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G = C23.7D30order 480 = 25·3·5

2nd non-split extension by C23 of D30 acting via D30/C15=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.7D30, C232Dic15, (C2×C4)⋊Dic15, (C2×C60)⋊13C4, (C6×D4).9D5, (C2×C30).2D4, (C22×C30)⋊2C4, (D4×C10).9S3, (C2×D4).3D15, (C2×C20)⋊7Dic3, (C2×C12)⋊2Dic5, C1513(C23⋊C4), (D4×C30).18C2, C32(C23⋊Dic5), (C22×C6)⋊2Dic5, C30.38D42C2, (C22×C10)⋊5Dic3, (C22×C10).31D6, (C22×C6).16D10, C54(C23.7D6), C22.2(C157D4), (C22×C30).7C22, C6.16(C23.D5), C22.3(C2×Dic15), C30.104(C22⋊C4), C2.5(C30.38D4), C10.27(C6.D4), (C2×C6).7(C5⋊D4), (C2×C30).174(C2×C4), (C2×C10).6(C3⋊D4), (C2×C6).31(C2×Dic5), (C2×C10).51(C2×Dic3), SmallGroup(480,194)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C23.7D30
C1C5C15C30C2×C30C22×C30C30.38D4 — C23.7D30
C15C30C2×C30 — C23.7D30
C1C2C23C2×D4

Generators and relations for C23.7D30
 G = < a,b,c,d,e | a2=b2=c2=d30=1, e2=ba=ab, dad-1=eae-1=ac=ca, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=bcd-1 >

Subgroups: 500 in 104 conjugacy classes, 39 normal (29 characteristic)
C1, C2, C2 [×4], C3, C4 [×3], C22, C22 [×2], C22 [×3], C5, C6, C6 [×4], C2×C4, C2×C4 [×2], D4 [×2], C23 [×2], C10, C10 [×4], Dic3 [×2], C12, C2×C6, C2×C6 [×2], C2×C6 [×3], C15, C22⋊C4 [×2], C2×D4, Dic5 [×2], C20, C2×C10, C2×C10 [×2], C2×C10 [×3], C2×Dic3 [×2], C2×C12, C3×D4 [×2], C22×C6 [×2], C30, C30 [×4], C23⋊C4, C2×Dic5 [×2], C2×C20, C5×D4 [×2], C22×C10 [×2], C6.D4 [×2], C6×D4, Dic15 [×2], C60, C2×C30, C2×C30 [×2], C2×C30 [×3], C23.D5 [×2], D4×C10, C23.7D6, C2×Dic15 [×2], C2×C60, D4×C15 [×2], C22×C30 [×2], C23⋊Dic5, C30.38D4 [×2], D4×C30, C23.7D30
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, Dic3 [×2], D6, C22⋊C4, Dic5 [×2], D10, C2×Dic3, C3⋊D4 [×2], D15, C23⋊C4, C2×Dic5, C5⋊D4 [×2], C6.D4, Dic15 [×2], D30, C23.D5, C23.7D6, C2×Dic15, C157D4 [×2], C23⋊Dic5, C30.38D4, C23.7D30

Smallest permutation representation of C23.7D30
On 120 points
Generators in S120
(1 44)(2 86)(3 46)(4 88)(5 48)(6 90)(7 50)(8 62)(9 52)(10 64)(11 54)(12 66)(13 56)(14 68)(15 58)(16 70)(17 60)(18 72)(19 32)(20 74)(21 34)(22 76)(23 36)(24 78)(25 38)(26 80)(27 40)(28 82)(29 42)(30 84)(31 100)(33 102)(35 104)(37 106)(39 108)(41 110)(43 112)(45 114)(47 116)(49 118)(51 120)(53 92)(55 94)(57 96)(59 98)(61 119)(63 91)(65 93)(67 95)(69 97)(71 99)(73 101)(75 103)(77 105)(79 107)(81 109)(83 111)(85 113)(87 115)(89 117)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(61 119)(62 120)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)(85 113)(86 114)(87 115)(88 116)(89 117)(90 118)
(1 113)(2 114)(3 115)(4 116)(5 117)(6 118)(7 119)(8 120)(9 91)(10 92)(11 93)(12 94)(13 95)(14 96)(15 97)(16 98)(17 99)(18 100)(19 101)(20 102)(21 103)(22 104)(23 105)(24 106)(25 107)(26 108)(27 109)(28 110)(29 111)(30 112)(31 72)(32 73)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 81)(41 82)(42 83)(43 84)(44 85)(45 86)(46 87)(47 88)(48 89)(49 90)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)
(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 15)(2 57 114 68)(3 13)(4 55 116 66)(5 11)(6 53 118 64)(7 9)(8 51 120 62)(10 49 92 90)(12 47 94 88)(14 45 96 86)(16 43 98 84)(17 29)(18 41 100 82)(19 27)(20 39 102 80)(21 25)(22 37 104 78)(24 35 106 76)(26 33 108 74)(28 31 110 72)(30 59 112 70)(32 81)(34 79)(36 77)(38 75)(40 73)(42 71)(44 69)(46 67)(48 65)(50 63)(52 61)(54 89)(56 87)(58 85)(60 83)(91 119)(93 117)(95 115)(97 113)(99 111)(101 109)(103 107)

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

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

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

81 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E5A5B6A6B6C6D6E6F6G10A···10F10G···10N12A12B15A15B15C15D20A20B20C20D30A···30L30M···30AB60A···60H
order12222234444455666666610···1010···101212151515152020202030···3030···3060···60
size11222424606060602222244442···24···444222244442···24···44···4

81 irreducible representations

dim1111122222222222222224444
type++++++--+--++--++
imageC1C2C2C4C4S3D4D5Dic3Dic3D6Dic5Dic5D10C3⋊D4D15C5⋊D4Dic15Dic15D30C157D4C23⋊C4C23.7D6C23⋊Dic5C23.7D30
kernelC23.7D30C30.38D4D4×C30C2×C60C22×C30D4×C10C2×C30C6×D4C2×C20C22×C10C22×C10C2×C12C22×C6C22×C6C2×C10C2×D4C2×C6C2×C4C23C23C22C15C5C3C1
# reps12122122111222448444161248

Matrix representation of C23.7D30 in GL4(𝔽61) generated by

144400
334700
59593117
204430
,
144400
334700
2803044
33331731
,
60000
06000
00600
00060
,
4392511
34114814
58351122
58231235
,
335600
592800
47345629
351585
G:=sub<GL(4,GF(61))| [14,33,59,2,44,47,59,0,0,0,31,44,0,0,17,30],[14,33,28,33,44,47,0,33,0,0,30,17,0,0,44,31],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[4,34,58,58,39,11,35,23,25,48,11,12,11,14,22,35],[33,59,47,3,56,28,34,51,0,0,56,58,0,0,29,5] >;

C23.7D30 in GAP, Magma, Sage, TeX

C_2^3._7D_{30}
% in TeX

G:=Group("C2^3.7D30");
// GroupNames label

G:=SmallGroup(480,194);
// by ID

G=gap.SmallGroup(480,194);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,219,675,2693,18822]);
// Polycyclic

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

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