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G = D303C4order 240 = 24·3·5

1st semidirect product of D30 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D303C4, C6.5D20, C2.2D60, C30.33D4, C10.5D12, C22.6D30, (C2×C60)⋊3C2, (C2×C20)⋊2S3, C53(D6⋊C4), (C2×C4)⋊1D15, (C2×C12)⋊2D5, C6.9(C4×D5), C2.5(C4×D15), C10.16(C4×S3), C157(C22⋊C4), C30.39(C2×C4), (C2×C6).24D10, (C2×C10).24D6, C32(D10⋊C4), (C2×Dic15)⋊1C2, C2.2(C157D4), C6.15(C5⋊D4), C10.15(C3⋊D4), (C2×C30).25C22, (C22×D15).1C2, SmallGroup(240,75)

Series: Derived Chief Lower central Upper central

C1C30 — D303C4
C1C5C15C30C2×C30C22×D15 — D303C4
C15C30 — D303C4
C1C22C2×C4

Generators and relations for D303C4
 G = < a,b,c | a30=b2=c4=1, bab=a-1, ac=ca, cbc-1=a15b >

Subgroups: 392 in 68 conjugacy classes, 29 normal (27 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4, C23, D5 [×2], C10 [×3], Dic3, C12, D6 [×4], C2×C6, C15, C22⋊C4, Dic5, C20, D10 [×4], C2×C10, C2×Dic3, C2×C12, C22×S3, D15 [×2], C30 [×3], C2×Dic5, C2×C20, C22×D5, D6⋊C4, Dic15, C60, D30 [×2], D30 [×2], C2×C30, D10⋊C4, C2×Dic15, C2×C60, C22×D15, D303C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, D15, C4×D5, D20, C5⋊D4, D6⋊C4, D30, D10⋊C4, C4×D15, D60, C157D4, D303C4

Smallest permutation representation of D303C4
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 60)(32 59)(33 58)(34 57)(35 56)(36 55)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(61 65)(62 64)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(91 95)(92 94)(96 120)(97 119)(98 118)(99 117)(100 116)(101 115)(102 114)(103 113)(104 112)(105 111)(106 110)(107 109)
(1 101 31 71)(2 102 32 72)(3 103 33 73)(4 104 34 74)(5 105 35 75)(6 106 36 76)(7 107 37 77)(8 108 38 78)(9 109 39 79)(10 110 40 80)(11 111 41 81)(12 112 42 82)(13 113 43 83)(14 114 44 84)(15 115 45 85)(16 116 46 86)(17 117 47 87)(18 118 48 88)(19 119 49 89)(20 120 50 90)(21 91 51 61)(22 92 52 62)(23 93 53 63)(24 94 54 64)(25 95 55 65)(26 96 56 66)(27 97 57 67)(28 98 58 68)(29 99 59 69)(30 100 60 70)

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

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

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

D303C4 is a maximal subgroup of
(C2×C20).D6  C4⋊Dic3⋊D5  C4⋊Dic5⋊S3  Dic3⋊C4⋊D5  Dic5.8D12  D6⋊Dic5⋊C2  Dic3.D20  D30.D4  C10.D4⋊S3  (C4×Dic5)⋊S3  Dic34D20  D30.C2⋊C4  D30.23(C2×C4)  Dic54D12  Dic3⋊D20  D30⋊Q8  D10.16D12  Dic5⋊D12  D302Q8  D303Q8  D6.D20  D304Q8  D30.7D4  C1520(C4×D4)  C1522(C4×D4)  D10⋊D12  D6⋊D20  (C2×Dic6)⋊D5  D5×D6⋊C4  S3×D10⋊C4  D64D20  D304D4  C422D15  C4×D60  C427D15  C423D15  C22⋊C4×D15  Dic1519D4  D3016D4  D30.28D4  D309D4  C23.11D30  C22.D60  C4⋊C47D15  D6011C4  D30.29D4  C4⋊D60  D305Q8  D306Q8  C4⋊C4⋊D15  C4×C157D4  C23.28D30  C6029D4  D3017D4  Dic1512D4  D307Q8  C60.23D4
D303C4 is a maximal quotient of
D607C4  C23.6D30  D609C4  Dic309C4  Dic308C4  D303C8  D608C4  M4(2)⋊D15  C4.D60  D6010C4  C30.29C42

66 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D5A5B6A6B6C10A···10F12A12B12C12D15A15B15C15D20A···20H30A···30L60A···60P
order122222344445566610···10121212121515151520···2030···3060···60
size111130302223030222222···2222222222···22···22···2

66 irreducible representations

dim111112222222222222222
type++++++++++++++
imageC1C2C2C2C4S3D4D5D6D10C4×S3D12C3⋊D4D15C4×D5D20C5⋊D4D30C4×D15D60C157D4
kernelD303C4C2×Dic15C2×C60C22×D15D30C2×C20C30C2×C12C2×C10C2×C6C10C10C10C2×C4C6C6C6C22C2C2C2
# reps111141221222244444888

Matrix representation of D303C4 in GL4(𝔽61) generated by

235300
84500
005224
003223
,
235300
53800
004531
003916
,
11000
01100
003016
00131
G:=sub<GL(4,GF(61))| [23,8,0,0,53,45,0,0,0,0,52,32,0,0,24,23],[23,5,0,0,53,38,0,0,0,0,45,39,0,0,31,16],[11,0,0,0,0,11,0,0,0,0,30,1,0,0,16,31] >;

D303C4 in GAP, Magma, Sage, TeX

D_{30}\rtimes_3C_4
% in TeX

G:=Group("D30:3C4");
// GroupNames label

G:=SmallGroup(240,75);
// by ID

G=gap.SmallGroup(240,75);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,121,31,964,6917]);
// Polycyclic

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

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