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## G = C5×A4⋊Q8order 480 = 25·3·5

### Direct product of C5 and A4⋊Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×A4 — C5×A4⋊Q8
 Chief series C1 — C22 — A4 — C2×A4 — C10×A4 — C5×A4⋊C4 — C5×A4⋊Q8
 Lower central A4 — C2×A4 — C5×A4⋊Q8
 Upper central C1 — C10 — C20

Generators and relations for C5×A4⋊Q8
G = < a,b,c,d,e,f | a5=b2=c2=d3=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, dbd-1=fbf-1=bc=cb, be=eb, dcd-1=b, ce=ec, cf=fc, de=ed, fdf-1=d-1, fef-1=e-1 >

Subgroups: 252 in 84 conjugacy classes, 24 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C2×C4, Q8, C23, C10, C10, Dic3, C12, A4, C15, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C20, C20, C2×C10, C2×C10, Dic6, C2×A4, C30, C22⋊Q8, C2×C20, C5×Q8, C22×C10, A4⋊C4, C4×A4, C5×Dic3, C60, C5×A4, C5×C22⋊C4, C5×C4⋊C4, C22×C20, Q8×C10, A4⋊Q8, C5×Dic6, C10×A4, C5×C22⋊Q8, C5×A4⋊C4, A4×C20, C5×A4⋊Q8
Quotients: C1, C2, C22, C5, S3, Q8, C10, D6, C2×C10, Dic6, S4, C5×S3, C5×Q8, C2×S4, S3×C10, A4⋊Q8, C5×Dic6, C5×S4, C10×S4, C5×A4⋊Q8

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

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

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

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

70 conjugacy classes

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

70 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 6 6 type + + + + - + - + + - image C1 C2 C2 C5 C10 C10 S3 Q8 D6 Dic6 C5×S3 C5×Q8 S3×C10 C5×Dic6 S4 C2×S4 C5×S4 C10×S4 A4⋊Q8 C5×A4⋊Q8 kernel C5×A4⋊Q8 C5×A4⋊C4 A4×C20 A4⋊Q8 A4⋊C4 C4×A4 C22×C20 C5×A4 C22×C10 C2×C10 C22×C4 A4 C23 C22 C20 C10 C4 C2 C5 C1 # reps 1 2 1 4 8 4 1 1 1 2 4 4 4 8 2 2 8 8 1 4

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

 1 0 0 0 0 0 1 0 0 0 0 0 20 0 0 0 0 0 20 0 0 0 0 0 20
,
 1 0 0 0 0 0 1 0 0 0 0 0 60 60 60 0 0 0 0 1 0 0 0 1 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
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 38 35 0 0 0 11 23 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 60
,
 12 22 0 0 0 35 49 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,20,0,0,0,0,0,20,0,0,0,0,0,20],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,60,0,1,0,0,60,1,0],[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],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[38,11,0,0,0,35,23,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[12,35,0,0,0,22,49,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;

C5×A4⋊Q8 in GAP, Magma, Sage, TeX

C_5\times A_4\rtimes Q_8
% in TeX

G:=Group("C5xA4:Q8");
// GroupNames label

G:=SmallGroup(480,1013);
// by ID

G=gap.SmallGroup(480,1013);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-3,-2,2,140,309,148,2804,10085,285,5886,475]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^3=e^4=1,f^2=e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,d*b*d^-1=f*b*f^-1=b*c=c*b,b*e=e*b,d*c*d^-1=b,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f^-1=d^-1,f*e*f^-1=e^-1>;
// generators/relations

׿
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