A4xC5:C8 - GroupNames

G = A₄×C₅⋊C₈ order 480 = 2⁵·3·5

Direct product of A₄ and C₅⋊C₈

direct product, metabelian, soluble, monomial, A-group

Aliases: A₄×C₅⋊C₈, C₅⋊(C₈×A₄), (C₂×C₁₀)⋊C₂₄, (C₅×A₄)⋊₂C₈, C₂.₁(A₄×F₅), C₁₀.₃(C₄×A₄), (C₂×A₄).₂F₅, (C₁₀×A₄).₂C₄, (C₂²×C₁₀).C₁₂, C₂³.₂(C₃×F₅), Dic₅.₅(C₂×A₄), (A₄×Dic₅).₄C₂, (C₂²×Dic₅).₂C₆, C₂²⋊(C₃×C₅⋊C₈), (C₂²×C₅⋊C₈)⋊C₃, SmallGroup(480,966)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — A₄×C₅⋊C₈

Chief series

C₁ — C₅ — C₂×C₁₀ — C₂²×C₁₀ — C₂²×Dic₅ — A₄×Dic₅ — A₄×C₅⋊C₈

Lower central

C₂×C₁₀ — A₄×C₅⋊C₈

Upper central

C₁ — C₂

Generators and relations for A₄×C₅⋊C₈
G = < a,b,c,d,e | a²=b²=c³=d⁵=e⁸=1, cac^-1=ab=ba, ad=da, ae=ea, cbc^-1=a, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d³ >

C₁

C₂

₃C₂

₄C₃

C₅

C₂²

₃C₂²

₅C₄

₁₅C₄

₄C₆

C₁₀

₃C₁₀

₄C₁₅

C₂³

₅C₈

₁₅C₂×C₄

₁₅C₈

₁₅C₂×C₄

A₄

₂₀C₁₂

C₂×C₁₀

Dic₅

₃C₂×C₁₀

₃Dic₅

₄C₃₀

₅C₂²×C₄

₁₅C₂×C₈

C₂×A₄

₂₀C₂₄

C₂²×C₁₀

C₅⋊C₈

₃C₂×Dic₅

₃C₅⋊C₈

C₅×A₄

₄C₃×Dic₅

₅C₂²×C₈

₅C₄×A₄

C₂²×Dic₅

₃C₂×C₅⋊C₈

C₁₀×A₄

₄C₃×C₅⋊C₈

₅C₈×A₄

C₂²×C₅⋊C₈

A₄×Dic₅

A₄×C₅⋊C₈

Smallest permutation representation of A₄×C₅⋊C₈
►On 120 points

Generators in S₁₂₀

(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)

(9 13)(10 14)(11 15)(12 16)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(113 117)(114 118)(115 119)(116 120)

(1 100 60)(2 101 61)(3 102 62)(4 103 63)(5 104 64)(6 97 57)(7 98 58)(8 99 59)(9 86 51)(10 87 52)(11 88 53)(12 81 54)(13 82 55)(14 83 56)(15 84 49)(16 85 50)(17 94 119)(18 95 120)(19 96 113)(20 89 114)(21 90 115)(22 91 116)(23 92 117)(24 93 118)(25 45 66)(26 46 67)(27 47 68)(28 48 69)(29 41 70)(30 42 71)(31 43 72)(32 44 65)(33 108 74)(34 109 75)(35 110 76)(36 111 77)(37 112 78)(38 105 79)(39 106 80)(40 107 73)

(1 53 23 46 109)(2 47 54 110 24)(3 111 48 17 55)(4 18 112 56 41)(5 49 19 42 105)(6 43 50 106 20)(7 107 44 21 51)(8 22 108 52 45)(9 98 73 65 90)(10 66 99 91 74)(11 92 67 75 100)(12 76 93 101 68)(13 102 77 69 94)(14 70 103 95 78)(15 96 71 79 104)(16 80 89 97 72)(25 59 116 33 87)(26 34 60 88 117)(27 81 35 118 61)(28 119 82 62 36)(29 63 120 37 83)(30 38 64 84 113)(31 85 39 114 57)(32 115 86 58 40)

(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)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)

G:=sub<Sym(120)| (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112), (9,13)(10,14)(11,15)(12,16)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(113,117)(114,118)(115,119)(116,120), (1,100,60)(2,101,61)(3,102,62)(4,103,63)(5,104,64)(6,97,57)(7,98,58)(8,99,59)(9,86,51)(10,87,52)(11,88,53)(12,81,54)(13,82,55)(14,83,56)(15,84,49)(16,85,50)(17,94,119)(18,95,120)(19,96,113)(20,89,114)(21,90,115)(22,91,116)(23,92,117)(24,93,118)(25,45,66)(26,46,67)(27,47,68)(28,48,69)(29,41,70)(30,42,71)(31,43,72)(32,44,65)(33,108,74)(34,109,75)(35,110,76)(36,111,77)(37,112,78)(38,105,79)(39,106,80)(40,107,73), (1,53,23,46,109)(2,47,54,110,24)(3,111,48,17,55)(4,18,112,56,41)(5,49,19,42,105)(6,43,50,106,20)(7,107,44,21,51)(8,22,108,52,45)(9,98,73,65,90)(10,66,99,91,74)(11,92,67,75,100)(12,76,93,101,68)(13,102,77,69,94)(14,70,103,95,78)(15,96,71,79,104)(16,80,89,97,72)(25,59,116,33,87)(26,34,60,88,117)(27,81,35,118,61)(28,119,82,62,36)(29,63,120,37,83)(30,38,64,84,113)(31,85,39,114,57)(32,115,86,58,40), (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)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)>;

G:=Group( (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112), (9,13)(10,14)(11,15)(12,16)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(113,117)(114,118)(115,119)(116,120), (1,100,60)(2,101,61)(3,102,62)(4,103,63)(5,104,64)(6,97,57)(7,98,58)(8,99,59)(9,86,51)(10,87,52)(11,88,53)(12,81,54)(13,82,55)(14,83,56)(15,84,49)(16,85,50)(17,94,119)(18,95,120)(19,96,113)(20,89,114)(21,90,115)(22,91,116)(23,92,117)(24,93,118)(25,45,66)(26,46,67)(27,47,68)(28,48,69)(29,41,70)(30,42,71)(31,43,72)(32,44,65)(33,108,74)(34,109,75)(35,110,76)(36,111,77)(37,112,78)(38,105,79)(39,106,80)(40,107,73), (1,53,23,46,109)(2,47,54,110,24)(3,111,48,17,55)(4,18,112,56,41)(5,49,19,42,105)(6,43,50,106,20)(7,107,44,21,51)(8,22,108,52,45)(9,98,73,65,90)(10,66,99,91,74)(11,92,67,75,100)(12,76,93,101,68)(13,102,77,69,94)(14,70,103,95,78)(15,96,71,79,104)(16,80,89,97,72)(25,59,116,33,87)(26,34,60,88,117)(27,81,35,118,61)(28,119,82,62,36)(29,63,120,37,83)(30,38,64,84,113)(31,85,39,114,57)(32,115,86,58,40), (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)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120) );

G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112)], [(9,13),(10,14),(11,15),(12,16),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(113,117),(114,118),(115,119),(116,120)], [(1,100,60),(2,101,61),(3,102,62),(4,103,63),(5,104,64),(6,97,57),(7,98,58),(8,99,59),(9,86,51),(10,87,52),(11,88,53),(12,81,54),(13,82,55),(14,83,56),(15,84,49),(16,85,50),(17,94,119),(18,95,120),(19,96,113),(20,89,114),(21,90,115),(22,91,116),(23,92,117),(24,93,118),(25,45,66),(26,46,67),(27,47,68),(28,48,69),(29,41,70),(30,42,71),(31,43,72),(32,44,65),(33,108,74),(34,109,75),(35,110,76),(36,111,77),(37,112,78),(38,105,79),(39,106,80),(40,107,73)], [(1,53,23,46,109),(2,47,54,110,24),(3,111,48,17,55),(4,18,112,56,41),(5,49,19,42,105),(6,43,50,106,20),(7,107,44,21,51),(8,22,108,52,45),(9,98,73,65,90),(10,66,99,91,74),(11,92,67,75,100),(12,76,93,101,68),(13,102,77,69,94),(14,70,103,95,78),(15,96,71,79,104),(16,80,89,97,72),(25,59,116,33,87),(26,34,60,88,117),(27,81,35,118,61),(28,119,82,62,36),(29,63,120,37,83),(30,38,64,84,113),(31,85,39,114,57),(32,115,86,58,40)], [(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),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120)]])

40 conjugacy classes

class	1	2A	2B	2C	3A	3B	4A	4B	4C	4D	5	6A	6B	8A	8B	8C	8D	8E	8F	8G	8H	10A	10B	10C	12A	12B	12C	12D	15A	15B	24A	···	24H	30A	30B
order	1	2	2	2	3	3	4	4	4	4	5	6	6	8	8	8	8	8	8	8	8	10	10	10	12	12	12	12	15	15	24	···	24	30	30
size	1	1	3	3	4	4	5	5	15	15	4	4	4	5	5	5	5	15	15	15	15	4	12	12	20	20	20	20	16	16	20	···	20	16	16

40 irreducible representations

dim	1	1	1	1	1	1	1	1	12	12	3	3	3	3	4	4	4	4
type	+	+							+	-	+	+			+	-
image	C₁	C₂	C₃	C₄	C₆	C₈	C₁₂	C₂₄	A₄×F₅	A₄×C₅⋊C₈	A₄	C₂×A₄	C₄×A₄	C₈×A₄	F₅	C₅⋊C₈	C₃×F₅	C₃×C₅⋊C₈
kernel	A₄×C₅⋊C₈	A₄×Dic₅	C₂²×C₅⋊C₈	C₁₀×A₄	C₂²×Dic₅	C₅×A₄	C₂²×C₁₀	C₂×C₁₀	C₂	C₁	C₅⋊C₈	Dic₅	C₁₀	C₅	C₂×A₄	A₄	C₂³	C₂²
# reps	1	1	2	2	2	4	4	8	1	1	1	1	2	4	1	1	2	2

Matrix representation of A₄×C₅⋊C₈ ►in GL₇(𝔽₂₄₁)

240	0	0	0	0	0	0
0	1	0	0	0	0	0
0	226	240	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

1	0	0	0	0	0	0
0	240	0	0	0	0	0
16	0	240	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

0	1	0	0	0	0	0
16	226	239	0	0	0	0
0	0	15	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	0	0	0	240
0	0	0	1	0	0	240
0	0	0	0	1	0	240
0	0	0	0	0	1	240

233	0	0	0	0	0	0
0	233	0	0	0	0	0
0	0	233	0	0	0	0
0	0	0	51	180	32	157
0	0	0	83	96	94	208
0	0	0	145	147	33	240
0	0	0	84	179	190	61

G:=sub<GL(7,GF(241))| [240,0,0,0,0,0,0,0,1,226,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,16,0,0,0,0,0,240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,16,0,0,0,0,0,1,226,0,0,0,0,0,0,239,15,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,240,240,240,240],[233,0,0,0,0,0,0,0,233,0,0,0,0,0,0,0,233,0,0,0,0,0,0,0,51,83,145,84,0,0,0,180,96,147,179,0,0,0,32,94,33,190,0,0,0,157,208,240,61] >;

A₄×C₅⋊C₈ in GAP, Magma, Sage, TeX

A_4\times C_5\rtimes C_8

% in TeX

G:=Group("A4xC5:C8");

// GroupNames label

G:=SmallGroup(480,966);

// by ID

G=gap.SmallGroup(480,966);

# by ID

G:=PCGroup([7,-2,-3,-2,-2,-2,2,-5,42,58,1271,516,9414,3156]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^5=e^8=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;

// generators/relations

Export

Subgroup lattice of A₄×C₅⋊C₈ in TeX

233	0	0	0	0	0	0
0	233	0	0	0	0	0
0	0	233	0	0	0	0
0	0	0	51	180	32	157
0	0	0	83	96	94	208
0	0	0	145	147	33	240
0	0	0	84	179	190	61

233	0	0	0	0	0	0
0	233	0	0	0	0	0
0	0	233	0	0	0	0
0	0	0	51	180	32	157
0	0	0	83	96	94	208
0	0	0	145	147	33	240
0	0	0	84	179	190	61

G = A4×C5⋊C8 order 480 = 25·3·5

Direct product of A4 and C5⋊C8

G = A₄×C₅⋊C₈ order 480 = 2⁵·3·5

Direct product of A₄ and C₅⋊C₈

233	0	0	0	0	0	0
0	233	0	0	0	0	0
0	0	233	0	0	0	0
0	0	0	51	180	32	157
0	0	0	83	96	94	208
0	0	0	145	147	33	240
0	0	0	84	179	190	61