○D₄, D₅×C₄○D₄, C₂₀.₇₂C₂⁴
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, D₅, C₂²×C₄, C₂⁴, D₁₀, C₂³×C₄, C₄×D₅, C₂²×D₅, Q₈○M₄(2), C₂×C₄×D₅, C₂³×D₅, D₅×C₂²×C₄, C₂₀.₇₂C₂⁴

Smallest permutation representation of C₂₀.₇₂C₂⁴
►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)

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

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

(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)

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

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

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

74 conjugacy classes

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

74 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+	+	+					+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	D₅	D₁₀	D₁₀	D₁₀	C₄×D₅	C₄×D₅	Q₈○M₄(2)	C₂₀.₇₂C₂⁴
kernel	C₂₀.₇₂C₂⁴	C₂×C₈⋊D₅	D₂₀.₃C₄	D₅×M₄(2)	D₂₀.₂C₄	D₄.Dic₅	C₅×C₈○D₄	D₅×C₄○D₄	D₄×D₅	D₄⋊₂D₅	Q₈×D₅	Q₈⋊₂D₅	C₈○D₄	C₂×C₈	M₄(2)	C₄○D₄	D₄	Q₈	C₅	C₁
# reps	1	3	3	3	3	1	1	1	6	6	2	2	2	6	6	2	12	4	2	8

Matrix representation of C₂₀.₇₂C₂⁴ ►in GL₆(𝔽₄₁)

6	35	0	0	0	0
6	1	0	0	0	0
0	0	0	39	0	0
0	0	25	0	0	0
0	0	40	0	40	2
0	0	20	40	36	1

35	6	0	0	0	0
1	6	0	0	0	0
0	0	1	0	0	0
0	0	0	40	0	0
0	0	0	0	1	0
0	0	1	0	1	40

40	0	0	0	0	0
0	40	0	0	0	0
0	0	0	0	1	0
0	0	20	0	20	1
0	0	1	0	0	0
0	0	21	1	21	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	0	0	40	0
0	0	1	0	0	40

G:=sub<GL(6,GF(41))| [6,6,0,0,0,0,35,1,0,0,0,0,0,0,0,25,40,20,0,0,39,0,0,40,0,0,0,0,40,36,0,0,0,0,2,1],[35,1,0,0,0,0,6,6,0,0,0,0,0,0,1,0,0,1,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,20,1,21,0,0,0,0,0,1,0,0,1,20,0,21,0,0,0,1,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;

C₂₀.₇₂C₂⁴ in GAP, Magma, Sage, TeX

C_{20}._{72}C_2^4

% in TeX

G:=Group("C20.72C2^4");

// GroupNames label

G:=SmallGroup(320,1422);

// by ID

G=gap.SmallGroup(320,1422);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,1123,80,102,12550]);

// Polycyclic

G:=Group<a,b,c,d|a^40=b^2=c^2=d^2=1,b*a*b=a^29,c*a*c=a^21,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^20*c>;

// generators/relations

G = C20.72C24 order 320 = 26·5

19th non-split extension by C20 of C24 acting via C24/C23=C2

G = C₂₀.₇₂C₂⁴ order 320 = 2⁶·5

19^th non-split extension by C₂₀ of C₂⁴ acting via C₂⁴/C₂³=C₂