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## G = D30⋊4D4order 480 = 25·3·5

### 4th semidirect product of D30 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — D30⋊4D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C22×S3×D5 — D30⋊4D4
 Lower central C15 — C2×C30 — D30⋊4D4
 Upper central C1 — C22 — C2×C4

Generators and relations for D304D4
G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, ac=ca, dad=a11, cbc-1=a15b, dbd=a25b, dcd=c-1 >

Subgroups: 1772 in 260 conjugacy classes, 54 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3, C3×D5, D15, C30, C22≀C2, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, D6⋊C4, C3×C22⋊C4, C2×D12, C2×D12, C2×C3⋊D4, S3×C23, C3×Dic5, Dic15, C60, S3×D5, C6×D5, C6×D5, S3×C10, S3×C10, D30, D30, C2×C30, D10⋊C4, D10⋊C4, C23.D5, C2×C5⋊D4, D4×C10, C23×D5, D6⋊D4, C15⋊D4, C5⋊D12, C6×Dic5, C5×D12, C2×Dic15, C2×C60, C2×S3×D5, D5×C2×C6, S3×C2×C10, C22×D15, C23⋊D10, D6⋊Dic5, C3×D10⋊C4, D303C4, C2×C15⋊D4, C2×C5⋊D12, C10×D12, C22×S3×D5, D304D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, D12, C22×S3, C22≀C2, C5⋊D4, C22×D5, C2×D12, S3×D4, S3×D5, D4×D5, C2×C5⋊D4, D6⋊D4, C2×S3×D5, C23⋊D10, D5×D12, C20⋊D6, S3×C5⋊D4, D304D4

Smallest permutation representation of D304D4
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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(80 90)(81 89)(82 88)(83 87)(84 86)(91 105)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)
(1 78 55 106)(2 79 56 107)(3 80 57 108)(4 81 58 109)(5 82 59 110)(6 83 60 111)(7 84 31 112)(8 85 32 113)(9 86 33 114)(10 87 34 115)(11 88 35 116)(12 89 36 117)(13 90 37 118)(14 61 38 119)(15 62 39 120)(16 63 40 91)(17 64 41 92)(18 65 42 93)(19 66 43 94)(20 67 44 95)(21 68 45 96)(22 69 46 97)(23 70 47 98)(24 71 48 99)(25 72 49 100)(26 73 50 101)(27 74 51 102)(28 75 52 103)(29 76 53 104)(30 77 54 105)
(1 106)(2 117)(3 98)(4 109)(5 120)(6 101)(7 112)(8 93)(9 104)(10 115)(11 96)(12 107)(13 118)(14 99)(15 110)(16 91)(17 102)(18 113)(19 94)(20 105)(21 116)(22 97)(23 108)(24 119)(25 100)(26 111)(27 92)(28 103)(29 114)(30 95)(31 84)(32 65)(33 76)(34 87)(35 68)(36 79)(37 90)(38 71)(39 82)(40 63)(41 74)(42 85)(43 66)(44 77)(45 88)(46 69)(47 80)(48 61)(49 72)(50 83)(51 64)(52 75)(53 86)(54 67)(55 78)(56 89)(57 70)(58 81)(59 62)(60 73)

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D5 D6 D6 D6 D10 D10 D12 C5⋊D4 S3×D4 S3×D5 D4×D5 C2×S3×D5 D5×D12 C20⋊D6 S3×C5⋊D4 kernel D30⋊4D4 D6⋊Dic5 C3×D10⋊C4 D30⋊3C4 C2×C15⋊D4 C2×C5⋊D12 C10×D12 C22×S3×D5 D10⋊C4 C6×D5 S3×C10 D30 C2×D12 C2×Dic5 C2×C20 C22×D5 C2×C12 C22×S3 D10 D6 C10 C2×C4 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 2 4 4 8 2 2 4 2 4 4 4

Matrix representation of D304D4 in GL4(𝔽61) generated by

 0 60 0 0 1 1 0 0 0 0 1 45 0 0 60 17
,
 0 60 0 0 60 0 0 0 0 0 0 17 0 0 18 0
,
 23 46 0 0 15 38 0 0 0 0 14 28 0 0 17 47
,
 23 46 0 0 23 38 0 0 0 0 14 28 0 0 17 47
G:=sub<GL(4,GF(61))| [0,1,0,0,60,1,0,0,0,0,1,60,0,0,45,17],[0,60,0,0,60,0,0,0,0,0,0,18,0,0,17,0],[23,15,0,0,46,38,0,0,0,0,14,17,0,0,28,47],[23,23,0,0,46,38,0,0,0,0,14,17,0,0,28,47] >;

D304D4 in GAP, Magma, Sage, TeX

D_{30}\rtimes_4D_4
% in TeX

G:=Group("D30:4D4");
// GroupNames label

G:=SmallGroup(480,551);
// by ID

G=gap.SmallGroup(480,551);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^11,c*b*c^-1=a^15*b,d*b*d=a^25*b,d*c*d=c^-1>;
// generators/relations

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