direct product, non-abelian, soluble, monomial
Aliases: C10×A4⋊C4, (C2×A4)⋊C20, (C10×A4)⋊5C4, C24.(C5×S3), A4⋊2(C2×C20), C2.2(C10×S4), C23⋊(C5×Dic3), C10.36(C2×S4), (C2×C10).12S4, (C22×A4).C10, C22.6(C5×S4), C22⋊(C10×Dic3), C23.4(S3×C10), (C23×C10).1S3, (C22×C10)⋊2Dic3, (C22×C10).11D6, (C10×A4).21C22, (A4×C2×C10).3C2, (C5×A4)⋊12(C2×C4), (C2×A4).4(C2×C10), (C2×C10)⋊4(C2×Dic3), SmallGroup(480,1022)
Series: Derived ►Chief ►Lower central ►Upper central
| A4 — C10×A4⋊C4 |
Subgroups: 376 in 126 conjugacy classes, 36 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×4], C22 [×2], C22 [×10], C5, C6 [×3], C2×C4 [×8], C23, C23 [×2], C23 [×4], C10, C10 [×2], C10 [×4], Dic3 [×2], A4, C2×C6, C15, C22⋊C4 [×4], C22×C4 [×2], C24, C20 [×4], C2×C10 [×2], C2×C10 [×10], C2×Dic3, C2×A4, C2×A4 [×2], C30 [×3], C2×C22⋊C4, C2×C20 [×8], C22×C10, C22×C10 [×2], C22×C10 [×4], A4⋊C4 [×2], C22×A4, C5×Dic3 [×2], C5×A4, C2×C30, C5×C22⋊C4 [×4], C22×C20 [×2], C23×C10, C2×A4⋊C4, C10×Dic3, C10×A4, C10×A4 [×2], C10×C22⋊C4, C5×A4⋊C4 [×2], A4×C2×C10, C10×A4⋊C4
Quotients:
C1, C2 [×3], C4 [×2], C22, C5, S3, C2×C4, C10 [×3], Dic3 [×2], D6, C20 [×2], C2×C10, C2×Dic3, S4, C5×S3, C2×C20, A4⋊C4 [×2], C2×S4, C5×Dic3 [×2], S3×C10, C2×A4⋊C4, C10×Dic3, C5×S4, C5×A4⋊C4 [×2], C10×S4, C10×A4⋊C4
Generators and relations
G = < a,b,c,d,e | a10=b2=c2=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=d-1 >
►On 120 points
Generators in S120
(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)
(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 59)(22 60)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(71 98)(72 99)(73 100)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 97)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)
(1 67)(2 68)(3 69)(4 70)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 111)(50 112)(71 98)(72 99)(73 100)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 97)
(1 85 92)(2 86 93)(3 87 94)(4 88 95)(5 89 96)(6 90 97)(7 81 98)(8 82 99)(9 83 100)(10 84 91)(11 111 53)(12 112 54)(13 113 55)(14 114 56)(15 115 57)(16 116 58)(17 117 59)(18 118 60)(19 119 51)(20 120 52)(21 37 45)(22 38 46)(23 39 47)(24 40 48)(25 31 49)(26 32 50)(27 33 41)(28 34 42)(29 35 43)(30 36 44)(61 109 79)(62 110 80)(63 101 71)(64 102 72)(65 103 73)(66 104 74)(67 105 75)(68 106 76)(69 107 77)(70 108 78)
(1 38 67 18)(2 39 68 19)(3 40 69 20)(4 31 70 11)(5 32 61 12)(6 33 62 13)(7 34 63 14)(8 35 64 15)(9 36 65 16)(10 37 66 17)(21 104 59 84)(22 105 60 85)(23 106 51 86)(24 107 52 87)(25 108 53 88)(26 109 54 89)(27 110 55 90)(28 101 56 81)(29 102 57 82)(30 103 58 83)(41 80 113 97)(42 71 114 98)(43 72 115 99)(44 73 116 100)(45 74 117 91)(46 75 118 92)(47 76 119 93)(48 77 120 94)(49 78 111 95)(50 79 112 96)
G:=sub<Sym(120)| (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), (11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,59)(22,60)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(71,98)(72,99)(73,100)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,97)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110), (1,67)(2,68)(3,69)(4,70)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,111)(50,112)(71,98)(72,99)(73,100)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,97), (1,85,92)(2,86,93)(3,87,94)(4,88,95)(5,89,96)(6,90,97)(7,81,98)(8,82,99)(9,83,100)(10,84,91)(11,111,53)(12,112,54)(13,113,55)(14,114,56)(15,115,57)(16,116,58)(17,117,59)(18,118,60)(19,119,51)(20,120,52)(21,37,45)(22,38,46)(23,39,47)(24,40,48)(25,31,49)(26,32,50)(27,33,41)(28,34,42)(29,35,43)(30,36,44)(61,109,79)(62,110,80)(63,101,71)(64,102,72)(65,103,73)(66,104,74)(67,105,75)(68,106,76)(69,107,77)(70,108,78), (1,38,67,18)(2,39,68,19)(3,40,69,20)(4,31,70,11)(5,32,61,12)(6,33,62,13)(7,34,63,14)(8,35,64,15)(9,36,65,16)(10,37,66,17)(21,104,59,84)(22,105,60,85)(23,106,51,86)(24,107,52,87)(25,108,53,88)(26,109,54,89)(27,110,55,90)(28,101,56,81)(29,102,57,82)(30,103,58,83)(41,80,113,97)(42,71,114,98)(43,72,115,99)(44,73,116,100)(45,74,117,91)(46,75,118,92)(47,76,119,93)(48,77,120,94)(49,78,111,95)(50,79,112,96)>;
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)(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), (11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,59)(22,60)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(71,98)(72,99)(73,100)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,97)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110), (1,67)(2,68)(3,69)(4,70)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,111)(50,112)(71,98)(72,99)(73,100)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,97), (1,85,92)(2,86,93)(3,87,94)(4,88,95)(5,89,96)(6,90,97)(7,81,98)(8,82,99)(9,83,100)(10,84,91)(11,111,53)(12,112,54)(13,113,55)(14,114,56)(15,115,57)(16,116,58)(17,117,59)(18,118,60)(19,119,51)(20,120,52)(21,37,45)(22,38,46)(23,39,47)(24,40,48)(25,31,49)(26,32,50)(27,33,41)(28,34,42)(29,35,43)(30,36,44)(61,109,79)(62,110,80)(63,101,71)(64,102,72)(65,103,73)(66,104,74)(67,105,75)(68,106,76)(69,107,77)(70,108,78), (1,38,67,18)(2,39,68,19)(3,40,69,20)(4,31,70,11)(5,32,61,12)(6,33,62,13)(7,34,63,14)(8,35,64,15)(9,36,65,16)(10,37,66,17)(21,104,59,84)(22,105,60,85)(23,106,51,86)(24,107,52,87)(25,108,53,88)(26,109,54,89)(27,110,55,90)(28,101,56,81)(29,102,57,82)(30,103,58,83)(41,80,113,97)(42,71,114,98)(43,72,115,99)(44,73,116,100)(45,74,117,91)(46,75,118,92)(47,76,119,93)(48,77,120,94)(49,78,111,95)(50,79,112,96) );
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),(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)], [(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,59),(22,60),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(71,98),(72,99),(73,100),(74,91),(75,92),(76,93),(77,94),(78,95),(79,96),(80,97),(81,101),(82,102),(83,103),(84,104),(85,105),(86,106),(87,107),(88,108),(89,109),(90,110)], [(1,67),(2,68),(3,69),(4,70),(5,61),(6,62),(7,63),(8,64),(9,65),(10,66),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(41,113),(42,114),(43,115),(44,116),(45,117),(46,118),(47,119),(48,120),(49,111),(50,112),(71,98),(72,99),(73,100),(74,91),(75,92),(76,93),(77,94),(78,95),(79,96),(80,97)], [(1,85,92),(2,86,93),(3,87,94),(4,88,95),(5,89,96),(6,90,97),(7,81,98),(8,82,99),(9,83,100),(10,84,91),(11,111,53),(12,112,54),(13,113,55),(14,114,56),(15,115,57),(16,116,58),(17,117,59),(18,118,60),(19,119,51),(20,120,52),(21,37,45),(22,38,46),(23,39,47),(24,40,48),(25,31,49),(26,32,50),(27,33,41),(28,34,42),(29,35,43),(30,36,44),(61,109,79),(62,110,80),(63,101,71),(64,102,72),(65,103,73),(66,104,74),(67,105,75),(68,106,76),(69,107,77),(70,108,78)], [(1,38,67,18),(2,39,68,19),(3,40,69,20),(4,31,70,11),(5,32,61,12),(6,33,62,13),(7,34,63,14),(8,35,64,15),(9,36,65,16),(10,37,66,17),(21,104,59,84),(22,105,60,85),(23,106,51,86),(24,107,52,87),(25,108,53,88),(26,109,54,89),(27,110,55,90),(28,101,56,81),(29,102,57,82),(30,103,58,83),(41,80,113,97),(42,71,114,98),(43,72,115,99),(44,73,116,100),(45,74,117,91),(46,75,118,92),(47,76,119,93),(48,77,120,94),(49,78,111,95),(50,79,112,96)])
Matrix representation ►G ⊆ GL5(𝔽61)
| 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 34 | 0 | 0 |
| 0 | 0 | 0 | 34 | 0 |
| 0 | 0 | 0 | 0 | 34 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 0 | 60 | 0 | 0 | 0 |
| 1 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 51 | 35 | 0 | 0 | 0 |
| 25 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 |
| 0 | 0 | 0 | 50 | 0 |
| 0 | 0 | 50 | 0 | 0 |
G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,34,0,0,0,0,0,34,0,0,0,0,0,34],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,60],[0,1,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[51,25,0,0,0,35,10,0,0,0,0,0,0,0,50,0,0,0,50,0,0,0,50,0,0] >;
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4H | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 10A | ··· | 10L | 10M | ··· | 10AB | 15A | 15B | 15C | 15D | 20A | ··· | 20AF | 30A | ··· | 30L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 8 | 6 | ··· | 6 | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 1 | ··· | 1 | 3 | ··· | 3 | 8 | 8 | 8 | 8 | 6 | ··· | 6 | 8 | ··· | 8 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | - | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | S3 | Dic3 | D6 | C5×S3 | C5×Dic3 | S3×C10 | S4 | A4⋊C4 | C2×S4 | C5×S4 | C5×A4⋊C4 | C10×S4 |
| kernel | C10×A4⋊C4 | C5×A4⋊C4 | A4×C2×C10 | C10×A4 | C2×A4⋊C4 | A4⋊C4 | C22×A4 | C2×A4 | C23×C10 | C22×C10 | C22×C10 | C24 | C23 | C23 | C2×C10 | C10 | C10 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 16 | 1 | 2 | 1 | 4 | 8 | 4 | 2 | 4 | 2 | 8 | 16 | 8 |
In GAP, Magma, Sage, TeX
C_{10}\times A_4\rtimes C_4 % in TeX
G:=Group("C10xA4:C4"); // GroupNames label
G:=SmallGroup(480,1022);
// by ID
G=gap.SmallGroup(480,1022);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-3,-2,2,140,2804,10085,285,5886,475]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^2=c^2=d^3=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀