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G = C23.6D30order 480 = 25·3·5

1st non-split extension by C23 of D30 acting via D30/C15=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.6D30, C22.2D60, (C2×C6).3D20, C22⋊C41D15, (C2×C10).3D12, C1510(C23⋊C4), (C2×Dic15)⋊1C4, (C2×C30).134D4, (C22×D15)⋊1C4, C22.3(C4×D15), C10.28(D6⋊C4), C30.38D41C2, (C22×C10).30D6, (C22×C6).15D10, C54(C23.6D6), C2.4(D303C4), C30.70(C22⋊C4), C32(C23.1D10), C22.8(C157D4), (C22×C30).6C22, C6.13(D10⋊C4), (C2×C6).4(C4×D5), (C5×C22⋊C4)⋊2S3, (C3×C22⋊C4)⋊2D5, (C15×C22⋊C4)⋊3C2, (C2×C10).27(C4×S3), (C2×C30).64(C2×C4), (C2×C157D4).1C2, (C2×C6).66(C5⋊D4), (C2×C10).66(C3⋊D4), SmallGroup(480,166)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C23.6D30
C1C5C15C30C2×C30C22×C30C2×C157D4 — C23.6D30
C15C30C2×C30 — C23.6D30
C1C2C23C22⋊C4

Generators and relations for C23.6D30
 G = < a,b,c,d,e | a2=b2=c2=1, d30=a, e2=abc, ab=ba, eae-1=ac=ca, ad=da, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=bcd29 >

Subgroups: 740 in 104 conjugacy classes, 33 normal (all characteristic)
C1, C2, C2 [×4], C3, C4 [×3], C22 [×3], C22 [×3], C5, S3, C6, C6 [×3], C2×C4 [×3], D4 [×2], C23, C23, D5, C10, C10 [×3], Dic3 [×2], C12, D6 [×2], C2×C6 [×3], C2×C6, C15, C22⋊C4, C22⋊C4, C2×D4, Dic5 [×2], C20, D10 [×2], C2×C10 [×3], C2×C10, C2×Dic3 [×2], C3⋊D4 [×2], C2×C12, C22×S3, C22×C6, D15, C30, C30 [×3], C23⋊C4, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20, C22×D5, C22×C10, C6.D4, C3×C22⋊C4, C2×C3⋊D4, Dic15 [×2], C60, D30 [×2], C2×C30 [×3], C2×C30, C23.D5, C5×C22⋊C4, C2×C5⋊D4, C23.6D6, C2×Dic15, C2×Dic15, C157D4 [×2], C2×C60, C22×D15, C22×C30, C23.1D10, C30.38D4, C15×C22⋊C4, C2×C157D4, C23.6D30
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, D15, C23⋊C4, C4×D5, D20, C5⋊D4, D6⋊C4, D30, D10⋊C4, C23.6D6, C4×D15, D60, C157D4, C23.1D10, D303C4, C23.6D30

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

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

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

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

81 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E5A5B6A6B6C6D6E10A···10F10G10H10I10J12A12B12C12D15A15B15C15D20A···20H30A···30L30M···30T60A···60P
order122222344444556666610···1010101010121212121515151520···2030···3030···3060···60
size112226024460606022222442···24444444422224···42···24···44···4

81 irreducible representations

dim11111122222222222222224444
type+++++++++++++++
imageC1C2C2C2C4C4S3D4D5D6D10C4×S3D12C3⋊D4D15C4×D5D20C5⋊D4D30C4×D15D60C157D4C23⋊C4C23.6D6C23.1D10C23.6D30
kernelC23.6D30C30.38D4C15×C22⋊C4C2×C157D4C2×Dic15C22×D15C5×C22⋊C4C2×C30C3×C22⋊C4C22×C10C22×C6C2×C10C2×C10C2×C10C22⋊C4C2×C6C2×C6C2×C6C23C22C22C22C15C5C3C1
# reps11112212212222444448881248

Matrix representation of C23.6D30 in GL4(𝔽61) generated by

474400
331400
4403044
45171731
,
141700
284700
16443044
1701731
,
60000
06000
00600
00060
,
2152577
38365060
1952569
2450569
,
285600
593300
3515582
2818563
G:=sub<GL(4,GF(61))| [47,33,44,45,44,14,0,17,0,0,30,17,0,0,44,31],[14,28,16,17,17,47,44,0,0,0,30,17,0,0,44,31],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[21,38,19,24,52,36,52,50,57,50,56,56,7,60,9,9],[28,59,35,28,56,33,15,18,0,0,58,56,0,0,2,3] >;

C23.6D30 in GAP, Magma, Sage, TeX

C_2^3._6D_{30}
% in TeX

G:=Group("C2^3.6D30");
// GroupNames label

G:=SmallGroup(480,166);
// by ID

G=gap.SmallGroup(480,166);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,422,346,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^30=a,e^2=a*b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^29>;
// generators/relations

׿
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