C4:C4:Dic5 - GroupNames

G = C₄⋊C₄⋊Dic₅ order 320 = 2⁶·5

1^st semidirect product of C₄⋊C₄ and Dic₅ acting via Dic₅/C₅=C₄

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄⋊C₄⋊₁Dic₅, (C₂×C₁₀).₃D₈, C₁₀.₂₈C₄≀C₂, (D₄×C₁₀)⋊₁₃C₄, (C₂×D₄)⋊₁Dic₅, C₄⋊D₄.₁D₅, (C₂×C₂₀).₂₂₉D₄, C₂².₂(D₄⋊D₅), (C₂×C₁₀).₁₀SD₁₆, (C₂²×C₁₀).₄₄D₄, (C₂²×C₄).₅₉D₁₀, C₅⋊₅(C₂².SD₁₆), C₂₀.₅₅D₄⋊₂₈C₂, C₁₀.₄₁(C₂³⋊C₄), C₂.₃(D₄⋊Dic₅), C₂³.₄₈(C₅⋊D₄), C₂².₂(D₄.D₅), C₂.₅(C₂³⋊Dic₅), C₁₀.₃₈(D₄⋊C₄), C₂.₄(D₄⋊₂Dic₅), C₁₀.₁₀C₄²⋊₄₁C₂, (C₂²×C₂₀).₃₇₁C₂², C₂².₃₇(C₂³.D₅), (C₅×C₄⋊C₄)⋊₈C₄, (C₂×C₄).₇(C₂×Dic₅), (C₂×C₂₀).₃₄₀(C₂×C₄), (C₅×C₄⋊D₄).₁₀C₂, (C₂×C₄).₁₆₃(C₅⋊D₄), (C₂×C₁₀).₁₅₉(C₂²⋊C₄), SmallGroup(320,95)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₀ — C₄⋊C₄⋊Dic₅

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂²×C₁₀ — C₂²×C₂₀ — C₁₀.₁₀C₄² — C₄⋊C₄⋊Dic₅

Lower central

C₅ — C₂×C₁₀ — C₂×C₂₀ — C₄⋊C₄⋊Dic₅

Upper central

C₁ — C₂² — C₂²×C₄ — C₄⋊D₄

Generators and relations for C₄⋊C₄⋊Dic₅
G = < a,b,c,d | a⁴=b⁴=c¹⁰=1, d²=c⁵, bab^-1=a^-1, ac=ca, dad^-1=a^-1b², cbc^-1=b^-1, dbd^-1=a^-1b, dcd^-1=c^-1 >

Subgroups: 334 in 90 conjugacy classes, 31 normal (all characteristic)
C₁, C₂^[×3], C₂^[×3], C₄^[×5], C₂²^[×3], C₂²^[×5], C₅, C₈, C₂×C₄^[×2], C₂×C₄^[×6], D₄^[×3], C₂³, C₂³, C₁₀^[×3], C₁₀^[×3], C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, Dic₅^[×2], C₂₀^[×3], C₂×C₁₀^[×3], C₂×C₁₀^[×5], C₂.C₄², C₂²⋊C₈, C₄⋊D₄, C₅⋊₂C₈, C₂×Dic₅^[×4], C₂×C₂₀^[×2], C₂×C₂₀^[×2], C₅×D₄^[×3], C₂²×C₁₀, C₂²×C₁₀, C₂².SD₁₆, C₂×C₅⋊₂C₈, C₅×C₂²⋊C₄, C₅×C₄⋊C₄, C₂²×Dic₅, C₂²×C₂₀, D₄×C₁₀, D₄×C₁₀, C₂₀.₅₅D₄, C₁₀.₁₀C₄², C₅×C₄⋊D₄, C₄⋊C₄⋊Dic₅
Quotients: C₁, C₂^[×3], C₄^[×2], C₂², C₂×C₄, D₄^[×2], D₅, C₂²⋊C₄, D₈, SD₁₆, Dic₅^[×2], D₁₀, C₂³⋊C₄, D₄⋊C₄, C₄≀C₂, C₂×Dic₅, C₅⋊D₄^[×2], C₂².SD₁₆, D₄⋊D₅, D₄.D₅, C₂³.D₅, D₄⋊Dic₅, C₂³⋊Dic₅, D₄⋊₂Dic₅, C₄⋊C₄⋊Dic₅

Smallest permutation representation of C₄⋊C₄⋊Dic₅
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

47 conjugacy classes

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

47 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4
type	+	+	+	+			+	+	+	+		-	+	-				+	+	-
image	C₁	C₂	C₂	C₂	C₄	C₄	D₄	D₄	D₅	D₈	SD₁₆	Dic₅	D₁₀	Dic₅	C₄≀C₂	C₅⋊D₄	C₅⋊D₄	C₂³⋊C₄	D₄⋊D₅	D₄.D₅	C₂³⋊Dic₅	D₄⋊₂Dic₅
kernel	C₄⋊C₄⋊Dic₅	C₂₀.₅₅D₄	C₁₀.₁₀C₄²	C₅×C₄⋊D₄	C₅×C₄⋊C₄	D₄×C₁₀	C₂×C₂₀	C₂²×C₁₀	C₄⋊D₄	C₂×C₁₀	C₂×C₁₀	C₄⋊C₄	C₂²×C₄	C₂×D₄	C₁₀	C₂×C₄	C₂³	C₁₀	C₂²	C₂²	C₂	C₂
# reps	1	1	1	1	2	2	1	1	2	2	2	2	2	2	4	4	4	1	2	2	4	4

Matrix representation of C₄⋊C₄⋊Dic₅ ►in GL₆(𝔽₄₁)

32	0	0	0	0	0
26	9	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	9	37
0	0	0	0	0	32

3	21	0	0	0	0
21	38	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	3	7
0	0	0	0	34	38

1	0	0	0	0	0
29	40	0	0	0	0
0	0	0	1	0	0
0	0	40	34	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
22	32	0	0	0	0
0	0	1	0	0	0
0	0	34	40	0	0
0	0	0	0	1	0
0	0	0	0	25	40

G:=sub<GL(6,GF(41))| [32,26,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,37,32],[3,21,0,0,0,0,21,38,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,3,34,0,0,0,0,7,38],[1,29,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,22,0,0,0,0,0,32,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,1,25,0,0,0,0,0,40] >;

C₄⋊C₄⋊Dic₅ in GAP, Magma, Sage, TeX

C_4\rtimes C_4\rtimes {\rm Dic}_5

% in TeX

G:=Group("C4:C4:Dic5");

// GroupNames label

G:=SmallGroup(320,95);

// by ID

G=gap.SmallGroup(320,95);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,1571,570,136,12550]);

// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=c^5,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;

// generators/relations

G = C4⋊C4⋊Dic5 order 320 = 26·5

1st semidirect product of C4⋊C4 and Dic5 acting via Dic5/C5=C4

G = C₄⋊C₄⋊Dic₅ order 320 = 2⁶·5

1^st semidirect product of C₄⋊C₄ and Dic₅ acting via Dic₅/C₅=C₄