(C2xQ8):4F5 - GroupNames

G = (C₂×Q₈)⋊₄F₅ order 320 = 2⁶·5

2^nd semidirect product of C₂×Q₈ and F₅ acting via F₅/D₅=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂×Q₈)⋊₄F₅, (Q₈×D₅)⋊₇C₄, (Q₈×C₁₀)⋊₄C₄, Q₈.₇(C₂×F₅), Q₈⋊F₅⋊₃C₂, (C₄×D₅).₄₁D₄, C₄⋊F₅.₈C₂², D₁₀.₉₈(C₂×D₄), D₅⋊C₈.₇C₂², Dic₅.₈(C₂×D₄), C₅⋊(C₂³.₃₈D₄), C₄.₁₈(C₂²×F₅), (C₂×Dic₁₀)⋊₁₂C₄, C₂₀.₁₈(C₂²×C₄), Dic₁₀.₈(C₂×C₄), D₅⋊M₄(2).₅C₂, (C₄×D₅).₄₀C₂³, C₄.₁₇(C₂²⋊F₅), (Q₈×D₅).₁₁C₂², C₂₀.₁₇(C₂²⋊C₄), (C₂×Dic₅).₁₂₂D₄, (C₂²×D₅).₁₄₈D₄, D₅.₄(C₈.C₂²), D₁₀.₄₆(C₂²⋊C₄), C₂².₂₈(C₂²⋊F₅), Dic₅.₁₃(C₂²⋊C₄), D₁₀.C₂³.₅C₂, (C₂×Q₈×D₅).₁₀C₂, (C₂×C₄).₃₉(C₂×F₅), (C₅×Q₈).₇(C₂×C₄), (C₂×C₂₀).₆₀(C₂×C₄), (C₄×D₅).₂₄(C₂×C₄), C₂.₂₇(C₂×C₂²⋊F₅), C₁₀.₂₆(C₂×C₂²⋊C₄), (C₂×C₄×D₅).₂₀₅C₂², (C₂×C₁₀).₅₈(C₂²⋊C₄), SmallGroup(320,1120)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — (C₂×Q₈)⋊₄F₅

Chief series

C₁ — C₅ — C₁₀ — D₁₀ — C₄×D₅ — C₄⋊F₅ — D₁₀.C₂³ — (C₂×Q₈)⋊₄F₅

Lower central

C₅ — C₁₀ — C₂₀ — (C₂×Q₈)⋊₄F₅

Upper central

C₁ — C₂ — C₂×C₄ — C₂×Q₈

Generators and relations for (C₂×Q₈)⋊₄F₅
G = < a,b,c,d | a¹⁰=b⁴=d⁴=1, c²=b², ab=ba, ac=ca, dad^-1=a³b², cbc^-1=b^-1, bd=db, dcd^-1=a⁵b^-1c >

Subgroups: 602 in 150 conjugacy classes, 50 normal (30 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, Q₈, Q₈, C₂³, D₅, D₅, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂×Q₈, C₂×Q₈, Dic₅, Dic₅, C₂₀, C₂₀, F₅, D₁₀, D₁₀, C₂×C₁₀, Q₈⋊C₄, C₄²⋊C₂, C₂×M₄(2), C₂²×Q₈, C₅⋊C₈, Dic₁₀, Dic₁₀, C₄×D₅, C₄×D₅, C₂×Dic₅, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₅×Q₈, C₅×Q₈, C₂×F₅, C₂²×D₅, C₂³.₃₈D₄, D₅⋊C₈, C₄.F₅, C₄×F₅, C₄⋊F₅, C₂².F₅, C₂²⋊F₅, C₂×Dic₁₀, C₂×Dic₁₀, C₂×C₄×D₅, C₂×C₄×D₅, Q₈×D₅, Q₈×D₅, Q₈×C₁₀, Q₈⋊F₅, D₅⋊M₄(2), D₁₀.C₂³, C₂×Q₈×D₅, (C₂×Q₈)⋊₄F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, F₅, C₂×C₂²⋊C₄, C₈.C₂², C₂×F₅, C₂³.₃₈D₄, C₂²⋊F₅, C₂²×F₅, C₂×C₂²⋊F₅, (C₂×Q₈)⋊₄F₅

Smallest permutation representation of (C₂×Q₈)⋊₄F₅
►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)

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	4	4	4	4	4	4	8
type	+	+	+	+	+				+	+	+	+	-	+	+	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	D₄	D₄	F₅	C₈.C₂²	C₂×F₅	C₂×F₅	C₂²⋊F₅	C₂²⋊F₅	(C₂×Q₈)⋊₄F₅
kernel	(C₂×Q₈)⋊₄F₅	Q₈⋊F₅	D₅⋊M₄(2)	D₁₀.C₂³	C₂×Q₈×D₅	C₂×Dic₁₀	Q₈×D₅	Q₈×C₁₀	C₄×D₅	C₂×Dic₅	C₂²×D₅	C₂×Q₈	D₅	C₂×C₄	Q₈	C₄	C₂²	C₁
# reps	1	4	1	1	1	2	4	2	2	1	1	1	2	1	2	2	2	2

Matrix representation of (C₂×Q₈)⋊₄F₅ ►in GL₈(𝔽₄₁)

0	40	1	0	0	0	0	0
0	40	0	1	0	0	0	0
0	40	0	0	0	0	0	0
1	40	0	0	0	0	0	0
0	0	0	0	0	34	4	37
0	0	0	0	7	0	37	37
0	0	0	0	37	4	0	34
0	0	0	0	4	4	7	0

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	34	0	4	4
0	0	0	0	0	34	4	37
0	0	0	0	4	4	7	0
0	0	0	0	4	37	0	7

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	37	4	0	34
0	0	0	0	4	4	7	0
0	0	0	0	0	7	37	4
0	0	0	0	34	0	4	4

0	0	1	0	0	0	0	0
1	0	0	0	0	0	0	0
0	0	0	1	0	0	0	0
0	1	0	0	0	0	0	0
0	0	0	0	9	0	0	0
0	0	0	0	0	32	0	0
0	0	0	0	0	0	0	9
0	0	0	0	0	0	9	0

G:=sub<GL(8,GF(41))| [0,0,0,1,0,0,0,0,40,40,40,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,7,37,4,0,0,0,0,34,0,4,4,0,0,0,0,4,37,0,7,0,0,0,0,37,37,34,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,34,0,4,4,0,0,0,0,0,34,4,37,0,0,0,0,4,4,7,0,0,0,0,0,4,37,0,7],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,37,4,0,34,0,0,0,0,4,4,7,0,0,0,0,0,0,7,37,4,0,0,0,0,34,0,4,4],[0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0] >;

(C₂×Q₈)⋊₄F₅ in GAP, Magma, Sage, TeX

(C_2\times Q_8)\rtimes_4F_5

% in TeX

G:=Group("(C2xQ8):4F5");

// GroupNames label

G:=SmallGroup(320,1120);

// by ID

G=gap.SmallGroup(320,1120);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,184,1684,438,102,6278,1595]);

// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=d^4=1,c^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^3*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^5*b^-1*c>;

// generators/relations

0	40	1	0	0	0	0	0
0	40	0	1	0	0	0	0
0	40	0	0	0	0	0	0
1	40	0	0	0	0	0	0
0	0	0	0	0	34	4	37
0	0	0	0	7	0	37	37
0	0	0	0	37	4	0	34
0	0	0	0	4	4	7	0

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	34	0	4	4
0	0	0	0	0	34	4	37
0	0	0	0	4	4	7	0
0	0	0	0	4	37	0	7

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	37	4	0	34
0	0	0	0	4	4	7	0
0	0	0	0	0	7	37	4
0	0	0	0	34	0	4	4

0	0	1	0	0	0	0	0
1	0	0	0	0	0	0	0
0	0	0	1	0	0	0	0
0	1	0	0	0	0	0	0
0	0	0	0	9	0	0	0
0	0	0	0	0	32	0	0
0	0	0	0	0	0	0	9
0	0	0	0	0	0	9	0

0	40	1	0	0	0	0	0
0	40	0	1	0	0	0	0
0	40	0	0	0	0	0	0
1	40	0	0	0	0	0	0
0	0	0	0	0	34	4	37
0	0	0	0	7	0	37	37
0	0	0	0	37	4	0	34
0	0	0	0	4	4	7	0

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	34	0	4	4
0	0	0	0	0	34	4	37
0	0	0	0	4	4	7	0
0	0	0	0	4	37	0	7

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	37	4	0	34
0	0	0	0	4	4	7	0
0	0	0	0	0	7	37	4
0	0	0	0	34	0	4	4

0	0	1	0	0	0	0	0
1	0	0	0	0	0	0	0
0	0	0	1	0	0	0	0
0	1	0	0	0	0	0	0
0	0	0	0	9	0	0	0
0	0	0	0	0	32	0	0
0	0	0	0	0	0	0	9
0	0	0	0	0	0	9	0

G = (C2×Q8)⋊4F5 order 320 = 26·5

2nd semidirect product of C2×Q8 and F5 acting via F5/D5=C2

G = (C₂×Q₈)⋊₄F₅ order 320 = 2⁶·5

2^nd semidirect product of C₂×Q₈ and F₅ acting via F₅/D₅=C₂

0	40	1	0	0	0	0	0
0	40	0	1	0	0	0	0
0	40	0	0	0	0	0	0
1	40	0	0	0	0	0	0
0	0	0	0	0	34	4	37
0	0	0	0	7	0	37	37
0	0	0	0	37	4	0	34
0	0	0	0	4	4	7	0

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	34	0	4	4
0	0	0	0	0	34	4	37
0	0	0	0	4	4	7	0
0	0	0	0	4	37	0	7

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	37	4	0	34
0	0	0	0	4	4	7	0
0	0	0	0	0	7	37	4
0	0	0	0	34	0	4	4

0	0	1	0	0	0	0	0
1	0	0	0	0	0	0	0
0	0	0	1	0	0	0	0
0	1	0	0	0	0	0	0
0	0	0	0	9	0	0	0
0	0	0	0	0	32	0	0
0	0	0	0	0	0	0	9
0	0	0	0	0	0	9	0