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G = D4:8D30order 480 = 25·3·5

4th semidirect product of D4 and D30 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4:8D30, Q8:8D30, D60:28C22, C60.89C23, C30.65C24, D30.30C23, C15:122+ 1+4, Dic30:39C22, Dic15.32C23, (C2xC4):4D30, C4oD4:5D15, (C5xD4):24D6, (C2xC20):14D6, C5:5(D4oD12), (C5xQ8):24D6, (C3xD4):24D10, (C2xD60):16C2, (D4xD15):12C2, (C2xC12):14D10, (C3xQ8):21D10, C3:5(D4:8D10), (C2xC60):10C22, Q8:3D15:12C2, C6.65(C23xD5), (C4xD15):11C22, (D4xC15):26C22, C15:7D4:12C22, C10.65(S3xC23), (C2xC30).11C23, D60:11C2:18C2, (Q8xC15):23C22, C4.32(C22xD15), C2.13(C23xD15), C20.139(C22xS3), C12.137(C22xD5), (C22xD15):4C22, C22.3(C22xD15), (C3xC4oD4):4D5, (C5xC4oD4):8S3, (C15xC4oD4):4C2, (C2xC6).18(C22xD5), (C2xC10).19(C22xS3), SmallGroup(480,1176)

Series: Derived Chief Lower central Upper central

C1C30 — D4:8D30
C1C5C15C30D30C22xD15D4xD15 — D4:8D30
C15C30 — D4:8D30
C1C2C4oD4

Generators and relations for D4:8D30
 G = < a,b,c,d | a4=b2=c30=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, bd=db, dcd=c-1 >

Subgroups: 2180 in 332 conjugacy classes, 119 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C5, S3, C6, C6, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, C2xC6, C15, C2xD4, C4oD4, C4oD4, Dic5, C20, C20, D10, C2xC10, Dic6, C4xS3, D12, C3:D4, C2xC12, C3xD4, C3xQ8, C22xS3, D15, C30, C30, 2+ 1+4, Dic10, C4xD5, D20, C5:D4, C2xC20, C5xD4, C5xQ8, C22xD5, C2xD12, C4oD12, S3xD4, Q8:3S3, C3xC4oD4, Dic15, C60, C60, D30, D30, C2xC30, C2xD20, C4oD20, D4xD5, Q8:2D5, C5xC4oD4, D4oD12, Dic30, C4xD15, D60, C15:7D4, C2xC60, D4xC15, Q8xC15, C22xD15, D4:8D10, C2xD60, D60:11C2, D4xD15, Q8:3D15, C15xC4oD4, D4:8D30
Quotients: C1, C2, C22, S3, C23, D5, D6, C24, D10, C22xS3, D15, 2+ 1+4, C22xD5, S3xC23, D30, C23xD5, D4oD12, C22xD15, D4:8D10, C23xD15, D4:8D30

Smallest permutation representation of D4:8D30
On 120 points
Generators in S120
(1 116 72 36)(2 117 73 37)(3 118 74 38)(4 119 75 39)(5 120 76 40)(6 91 77 41)(7 92 78 42)(8 93 79 43)(9 94 80 44)(10 95 81 45)(11 96 82 46)(12 97 83 47)(13 98 84 48)(14 99 85 49)(15 100 86 50)(16 101 87 51)(17 102 88 52)(18 103 89 53)(19 104 90 54)(20 105 61 55)(21 106 62 56)(22 107 63 57)(23 108 64 58)(24 109 65 59)(25 110 66 60)(26 111 67 31)(27 112 68 32)(28 113 69 33)(29 114 70 34)(30 115 71 35)
(1 36)(2 117)(3 38)(4 119)(5 40)(6 91)(7 42)(8 93)(9 44)(10 95)(11 46)(12 97)(13 48)(14 99)(15 50)(16 101)(17 52)(18 103)(19 54)(20 105)(21 56)(22 107)(23 58)(24 109)(25 60)(26 111)(27 32)(28 113)(29 34)(30 115)(31 67)(33 69)(35 71)(37 73)(39 75)(41 77)(43 79)(45 81)(47 83)(49 85)(51 87)(53 89)(55 61)(57 63)(59 65)(62 106)(64 108)(66 110)(68 112)(70 114)(72 116)(74 118)(76 120)(78 92)(80 94)(82 96)(84 98)(86 100)(88 102)(90 104)
(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)(31 120)(32 119)(33 118)(34 117)(35 116)(36 115)(37 114)(38 113)(39 112)(40 111)(41 110)(42 109)(43 108)(44 107)(45 106)(46 105)(47 104)(48 103)(49 102)(50 101)(51 100)(52 99)(53 98)(54 97)(55 96)(56 95)(57 94)(58 93)(59 92)(60 91)(61 82)(62 81)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(83 90)(84 89)(85 88)(86 87)

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

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

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

87 conjugacy classes

class 1 2A2B2C2D2E···2J 3 4A4B4C4D4E4F5A5B6A6B6C6D10A10B10C···10H12A12B12C12D12E15A15B15C15D20A20B20C20D20E···20J30A30B30C30D30E···30P60A···60H60I···60T
order122222···23444444556666101010···101212121212151515152020202020···203030303030···3060···6060···60
size1122230···30222223030222444224···422444222222224···422224···42···24···4

87 irreducible representations

dim1111112222222222224444
type++++++++++++++++++++++
imageC1C2C2C2C2C2S3D5D6D6D6D10D10D10D15D30D30D302+ 1+4D4oD12D4:8D10D4:8D30
kernelD4:8D30C2xD60D60:11C2D4xD15Q8:3D15C15xC4oD4C5xC4oD4C3xC4oD4C2xC20C5xD4C5xQ8C2xC12C3xD4C3xQ8C4oD4C2xC4D4Q8C15C5C3C1
# reps133621123316624121241248

Matrix representation of D4:8D30 in GL6(F61)

6000000
0600000
0000327
00005429
0032700
00542900
,
100000
010000
0000327
00005429
00295400
0073200
,
1600000
100000
00001760
000010
00176000
001000
,
100000
1600000
00001760
00004444
00176000
00444400

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,32,54,0,0,0,0,7,29,0,0,32,54,0,0,0,0,7,29,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,29,7,0,0,0,0,54,32,0,0,32,54,0,0,0,0,7,29,0,0],[1,1,0,0,0,0,60,0,0,0,0,0,0,0,0,0,17,1,0,0,0,0,60,0,0,0,17,1,0,0,0,0,60,0,0,0],[1,1,0,0,0,0,0,60,0,0,0,0,0,0,0,0,17,44,0,0,0,0,60,44,0,0,17,44,0,0,0,0,60,44,0,0] >;

D4:8D30 in GAP, Magma, Sage, TeX

D_4\rtimes_8D_{30}
% in TeX

G:=Group("D4:8D30");
// GroupNames label

G:=SmallGroup(480,1176);
// by ID

G=gap.SmallGroup(480,1176);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,675,80,2693,18822]);
// Polycyclic

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

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