C4^2:9F5 - GroupNames

G = C₄²⋊₉F₅ order 320 = 2⁶·5

6^th semidirect product of C₄² and F₅ acting via F₅/D₅=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄²⋊₉F₅, (C₄×C₂₀)⋊₆C₄, C₅⋊(C₄²⋊₈C₄), (C₄×D₅).₈₁D₄, C₄.₁₁(C₄⋊F₅), C₂₀.₁₈(C₄⋊C₄), D₁₀.₉(C₂×Q₈), (C₄×D₅).₂₃Q₈, (C₄×Dic₅)⋊₂₁C₄, D₁₀.₂₆(C₂×D₄), (D₅×C₄²).₂₂C₂, D₁₀.₂₃(C₄○D₄), Dic₅.₂₇(C₄⋊C₄), D₅.₁(C₄.₄D₄), D₁₀.₃Q₈.₄C₂, D₅.₁(C₄².C₂), C₁₀.₇(C₄²⋊C₂), (C₂²×F₅).₂C₂², C₂².₆₇(C₂²×F₅), (C₂²×D₅).₂₆₆C₂³, C₂.₁₀(D₁₀.C₂³), C₂.₉(C₂×C₄⋊F₅), C₁₀.₆(C₂×C₄⋊C₄), (C₂×C₄⋊F₅).₁₀C₂, (C₂×C₄).₁₀₁(C₂×F₅), (C₂×C₂₀).₁₂₅(C₂×C₄), (C₂×C₄×D₅).₃₆₁C₂², (C₂×C₁₀).₂₇(C₂²×C₄), (C₂×Dic₅).₁₇₅(C₂×C₄), SmallGroup(320,1027)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — C₄²⋊₉F₅

Chief series

C₁ — C₅ — D₅ — D₁₀ — C₂²×D₅ — C₂²×F₅ — D₁₀.₃Q₈ — C₄²⋊₉F₅

Lower central

C₅ — C₂×C₁₀ — C₄²⋊₉F₅

Upper central

C₁ — C₂² — C₄²

Generators and relations for C₄²⋊₉F₅
G = < a,b,c,d | a⁴=b⁴=c⁵=d⁴=1, ab=ba, ac=ca, dad^-1=ab², bc=cb, dbd^-1=a²b, dcd^-1=c³ >

Subgroups: 618 in 154 conjugacy classes, 60 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₂×C₄, C₂×C₄, C₂×C₄, C₂³, D₅, C₁₀, C₁₀, C₄², C₄², C₄⋊C₄, C₂²×C₄, Dic₅, Dic₅, C₂₀, C₂₀, F₅, D₁₀, D₁₀, C₂×C₁₀, C₂.C₄², C₂×C₄², C₂×C₄⋊C₄, C₄×D₅, C₄×D₅, C₂×Dic₅, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₂×F₅, C₂²×D₅, C₄²⋊₈C₄, C₄×Dic₅, C₄×Dic₅, C₄×C₂₀, C₄⋊F₅, C₂×C₄×D₅, C₂×C₄×D₅, C₂²×F₅, D₁₀.₃Q₈, D₅×C₄², C₂×C₄⋊F₅, C₄²⋊₉F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₂³, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, F₅, C₂×C₄⋊C₄, C₄²⋊C₂, C₄.₄D₄, C₄².C₂, C₂×F₅, C₄²⋊₈C₄, C₄⋊F₅, C₂²×F₅, C₂×C₄⋊F₅, D₁₀.C₂³, C₄²⋊₉F₅

Smallest permutation representation of C₄²⋊₉F₅
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim	1	1	1	1	1	1	2	2	2	4	4	4	4
type	+	+	+	+			+	-		+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	D₄	Q₈	C₄○D₄	F₅	C₂×F₅	C₄⋊F₅	D₁₀.C₂³
kernel	C₄²⋊₉F₅	D₁₀.₃Q₈	D₅×C₄²	C₂×C₄⋊F₅	C₄×Dic₅	C₄×C₂₀	C₄×D₅	C₄×D₅	D₁₀	C₄²	C₂×C₄	C₄	C₂
# reps	1	4	1	2	6	2	2	2	8	1	3	4	8

Matrix representation of C₄²⋊₉F₅ ►in GL₆(𝔽₄₁)

11	2	0	0	0	0
21	30	0	0	0	0
0	0	9	0	0	0
0	0	0	9	0	0
0	0	0	0	9	0
0	0	0	0	0	9

11	2	0	0	0	0
21	30	0	0	0	0
0	0	22	38	0	3
0	0	0	19	38	3
0	0	3	38	19	0
0	0	3	0	38	22

1	0	0	0	0	0
0	1	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

17	18	0	0	0	0
34	24	0	0	0	0
0	0	0	0	32	0
0	0	32	0	0	0
0	0	0	0	0	32
0	0	0	32	0	0

G:=sub<GL(6,GF(41))| [11,21,0,0,0,0,2,30,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[11,21,0,0,0,0,2,30,0,0,0,0,0,0,22,0,3,3,0,0,38,19,38,0,0,0,0,38,19,38,0,0,3,3,0,22],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[17,34,0,0,0,0,18,24,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,32,0,0,32,0,0,0,0,0,0,0,32,0] >;

C₄²⋊₉F₅ in GAP, Magma, Sage, TeX

C_4^2\rtimes_9F_5

% in TeX

G:=Group("C4^2:9F5");

// GroupNames label

G:=SmallGroup(320,1027);

// by ID

G=gap.SmallGroup(320,1027);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,120,422,184,6278,1595]);

// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^5=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;

// generators/relations

G = C42⋊9F5 order 320 = 26·5

6th semidirect product of C42 and F5 acting via F5/D5=C2

G = C₄²⋊₉F₅ order 320 = 2⁶·5

6^th semidirect product of C₄² and F₅ acting via F₅/D₅=C₂