direct product, non-abelian, soluble
Aliases: C5×Q8⋊A4, (C5×Q8)⋊1A4, Q8⋊1(C5×A4), C23.5(C5×A4), (C22×Q8)⋊3C15, (C2×C10)⋊SL2(𝔽3), (C22×C10).5A4, C10.1(C22⋊A4), C22⋊(C5×SL2(𝔽3)), (Q8×C2×C10)⋊3C3, C2.1(C5×C22⋊A4), SmallGroup(480,1133)
Series: Derived ►Chief ►Lower central ►Upper central
| C22×Q8 — C5×Q8⋊A4 |
Subgroups: 270 in 82 conjugacy classes, 20 normal (12 characteristic)
C1, C2, C2 [×2], C3, C4 [×4], C22, C22 [×2], C5, C6, C2×C4 [×6], Q8 [×4], Q8 [×4], C23, C10, C10 [×2], A4, C15, C22×C4, C2×Q8 [×4], C20 [×4], C2×C10, C2×C10 [×2], SL2(𝔽3) [×4], C2×A4, C30, C22×Q8, C2×C20 [×6], C5×Q8 [×4], C5×Q8 [×4], C22×C10, C5×A4, C22×C20, Q8×C10 [×4], Q8⋊A4, C5×SL2(𝔽3) [×4], C10×A4, Q8×C2×C10, C5×Q8⋊A4
Quotients:
C1, C3, C5, A4 [×5], C15, SL2(𝔽3), C22⋊A4, C5×A4 [×5], Q8⋊A4, C5×SL2(𝔽3), C5×C22⋊A4, C5×Q8⋊A4
Generators and relations
G = < a,b,c,d,e,f | a5=b4=d2=e2=f3=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc-1=b-1, bd=db, be=eb, fbf-1=c, cd=dc, ce=ec, fcf-1=bc, fdf-1=de=ed, fef-1=d >
►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)
(1 42 12 47)(2 43 13 48)(3 44 14 49)(4 45 15 50)(5 41 11 46)(6 116 104 23)(7 117 105 24)(8 118 101 25)(9 119 102 21)(10 120 103 22)(16 33 97 114)(17 34 98 115)(18 35 99 111)(19 31 100 112)(20 32 96 113)(26 68 51 39)(27 69 52 40)(28 70 53 36)(29 66 54 37)(30 67 55 38)(56 65 94 81)(57 61 95 82)(58 62 91 83)(59 63 92 84)(60 64 93 85)(71 80 90 109)(72 76 86 110)(73 77 87 106)(74 78 88 107)(75 79 89 108)
(1 29 12 54)(2 30 13 55)(3 26 14 51)(4 27 15 52)(5 28 11 53)(6 112 104 31)(7 113 105 32)(8 114 101 33)(9 115 102 34)(10 111 103 35)(16 118 97 25)(17 119 98 21)(18 120 99 22)(19 116 100 23)(20 117 96 24)(36 46 70 41)(37 47 66 42)(38 48 67 43)(39 49 68 44)(40 50 69 45)(56 73 94 87)(57 74 95 88)(58 75 91 89)(59 71 92 90)(60 72 93 86)(61 107 82 78)(62 108 83 79)(63 109 84 80)(64 110 85 76)(65 106 81 77)
(1 12)(2 13)(3 14)(4 15)(5 11)(26 51)(27 52)(28 53)(29 54)(30 55)(36 70)(37 66)(38 67)(39 68)(40 69)(41 46)(42 47)(43 48)(44 49)(45 50)(56 94)(57 95)(58 91)(59 92)(60 93)(61 82)(62 83)(63 84)(64 85)(65 81)(71 90)(72 86)(73 87)(74 88)(75 89)(76 110)(77 106)(78 107)(79 108)(80 109)
(6 104)(7 105)(8 101)(9 102)(10 103)(16 97)(17 98)(18 99)(19 100)(20 96)(21 119)(22 120)(23 116)(24 117)(25 118)(31 112)(32 113)(33 114)(34 115)(35 111)(56 94)(57 95)(58 91)(59 92)(60 93)(61 82)(62 83)(63 84)(64 85)(65 81)(71 90)(72 86)(73 87)(74 88)(75 89)(76 110)(77 106)(78 107)(79 108)(80 109)
(1 81 7)(2 82 8)(3 83 9)(4 84 10)(5 85 6)(11 64 104)(12 65 105)(13 61 101)(14 62 102)(15 63 103)(16 55 95)(17 51 91)(18 52 92)(19 53 93)(20 54 94)(21 68 108)(22 69 109)(23 70 110)(24 66 106)(25 67 107)(26 58 98)(27 59 99)(28 60 100)(29 56 96)(30 57 97)(31 46 86)(32 47 87)(33 48 88)(34 49 89)(35 50 90)(36 76 116)(37 77 117)(38 78 118)(39 79 119)(40 80 120)(41 72 112)(42 73 113)(43 74 114)(44 75 115)(45 71 111)
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), (1,42,12,47)(2,43,13,48)(3,44,14,49)(4,45,15,50)(5,41,11,46)(6,116,104,23)(7,117,105,24)(8,118,101,25)(9,119,102,21)(10,120,103,22)(16,33,97,114)(17,34,98,115)(18,35,99,111)(19,31,100,112)(20,32,96,113)(26,68,51,39)(27,69,52,40)(28,70,53,36)(29,66,54,37)(30,67,55,38)(56,65,94,81)(57,61,95,82)(58,62,91,83)(59,63,92,84)(60,64,93,85)(71,80,90,109)(72,76,86,110)(73,77,87,106)(74,78,88,107)(75,79,89,108), (1,29,12,54)(2,30,13,55)(3,26,14,51)(4,27,15,52)(5,28,11,53)(6,112,104,31)(7,113,105,32)(8,114,101,33)(9,115,102,34)(10,111,103,35)(16,118,97,25)(17,119,98,21)(18,120,99,22)(19,116,100,23)(20,117,96,24)(36,46,70,41)(37,47,66,42)(38,48,67,43)(39,49,68,44)(40,50,69,45)(56,73,94,87)(57,74,95,88)(58,75,91,89)(59,71,92,90)(60,72,93,86)(61,107,82,78)(62,108,83,79)(63,109,84,80)(64,110,85,76)(65,106,81,77), (1,12)(2,13)(3,14)(4,15)(5,11)(26,51)(27,52)(28,53)(29,54)(30,55)(36,70)(37,66)(38,67)(39,68)(40,69)(41,46)(42,47)(43,48)(44,49)(45,50)(56,94)(57,95)(58,91)(59,92)(60,93)(61,82)(62,83)(63,84)(64,85)(65,81)(71,90)(72,86)(73,87)(74,88)(75,89)(76,110)(77,106)(78,107)(79,108)(80,109), (6,104)(7,105)(8,101)(9,102)(10,103)(16,97)(17,98)(18,99)(19,100)(20,96)(21,119)(22,120)(23,116)(24,117)(25,118)(31,112)(32,113)(33,114)(34,115)(35,111)(56,94)(57,95)(58,91)(59,92)(60,93)(61,82)(62,83)(63,84)(64,85)(65,81)(71,90)(72,86)(73,87)(74,88)(75,89)(76,110)(77,106)(78,107)(79,108)(80,109), (1,81,7)(2,82,8)(3,83,9)(4,84,10)(5,85,6)(11,64,104)(12,65,105)(13,61,101)(14,62,102)(15,63,103)(16,55,95)(17,51,91)(18,52,92)(19,53,93)(20,54,94)(21,68,108)(22,69,109)(23,70,110)(24,66,106)(25,67,107)(26,58,98)(27,59,99)(28,60,100)(29,56,96)(30,57,97)(31,46,86)(32,47,87)(33,48,88)(34,49,89)(35,50,90)(36,76,116)(37,77,117)(38,78,118)(39,79,119)(40,80,120)(41,72,112)(42,73,113)(43,74,114)(44,75,115)(45,71,111)>;
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), (1,42,12,47)(2,43,13,48)(3,44,14,49)(4,45,15,50)(5,41,11,46)(6,116,104,23)(7,117,105,24)(8,118,101,25)(9,119,102,21)(10,120,103,22)(16,33,97,114)(17,34,98,115)(18,35,99,111)(19,31,100,112)(20,32,96,113)(26,68,51,39)(27,69,52,40)(28,70,53,36)(29,66,54,37)(30,67,55,38)(56,65,94,81)(57,61,95,82)(58,62,91,83)(59,63,92,84)(60,64,93,85)(71,80,90,109)(72,76,86,110)(73,77,87,106)(74,78,88,107)(75,79,89,108), (1,29,12,54)(2,30,13,55)(3,26,14,51)(4,27,15,52)(5,28,11,53)(6,112,104,31)(7,113,105,32)(8,114,101,33)(9,115,102,34)(10,111,103,35)(16,118,97,25)(17,119,98,21)(18,120,99,22)(19,116,100,23)(20,117,96,24)(36,46,70,41)(37,47,66,42)(38,48,67,43)(39,49,68,44)(40,50,69,45)(56,73,94,87)(57,74,95,88)(58,75,91,89)(59,71,92,90)(60,72,93,86)(61,107,82,78)(62,108,83,79)(63,109,84,80)(64,110,85,76)(65,106,81,77), (1,12)(2,13)(3,14)(4,15)(5,11)(26,51)(27,52)(28,53)(29,54)(30,55)(36,70)(37,66)(38,67)(39,68)(40,69)(41,46)(42,47)(43,48)(44,49)(45,50)(56,94)(57,95)(58,91)(59,92)(60,93)(61,82)(62,83)(63,84)(64,85)(65,81)(71,90)(72,86)(73,87)(74,88)(75,89)(76,110)(77,106)(78,107)(79,108)(80,109), (6,104)(7,105)(8,101)(9,102)(10,103)(16,97)(17,98)(18,99)(19,100)(20,96)(21,119)(22,120)(23,116)(24,117)(25,118)(31,112)(32,113)(33,114)(34,115)(35,111)(56,94)(57,95)(58,91)(59,92)(60,93)(61,82)(62,83)(63,84)(64,85)(65,81)(71,90)(72,86)(73,87)(74,88)(75,89)(76,110)(77,106)(78,107)(79,108)(80,109), (1,81,7)(2,82,8)(3,83,9)(4,84,10)(5,85,6)(11,64,104)(12,65,105)(13,61,101)(14,62,102)(15,63,103)(16,55,95)(17,51,91)(18,52,92)(19,53,93)(20,54,94)(21,68,108)(22,69,109)(23,70,110)(24,66,106)(25,67,107)(26,58,98)(27,59,99)(28,60,100)(29,56,96)(30,57,97)(31,46,86)(32,47,87)(33,48,88)(34,49,89)(35,50,90)(36,76,116)(37,77,117)(38,78,118)(39,79,119)(40,80,120)(41,72,112)(42,73,113)(43,74,114)(44,75,115)(45,71,111) );
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)], [(1,42,12,47),(2,43,13,48),(3,44,14,49),(4,45,15,50),(5,41,11,46),(6,116,104,23),(7,117,105,24),(8,118,101,25),(9,119,102,21),(10,120,103,22),(16,33,97,114),(17,34,98,115),(18,35,99,111),(19,31,100,112),(20,32,96,113),(26,68,51,39),(27,69,52,40),(28,70,53,36),(29,66,54,37),(30,67,55,38),(56,65,94,81),(57,61,95,82),(58,62,91,83),(59,63,92,84),(60,64,93,85),(71,80,90,109),(72,76,86,110),(73,77,87,106),(74,78,88,107),(75,79,89,108)], [(1,29,12,54),(2,30,13,55),(3,26,14,51),(4,27,15,52),(5,28,11,53),(6,112,104,31),(7,113,105,32),(8,114,101,33),(9,115,102,34),(10,111,103,35),(16,118,97,25),(17,119,98,21),(18,120,99,22),(19,116,100,23),(20,117,96,24),(36,46,70,41),(37,47,66,42),(38,48,67,43),(39,49,68,44),(40,50,69,45),(56,73,94,87),(57,74,95,88),(58,75,91,89),(59,71,92,90),(60,72,93,86),(61,107,82,78),(62,108,83,79),(63,109,84,80),(64,110,85,76),(65,106,81,77)], [(1,12),(2,13),(3,14),(4,15),(5,11),(26,51),(27,52),(28,53),(29,54),(30,55),(36,70),(37,66),(38,67),(39,68),(40,69),(41,46),(42,47),(43,48),(44,49),(45,50),(56,94),(57,95),(58,91),(59,92),(60,93),(61,82),(62,83),(63,84),(64,85),(65,81),(71,90),(72,86),(73,87),(74,88),(75,89),(76,110),(77,106),(78,107),(79,108),(80,109)], [(6,104),(7,105),(8,101),(9,102),(10,103),(16,97),(17,98),(18,99),(19,100),(20,96),(21,119),(22,120),(23,116),(24,117),(25,118),(31,112),(32,113),(33,114),(34,115),(35,111),(56,94),(57,95),(58,91),(59,92),(60,93),(61,82),(62,83),(63,84),(64,85),(65,81),(71,90),(72,86),(73,87),(74,88),(75,89),(76,110),(77,106),(78,107),(79,108),(80,109)], [(1,81,7),(2,82,8),(3,83,9),(4,84,10),(5,85,6),(11,64,104),(12,65,105),(13,61,101),(14,62,102),(15,63,103),(16,55,95),(17,51,91),(18,52,92),(19,53,93),(20,54,94),(21,68,108),(22,69,109),(23,70,110),(24,66,106),(25,67,107),(26,58,98),(27,59,99),(28,60,100),(29,56,96),(30,57,97),(31,46,86),(32,47,87),(33,48,88),(34,49,89),(35,50,90),(36,76,116),(37,77,117),(38,78,118),(39,79,119),(40,80,120),(41,72,112),(42,73,113),(43,74,114),(44,75,115),(45,71,111)])
Matrix representation ►G ⊆ GL5(𝔽61)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 58 | 0 | 0 |
| 0 | 0 | 0 | 58 | 0 |
| 0 | 0 | 0 | 0 | 58 |
| 14 | 48 | 0 | 0 | 0 |
| 48 | 47 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 60 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 60 | 60 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 60 | 60 |
| 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 48 | 47 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 13 | 0 | 0 |
G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,58,0,0,0,0,0,58,0,0,0,0,0,58],[14,48,0,0,0,48,47,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,60,0,0,1,0,60,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,60,0,0,0,1,60,0],[1,48,0,0,0,0,47,0,0,0,0,0,0,0,13,0,0,13,0,0,0,0,0,13,0] >;
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 15A | ··· | 15H | 20A | ··· | 20P | 30A | ··· | 30H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 3 | 3 | 16 | 16 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 16 | 16 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 16 | ··· | 16 | 6 | ··· | 6 | 16 | ··· | 16 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 |
| type | + | - | + | + | - | ||||||||
| image | C1 | C3 | C5 | C15 | SL2(𝔽3) | SL2(𝔽3) | C5×SL2(𝔽3) | A4 | A4 | C5×A4 | C5×A4 | Q8⋊A4 | C5×Q8⋊A4 |
| kernel | C5×Q8⋊A4 | Q8×C2×C10 | Q8⋊A4 | C22×Q8 | C2×C10 | C2×C10 | C22 | C5×Q8 | C22×C10 | Q8 | C23 | C5 | C1 |
| # reps | 1 | 2 | 4 | 8 | 1 | 2 | 12 | 4 | 1 | 16 | 4 | 1 | 4 |
In GAP, Magma, Sage, TeX
C_5\times Q_8\rtimes A_4
% in TeX
G:=Group("C5xQ8:A4"); // GroupNames label
G:=SmallGroup(480,1133);
// by ID
G=gap.SmallGroup(480,1133);
# by ID
G:=PCGroup([7,-3,-5,-2,2,-2,2,-2,632,1263,4204,172,7565,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^4=d^2=e^2=f^3=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,f*b*f^-1=c,c*d=d*c,c*e=e*c,f*c*f^-1=b*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀