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## G = C5×Q8⋊A4order 480 = 25·3·5

### Direct product of C5 and Q8⋊A4

Aliases: C5×Q8⋊A4, Q81(C5×A4), (C5×Q8)⋊1A4, 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

 Derived series C1 — C2 — C22×Q8 — C5×Q8⋊A4
 Chief series C1 — C2 — Q8 — C22×Q8 — Q8×C2×C10 — C5×Q8⋊A4
 Lower central C22×Q8 — C5×Q8⋊A4
 Upper central C1 — C10

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, fbf-1=c, cd=dc, ce=ec, fcf-1=bc, fdf-1=de=ed, fef-1=d >

Subgroups: 270 in 82 conjugacy classes, 20 normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C2×C4, Q8, Q8, C23, C10, C10, A4, C15, C22×C4, C2×Q8, C20, C2×C10, C2×C10, SL2(𝔽3), C2×A4, C30, C22×Q8, C2×C20, C5×Q8, C5×Q8, C22×C10, C5×A4, C22×C20, Q8×C10, Q8⋊A4, C5×SL2(𝔽3), C10×A4, Q8×C2×C10, C5×Q8⋊A4
Quotients: C1, C3, C5, A4, C15, SL2(𝔽3), C22⋊A4, C5×A4, Q8⋊A4, C5×SL2(𝔽3), C5×C22⋊A4, 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 28 12 47)(2 29 13 48)(3 30 14 49)(4 26 15 50)(5 27 11 46)(6 116 96 20)(7 117 97 16)(8 118 98 17)(9 119 99 18)(10 120 100 19)(21 35 102 111)(22 31 103 112)(23 32 104 113)(24 33 105 114)(25 34 101 115)(36 41 70 53)(37 42 66 54)(38 43 67 55)(39 44 68 51)(40 45 69 52)(56 93 85 63)(57 94 81 64)(58 95 82 65)(59 91 83 61)(60 92 84 62)(71 80 90 109)(72 76 86 110)(73 77 87 106)(74 78 88 107)(75 79 89 108)
(1 42 12 54)(2 43 13 55)(3 44 14 51)(4 45 15 52)(5 41 11 53)(6 112 96 31)(7 113 97 32)(8 114 98 33)(9 115 99 34)(10 111 100 35)(16 23 117 104)(17 24 118 105)(18 25 119 101)(19 21 120 102)(20 22 116 103)(26 40 50 69)(27 36 46 70)(28 37 47 66)(29 38 48 67)(30 39 49 68)(56 110 85 76)(57 106 81 77)(58 107 82 78)(59 108 83 79)(60 109 84 80)(61 75 91 89)(62 71 92 90)(63 72 93 86)(64 73 94 87)(65 74 95 88)
(1 12)(2 13)(3 14)(4 15)(5 11)(26 50)(27 46)(28 47)(29 48)(30 49)(36 70)(37 66)(38 67)(39 68)(40 69)(41 53)(42 54)(43 55)(44 51)(45 52)(56 85)(57 81)(58 82)(59 83)(60 84)(61 91)(62 92)(63 93)(64 94)(65 95)(71 90)(72 86)(73 87)(74 88)(75 89)(76 110)(77 106)(78 107)(79 108)(80 109)
(6 96)(7 97)(8 98)(9 99)(10 100)(16 117)(17 118)(18 119)(19 120)(20 116)(21 102)(22 103)(23 104)(24 105)(25 101)(31 112)(32 113)(33 114)(34 115)(35 111)(56 85)(57 81)(58 82)(59 83)(60 84)(61 91)(62 92)(63 93)(64 94)(65 95)(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 56 96)(12 57 97)(13 58 98)(14 59 99)(15 60 100)(16 66 106)(17 67 107)(18 68 108)(19 69 109)(20 70 110)(21 52 92)(22 53 93)(23 54 94)(24 55 95)(25 51 91)(26 71 111)(27 72 112)(28 73 113)(29 74 114)(30 75 115)(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 63 103)(42 64 104)(43 65 105)(44 61 101)(45 62 102)

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

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

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

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

Matrix representation of C5×Q8⋊A4 in 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] >;

C5×Q8⋊A4 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

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