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## G = (C2×C30)⋊D4order 480 = 25·3·5

### 4th semidirect product of C2×C30 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — (C2×C30)⋊D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×C15⋊D4 — (C2×C30)⋊D4
 Lower central C15 — C2×C30 — (C2×C30)⋊D4
 Upper central C1 — C22 — C23

Generators and relations for (C2×C30)⋊D4
G = < a,b,c,d | a2=b30=c4=d2=1, ab=ba, cac-1=ab15, ad=da, cbc-1=b-1, dbd=b19, dcd=c-1 >

Subgroups: 1308 in 260 conjugacy classes, 60 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, C10, Dic3, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C22×C6, C5×S3, C3×D5, C30, C30, C30, C22≀C2, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22×C10, C6.D4, C2×C3⋊D4, C2×C3⋊D4, C23×C6, C5×Dic3, Dic15, C6×D5, C6×D5, S3×C10, C2×C30, C2×C30, C2×C30, D10⋊C4, C23.D5, C2×C5⋊D4, D4×C10, C23×D5, C244S3, C15⋊D4, C10×Dic3, C5×C3⋊D4, C2×Dic15, D5×C2×C6, D5×C2×C6, S3×C2×C10, C22×C30, C23⋊D10, D10⋊Dic3, C30.38D4, C2×C15⋊D4, C10×C3⋊D4, D5×C22×C6, (C2×C30)⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C22≀C2, C5⋊D4, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, C2×C5⋊D4, C244S3, C15⋊D4, C2×S3×D5, C23⋊D10, C2×C15⋊D4, D5×C3⋊D4, (C2×C30)⋊D4

Smallest permutation representation of (C2×C30)⋊D4
On 120 points
Generators in S120
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 50)(30 51)(61 112)(62 113)(63 114)(64 115)(65 116)(66 117)(67 118)(68 119)(69 120)(70 91)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)
(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 68 52 104)(2 67 53 103)(3 66 54 102)(4 65 55 101)(5 64 56 100)(6 63 57 99)(7 62 58 98)(8 61 59 97)(9 90 60 96)(10 89 31 95)(11 88 32 94)(12 87 33 93)(13 86 34 92)(14 85 35 91)(15 84 36 120)(16 83 37 119)(17 82 38 118)(18 81 39 117)(19 80 40 116)(20 79 41 115)(21 78 42 114)(22 77 43 113)(23 76 44 112)(24 75 45 111)(25 74 46 110)(26 73 47 109)(27 72 48 108)(28 71 49 107)(29 70 50 106)(30 69 51 105)
(1 16)(2 5)(3 24)(4 13)(6 21)(7 10)(8 29)(9 18)(11 26)(12 15)(14 23)(17 20)(19 28)(22 25)(27 30)(31 58)(32 47)(33 36)(34 55)(35 44)(37 52)(38 41)(39 60)(40 49)(42 57)(43 46)(45 54)(48 51)(50 59)(53 56)(61 106)(62 95)(63 114)(64 103)(65 92)(66 111)(67 100)(68 119)(69 108)(70 97)(71 116)(72 105)(73 94)(74 113)(75 102)(76 91)(77 110)(78 99)(79 118)(80 107)(81 96)(82 115)(83 104)(84 93)(85 112)(86 101)(87 120)(88 109)(89 98)(90 117)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 3 4A 4B 4C 5A 5B 6A ··· 6G 6H ··· 6O 10A ··· 10F 10G 10H 10I 10J 10K 10L 10M 10N 15A 15B 20A 20B 20C 20D 30A ··· 30N order 1 2 2 2 2 2 2 2 2 2 2 3 4 4 4 5 5 6 ··· 6 6 ··· 6 10 ··· 10 10 10 10 10 10 10 10 10 15 15 20 20 20 20 30 ··· 30 size 1 1 1 1 2 2 10 10 10 10 12 2 12 60 60 2 2 2 ··· 2 10 ··· 10 2 ··· 2 4 4 4 4 12 12 12 12 4 4 12 12 12 12 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D10 D10 D10 C3⋊D4 C3⋊D4 C5⋊D4 S3×D5 D4×D5 C15⋊D4 C2×S3×D5 D5×C3⋊D4 kernel (C2×C30)⋊D4 D10⋊Dic3 C30.38D4 C2×C15⋊D4 C10×C3⋊D4 D5×C22×C6 C23×D5 C6×D5 C2×C30 C2×C3⋊D4 C22×D5 C22×C10 C2×Dic3 C22×S3 C22×C6 D10 C2×C10 C2×C6 C23 C6 C22 C22 C2 # reps 1 2 1 2 1 1 1 4 2 2 2 1 2 2 2 8 4 8 2 4 4 2 8

Matrix representation of (C2×C30)⋊D4 in GL4(𝔽61) generated by

 60 0 0 0 0 60 0 0 0 0 60 0 0 0 17 1
,
 44 1 0 0 16 60 0 0 0 0 14 0 0 0 45 48
,
 39 8 0 0 8 22 0 0 0 0 48 20 0 0 22 13
,
 17 18 0 0 45 44 0 0 0 0 60 0 0 0 17 1
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,60,17,0,0,0,1],[44,16,0,0,1,60,0,0,0,0,14,45,0,0,0,48],[39,8,0,0,8,22,0,0,0,0,48,22,0,0,20,13],[17,45,0,0,18,44,0,0,0,0,60,17,0,0,0,1] >;

(C2×C30)⋊D4 in GAP, Magma, Sage, TeX

(C_2\times C_{30})\rtimes D_4
% in TeX

G:=Group("(C2xC30):D4");
// GroupNames label

G:=SmallGroup(480,639);
// by ID

G=gap.SmallGroup(480,639);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,219,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^30=c^4=d^2=1,a*b=b*a,c*a*c^-1=a*b^15,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^19,d*c*d=c^-1>;
// generators/relations

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