(C2xD4):8F5 - GroupNames

G = (C₂×D₄)⋊₈F₅ order 320 = 2⁶·5

6^th semidirect product of C₂×D₄ and F₅ acting via F₅/D₅=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂×D₄)⋊₈F₅, (D₄×C₁₀)⋊₉C₄, C₂³⋊F₅⋊₄C₂, D₁₀.₆(C₂×D₄), (C₄×D₅).₃₆D₄, C₂³.₃(C₂×F₅), (C₂×Dic₁₀)⋊₁₀C₄, (C₂²×Dic₅)⋊₈C₄, C₄.₁₃(C₂²⋊F₅), C₂₀.₁₃(C₂²⋊C₄), C₂²⋊F₅.₂C₂², C₂².₁₂(C₂²×F₅), C₅⋊₂(C₂³.C₂³), D₁₀.C₂³⋊₃C₂, Dic₅.₄₄(C₂²⋊C₄), (C₂²×D₅).₁₄₈C₂³, (C₂×C₄).₃₅(C₂×F₅), (C₂×C₂₀).₅₃(C₂×C₄), C₂.₁₉(C₂×C₂²⋊F₅), C₁₀.₁₈(C₂×C₂²⋊C₄), (C₂×C₄×D₅).₂₀₀C₂², (C₂×D₄⋊₂D₅).₁₅C₂, (C₂×C₁₀).₇₄(C₂²×C₄), (C₂²×C₁₀).₂₂(C₂×C₄), (C₂×C₅⋊D₄).₈₈C₂², (C₂×Dic₅).₁₉₂(C₂×C₄), SmallGroup(320,1109)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — (C₂×D₄)⋊₈F₅

Chief series

C₁ — C₅ — C₁₀ — D₁₀ — C₂²×D₅ — C₂²⋊F₅ — D₁₀.C₂³ — (C₂×D₄)⋊₈F₅

Lower central

C₅ — C₁₀ — C₂×C₁₀ — (C₂×D₄)⋊₈F₅

Upper central

C₁ — C₂ — C₂×C₄ — C₂×D₄

Generators and relations for (C₂×D₄)⋊₈F₅
G = < a,b,c,d,e | a²=b⁴=c²=d⁵=e⁴=1, ab=ba, ac=ca, ad=da, eae^-1=ab², cbc=b^-1, bd=db, be=eb, cd=dc, ece^-1=ab²c, ede^-1=d³ >

Subgroups: 682 in 158 conjugacy classes, 48 normal (18 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, D₅, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, Dic₅, Dic₅, C₂₀, F₅, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂³⋊C₄, C₄²⋊C₂, C₂×C₄○D₄, Dic₁₀, C₄×D₅, C₂×Dic₅, C₂×Dic₅, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₅×D₄, C₂×F₅, C₂²×D₅, C₂²×C₁₀, C₂³.C₂³, C₄×F₅, C₄⋊F₅, C₂²⋊F₅, C₂×Dic₁₀, C₂×C₄×D₅, D₄⋊₂D₅, C₂²×Dic₅, C₂×C₅⋊D₄, D₄×C₁₀, C₂³⋊F₅, D₁₀.C₂³, C₂×D₄⋊₂D₅, (C₂×D₄)⋊₈F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, F₅, C₂×C₂²⋊C₄, C₂×F₅, C₂³.C₂³, C₂²⋊F₅, C₂²×F₅, C₂×C₂²⋊F₅, (C₂×D₄)⋊₈F₅

Smallest permutation representation of (C₂×D₄)⋊₈F₅
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim	1	1	1	1	1	1	1	2	4	4	4	4	4	8
type	+	+	+	+				+	+	+	+		+	-
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	D₄	F₅	C₂×F₅	C₂×F₅	C₂³.C₂³	C₂²⋊F₅	(C₂×D₄)⋊₈F₅
kernel	(C₂×D₄)⋊₈F₅	C₂³⋊F₅	D₁₀.C₂³	C₂×D₄⋊₂D₅	C₂×Dic₁₀	C₂²×Dic₅	D₄×C₁₀	C₄×D₅	C₂×D₄	C₂×C₄	C₂³	C₅	C₄	C₁
# reps	1	4	2	1	2	4	2	4	1	1	2	2	4	2

Matrix representation of (C₂×D₄)⋊₈F₅ ►in GL₈(𝔽₄₁)

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	0	40	0	0
0	0	0	0	40	0	0	0
0	0	0	0	0	0	0	40
0	0	0	0	0	0	40	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	0	32	0	0
0	0	0	0	32	0	0	0
0	0	0	0	0	0	0	9
0	0	0	0	0	0	9	0

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

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

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

(C₂×D₄)⋊₈F₅ in GAP, Magma, Sage, TeX

(C_2\times D_4)\rtimes_8F_5

% in TeX

G:=Group("(C2xD4):8F5");

// GroupNames label

G:=SmallGroup(320,1109);

// by ID

G=gap.SmallGroup(320,1109);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,297,1684,6278,1595]);

// Polycyclic

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

// generators/relations

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	0	40	0	0
0	0	0	0	40	0	0	0
0	0	0	0	0	0	0	40
0	0	0	0	0	0	40	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	0	32	0	0
0	0	0	0	32	0	0	0
0	0	0	0	0	0	0	9
0	0	0	0	0	0	9	0

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

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

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

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

G = (C2×D4)⋊8F5 order 320 = 26·5

6th semidirect product of C2×D4 and F5 acting via F5/D5=C2

G = (C₂×D₄)⋊₈F₅ order 320 = 2⁶·5

6^th semidirect product of C₂×D₄ and F₅ acting via F₅/D₅=C₂

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	0	40	0	0
0	0	0	0	40	0	0	0
0	0	0	0	0	0	0	40
0	0	0	0	0	0	40	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	0	32	0	0
0	0	0	0	32	0	0	0
0	0	0	0	0	0	0	9
0	0	0	0	0	0	9	0

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

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