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## G = C5×Q8×A4order 480 = 25·3·5

### Direct product of C5, Q8 and A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — C5×Q8×A4
 Chief series C1 — C22 — C23 — C22×C10 — C10×A4 — A4×C20 — C5×Q8×A4
 Lower central C22 — C23 — C5×Q8×A4
 Upper central C1 — C10 — C5×Q8

Generators and relations for C5×Q8×A4
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, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 216 in 92 conjugacy classes, 36 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C2×C4, Q8, Q8, C23, C10, C10, C12, A4, C15, C22×C4, C2×Q8, C20, C20, C2×C10, C2×C10, C3×Q8, C2×A4, C30, C22×Q8, C2×C20, C5×Q8, C5×Q8, C22×C10, C4×A4, C60, C5×A4, C22×C20, Q8×C10, Q8×A4, Q8×C15, C10×A4, Q8×C2×C10, A4×C20, C5×Q8×A4
Quotients: C1, C2, C3, C22, C5, C6, Q8, C10, A4, C2×C6, C15, C2×C10, C3×Q8, C2×A4, C30, C5×Q8, C22×A4, C5×A4, C2×C30, Q8×A4, Q8×C15, C10×A4, A4×C2×C10, C5×Q8×A4

Smallest permutation representation of C5×Q8×A4
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 41 11 37)(2 42 12 38)(3 43 13 39)(4 44 14 40)(5 45 15 36)(6 90 120 94)(7 86 116 95)(8 87 117 91)(9 88 118 92)(10 89 119 93)(16 55 25 46)(17 51 21 47)(18 52 22 48)(19 53 23 49)(20 54 24 50)(26 65 35 56)(27 61 31 57)(28 62 32 58)(29 63 33 59)(30 64 34 60)(66 96 75 105)(67 97 71 101)(68 98 72 102)(69 99 73 103)(70 100 74 104)(76 106 85 115)(77 107 81 111)(78 108 82 112)(79 109 83 113)(80 110 84 114)
(1 71 11 67)(2 72 12 68)(3 73 13 69)(4 74 14 70)(5 75 15 66)(6 64 120 60)(7 65 116 56)(8 61 117 57)(9 62 118 58)(10 63 119 59)(16 85 25 76)(17 81 21 77)(18 82 22 78)(19 83 23 79)(20 84 24 80)(26 95 35 86)(27 91 31 87)(28 92 32 88)(29 93 33 89)(30 94 34 90)(36 105 45 96)(37 101 41 97)(38 102 42 98)(39 103 43 99)(40 104 44 100)(46 115 55 106)(47 111 51 107)(48 112 52 108)(49 113 53 109)(50 114 54 110)
(1 11)(2 12)(3 13)(4 14)(5 15)(16 25)(17 21)(18 22)(19 23)(20 24)(36 45)(37 41)(38 42)(39 43)(40 44)(46 55)(47 51)(48 52)(49 53)(50 54)(66 75)(67 71)(68 72)(69 73)(70 74)(76 85)(77 81)(78 82)(79 83)(80 84)(96 105)(97 101)(98 102)(99 103)(100 104)(106 115)(107 111)(108 112)(109 113)(110 114)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 120)(7 116)(8 117)(9 118)(10 119)(26 35)(27 31)(28 32)(29 33)(30 34)(36 45)(37 41)(38 42)(39 43)(40 44)(56 65)(57 61)(58 62)(59 63)(60 64)(66 75)(67 71)(68 72)(69 73)(70 74)(86 95)(87 91)(88 92)(89 93)(90 94)(96 105)(97 101)(98 102)(99 103)(100 104)
(1 27 17)(2 28 18)(3 29 19)(4 30 20)(5 26 16)(6 114 104)(7 115 105)(8 111 101)(9 112 102)(10 113 103)(11 31 21)(12 32 22)(13 33 23)(14 34 24)(15 35 25)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)(41 61 51)(42 62 52)(43 63 53)(44 64 54)(45 65 55)(66 86 76)(67 87 77)(68 88 78)(69 89 79)(70 90 80)(71 91 81)(72 92 82)(73 93 83)(74 94 84)(75 95 85)(96 116 106)(97 117 107)(98 118 108)(99 119 109)(100 120 110)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 6A 6B 10A 10B 10C 10D 10E ··· 10L 12A ··· 12F 15A ··· 15H 20A ··· 20L 20M ··· 20X 30A ··· 30H 60A ··· 60X order 1 2 2 2 3 3 4 4 4 4 4 4 5 5 5 5 6 6 10 10 10 10 10 ··· 10 12 ··· 12 15 ··· 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 3 3 4 4 2 2 2 6 6 6 1 1 1 1 4 4 1 1 1 1 3 ··· 3 8 ··· 8 4 ··· 4 2 ··· 2 6 ··· 6 4 ··· 4 8 ··· 8

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 6 6 type + + - + + - image C1 C2 C3 C5 C6 C10 C15 C30 Q8 C3×Q8 C5×Q8 Q8×C15 A4 C2×A4 C5×A4 C10×A4 Q8×A4 C5×Q8×A4 kernel C5×Q8×A4 A4×C20 Q8×C2×C10 Q8×A4 C22×C20 C4×A4 C22×Q8 C22×C4 C5×A4 C2×C10 A4 C22 C5×Q8 C20 Q8 C4 C5 C1 # reps 1 3 2 4 6 12 8 24 1 2 4 8 1 3 4 12 1 4

Matrix representation of C5×Q8×A4 in GL5(𝔽61)

 34 0 0 0 0 0 34 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 54 28 0 0 0 20 7 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 21 6 0 0 0 28 40 0 0 0 0 0 60 0 0 0 0 0 60 0 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 0 0 0 0 0 1 0 0 0 0 0 0 1 60 0 0 1 0 60 0 0 0 0 60
,
 13 0 0 0 0 0 13 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0

G:=sub<GL(5,GF(61))| [34,0,0,0,0,0,34,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[54,20,0,0,0,28,7,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[21,28,0,0,0,6,40,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[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],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,60,60,60],[13,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0] >;

C5×Q8×A4 in GAP, Magma, Sage, TeX

C_5\times Q_8\times A_4
% in TeX

G:=Group("C5xQ8xA4");
// GroupNames label

G:=SmallGroup(480,1129);
// by ID

G=gap.SmallGroup(480,1129);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,-2,2,420,869,428,2539,4430]);
// 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,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

׿
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