A4xC2^2xC10 - GroupNames

G = A₄×C₂²×C₁₀ order 480 = 2⁵·3·5

Direct product of C₂²×C₁₀ and A₄

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

Aliases: A₄×C₂²×C₁₀, C₂⁵⋊₃C₁₅, C₂⁴⋊₅C₃₀, (C₂⁴×C₁₀)⋊₁C₃, (C₂³×C₁₀)⋊₇C₆, C₂³⋊₃(C₂×C₃₀), C₂²⋊(C₂²×C₃₀), (C₂²×C₁₀)⋊₇(C₂×C₆), (C₂×C₁₀)⋊₃(C₂²×C₆), SmallGroup(480,1208)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — A₄×C₂²×C₁₀

Chief series

C₁ — C₂² — C₂×C₁₀ — C₅×A₄ — C₁₀×A₄ — A₄×C₂×C₁₀ — A₄×C₂²×C₁₀

Lower central

C₂² — A₄×C₂²×C₁₀

Upper central

C₁ — C₂²×C₁₀

Generators and relations for A₄×C₂²×C₁₀
G = < a,b,c,d,e,f | a²=b²=c¹⁰=d²=e²=f³=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf^-1=de=ed, fef^-1=d >

Subgroups: 908 in 356 conjugacy classes, 96 normal (12 characteristic)
C₁, C₂, C₂, C₃, C₂², C₂², C₂², C₅, C₆, C₂³, C₂³, C₂³, C₁₀, C₁₀, A₄, C₂×C₆, C₁₅, C₂⁴, C₂⁴, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂×A₄, C₂²×C₆, C₃₀, C₂⁵, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₂²×A₄, C₅×A₄, C₂×C₃₀, C₂³×C₁₀, C₂³×C₁₀, C₂³×A₄, C₁₀×A₄, C₂²×C₃₀, C₂⁴×C₁₀, A₄×C₂×C₁₀, A₄×C₂²×C₁₀
Quotients: C₁, C₂, C₃, C₂², C₅, C₆, C₂³, C₁₀, A₄, C₂×C₆, C₁₅, C₂×C₁₀, C₂×A₄, C₂²×C₆, C₃₀, C₂²×C₁₀, C₂²×A₄, C₅×A₄, C₂×C₃₀, C₂³×A₄, C₁₀×A₄, C₂²×C₃₀, A₄×C₂×C₁₀, A₄×C₂²×C₁₀

Smallest permutation representation of A₄×C₂²×C₁₀
►On 120 points

Generators in S₁₂₀

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

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

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

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

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

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

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

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

160 conjugacy classes

class	1	2A	···	2G	2H	···	2O	3A	3B	5A	5B	5C	5D	6A	···	6N	10A	···	10AB	10AC	···	10BH	15A	···	15H	30A	···	30BD
order	1	2	···	2	2	···	2	3	3	5	5	5	5	6	···	6	10	···	10	10	···	10	15	···	15	30	···	30
size	1	1	···	1	3	···	3	4	4	1	1	1	1	4	···	4	1	···	1	3	···	3	4	···	4	4	···	4

160 irreducible representations

dim	1	1	1	1	1	1	1	1	3	3	3	3
type	+	+							+	+
image	C₁	C₂	C₃	C₅	C₆	C₁₀	C₁₅	C₃₀	A₄	C₂×A₄	C₅×A₄	C₁₀×A₄
kernel	A₄×C₂²×C₁₀	A₄×C₂×C₁₀	C₂⁴×C₁₀	C₂³×A₄	C₂³×C₁₀	C₂²×A₄	C₂⁵	C₂⁴	C₂²×C₁₀	C₂×C₁₀	C₂³	C₂²
# reps	1	7	2	4	14	28	8	56	1	7	4	28

Matrix representation of A₄×C₂²×C₁₀ ►in GL₅(𝔽₃₁)

30	0	0	0	0
0	1	0	0	0
0	0	30	0	0
0	0	0	30	0
0	0	0	0	30

1	0	0	0	0
0	1	0	0	0
0	0	30	0	0
0	0	0	30	0
0	0	0	0	30

1	0	0	0	0
0	30	0	0	0
0	0	27	0	0
0	0	0	27	0
0	0	0	0	27

1	0	0	0	0
0	1	0	0	0
0	0	30	0	0
0	0	0	30	0
0	0	28	0	1

1	0	0	0	0
0	1	0	0	0
0	0	30	0	0
0	0	6	1	0
0	0	0	0	30

25	0	0	0	0
0	1	0	0	0
0	0	25	29	0
0	0	1	6	1
0	0	6	28	0

G:=sub<GL(5,GF(31))| [30,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,0,30,0,0,0,0,0,30],[1,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,0,30,0,0,0,0,0,30],[1,0,0,0,0,0,30,0,0,0,0,0,27,0,0,0,0,0,27,0,0,0,0,0,27],[1,0,0,0,0,0,1,0,0,0,0,0,30,0,28,0,0,0,30,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,30,6,0,0,0,0,1,0,0,0,0,0,30],[25,0,0,0,0,0,1,0,0,0,0,0,25,1,6,0,0,29,6,28,0,0,0,1,0] >;

A₄×C₂²×C₁₀ in GAP, Magma, Sage, TeX

A_4\times C_2^2\times C_{10}

% in TeX

G:=Group("A4xC2^2xC10");

// GroupNames label

G:=SmallGroup(480,1208);

// by ID

G=gap.SmallGroup(480,1208);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,1286,2232]);

// Polycyclic

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

// generators/relations

G = A4×C22×C10 order 480 = 25·3·5

Direct product of C22×C10 and A4

G = A₄×C₂²×C₁₀ order 480 = 2⁵·3·5

Direct product of C₂²×C₁₀ and A₄