C4oD20:C4 - GroupNames

G = C₄○D₂₀⋊C₄ order 320 = 2⁶·5

3^rd semidirect product of C₄○D₂₀ and C₄ acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄○D₄⋊₃F₅, C₄○D₂₀⋊₃C₄, D₄.₉(C₂×F₅), Q₈.₉(C₂×F₅), D₄⋊₂D₅⋊₉C₄, D₂₀⋊C₄⋊₅C₂, Q₈⋊₂D₅⋊₉C₄, Q₈⋊F₅⋊₅C₂, D₂₀.₉(C₂×C₄), (C₄×D₅).₅₆D₄, D₅.₅(C₄○D₈), C₅⋊(C₂³.₂₄D₄), C₄⋊F₅.₁₀C₂², C₄.₂₃(C₂²×F₅), D₁₀.₁₀₁(C₂×D₄), C₂₀.₂₃(C₂²×C₄), D₅⋊C₈.₁₆C₂², Dic₁₀.₉(C₂×C₄), Dic₅.₁₁(C₂×D₄), (C₂²×D₅).₇₃D₄, (D₄×D₅).₁₅C₂², (C₄×D₅).₄₅C₂³, C₄.₄₇(C₂²⋊F₅), (Q₈×D₅).₁₃C₂², C₂₀.₄₇(C₂²⋊C₄), (C₂×Dic₅).₁₂₅D₄, C₂².₆(C₂²⋊F₅), D₁₀.₁₈(C₂²⋊C₄), D₁₀.C₂³⋊₆C₂, Dic₅.₅₀(C₂²⋊C₄), (C₂×D₅⋊C₈)⋊₄C₂, (C₅×C₄○D₄)⋊₃C₄, (D₅×C₄○D₄).₇C₂, (C₅×D₄).₉(C₂×C₄), (C₂×C₄).₉₀(C₂×F₅), (C₅×Q₈).₉(C₂×C₄), (C₂×C₂₀).₆₈(C₂×C₄), (C₄×D₅).₂₉(C₂×C₄), C₂.₃₆(C₂×C₂²⋊F₅), C₁₀.₃₅(C₂×C₂²⋊C₄), (C₂×C₄×D₅).₂₁₂C₂², (C₂×C₁₀).₆(C₂²⋊C₄), SmallGroup(320,1132)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — C₄○D₂₀⋊C₄

Chief series

C₁ — C₅ — C₁₀ — Dic₅ — C₄×D₅ — D₅⋊C₈ — C₂×D₅⋊C₈ — C₄○D₂₀⋊C₄

Lower central

C₅ — C₁₀ — C₂₀ — C₄○D₂₀⋊C₄

Upper central

C₁ — C₄ — C₂×C₄ — C₄○D₄

Generators and relations for C₄○D₂₀⋊C₄
G = < a,b,c,d | a⁴=c²=d⁴=1, b¹⁰=a², ab=ba, ac=ca, ad=da, cbc=a²b⁹, dbd^-1=b⁷, dcd^-1=a²bc >

Subgroups: 682 in 158 conjugacy classes, 50 normal (40 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, D₅, D₅, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, Dic₅, Dic₅, C₂₀, C₂₀, F₅, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, D₄⋊C₄, Q₈⋊C₄, C₄²⋊C₂, C₂²×C₈, C₂×C₄○D₄, C₅⋊C₈, Dic₁₀, Dic₁₀, C₄×D₅, C₄×D₅, D₂₀, D₂₀, C₂×Dic₅, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₅×D₄, C₅×Q₈, C₂×F₅, C₂²×D₅, C₂²×D₅, C₂³.₂₄D₄, D₅⋊C₈, D₅⋊C₈, C₄×F₅, C₄⋊F₅, C₂×C₅⋊C₈, C₂²⋊F₅, C₂×C₄×D₅, C₂×C₄×D₅, C₄○D₂₀, C₄○D₂₀, D₄×D₅, D₄×D₅, D₄⋊₂D₅, D₄⋊₂D₅, Q₈×D₅, Q₈⋊₂D₅, C₅×C₄○D₄, D₂₀⋊C₄, Q₈⋊F₅, C₂×D₅⋊C₈, D₁₀.C₂³, D₅×C₄○D₄, C₄○D₂₀⋊C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, F₅, C₂×C₂²⋊C₄, C₄○D₈, C₂×F₅, C₂³.₂₄D₄, C₂²⋊F₅, C₂²×F₅, C₂×C₂²⋊F₅, C₄○D₂₀⋊C₄

Smallest permutation representation of C₄○D₂₀⋊C₄
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	4H	···	4L	5	8A	···	8H	10A	10B	10C	10D	20A	20B	20C	20D	20E
order	1	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	···	4	5	8	···	8	10	10	10	10	20	20	20	20	20
size	1	1	2	4	5	5	10	20	1	1	2	4	5	5	10	20	···	20	4	10	···	10	4	8	8	8	4	4	8	8	8

38 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4	4	4	4	4	8
type	+	+	+	+	+	+					+	+	+		+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	D₄	D₄	D₄	C₄○D₈	F₅	C₂×F₅	C₂×F₅	C₂×F₅	C₂²⋊F₅	C₂²⋊F₅	C₄○D₂₀⋊C₄
kernel	C₄○D₂₀⋊C₄	D₂₀⋊C₄	Q₈⋊F₅	C₂×D₅⋊C₈	D₁₀.C₂³	D₅×C₄○D₄	C₄○D₂₀	D₄⋊₂D₅	Q₈⋊₂D₅	C₅×C₄○D₄	C₄×D₅	C₂×Dic₅	C₂²×D₅	D₅	C₄○D₄	C₂×C₄	D₄	Q₈	C₄	C₂²	C₁
# reps	1	2	2	1	1	1	2	2	2	2	2	1	1	8	1	1	1	1	2	2	2

Matrix representation of C₄○D₂₀⋊C₄ ►in GL₆(𝔽₄₁)

32	0	0	0	0	0
0	32	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	16	0	0	0	0
5	40	0	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	40	40	40	40

0	28	0	0	0	0
22	0	0	0	0	0
0	0	0	40	0	0
0	0	40	0	0	0
0	0	1	1	1	1
0	0	0	0	0	40

32	20	0	0	0	0
0	9	0	0	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	40	40	40	40
0	0	0	1	0	0

G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,5,0,0,0,0,16,40,0,0,0,0,0,0,0,0,0,40,0,0,1,0,0,40,0,0,0,1,0,40,0,0,0,0,1,40],[0,22,0,0,0,0,28,0,0,0,0,0,0,0,0,40,1,0,0,0,40,0,1,0,0,0,0,0,1,0,0,0,0,0,1,40],[32,0,0,0,0,0,20,9,0,0,0,0,0,0,1,0,40,0,0,0,0,0,40,1,0,0,0,1,40,0,0,0,0,0,40,0] >;

C₄○D₂₀⋊C₄ in GAP, Magma, Sage, TeX

C_4\circ D_{20}\rtimes C_4

% in TeX

G:=Group("C4oD20:C4");

// GroupNames label

G:=SmallGroup(320,1132);

// by ID

G=gap.SmallGroup(320,1132);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,297,136,438,102,6278,1595]);

// Polycyclic

G:=Group<a,b,c,d|a^4=c^2=d^4=1,b^10=a^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=a^2*b^9,d*b*d^-1=b^7,d*c*d^-1=a^2*b*c>;

// generators/relations

G = C4○D20⋊C4 order 320 = 26·5

3rd semidirect product of C4○D20 and C4 acting faithfully

G = C₄○D₂₀⋊C₄ order 320 = 2⁶·5

3^rd semidirect product of C₄○D₂₀ and C₄ acting faithfully