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, C₂, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₁₀, C₁₀, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, Dic₅, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂.C₄², C₂²⋊C₈, C₄⋊D₄, C₅⋊₂C₈, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₅×D₄, 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, C₄, C₂², C₂×C₄, D₄, D₅, C₂²⋊C₄, D₈, SD₁₆, Dic₅, D₁₀, C₂³⋊C₄, D₄⋊C₄, C₄≀C₂, C₂×Dic₅, C₅⋊D₄, 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 24)(12 27 17 25)(13 28 18 21)(14 29 19 22)(15 30 20 23)(41 55 78 63)(42 56 79 64)(43 57 80 65)(44 58 71 66)(45 59 72 67)(46 60 73 68)(47 51 74 69)(48 52 75 70)(49 53 76 61)(50 54 77 62)

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

(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 40)(22 39)(23 38)(24 37)(25 36)(26 32)(27 31)(28 35)(29 34)(30 33)(41 63 46 68)(42 62 47 67)(43 61 48 66)(44 70 49 65)(45 69 50 64)(51 77 56 72)(52 76 57 71)(53 75 58 80)(54 74 59 79)(55 73 60 78)

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

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

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

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