Copied to
clipboard

G = (C2×C6).D20order 480 = 25·3·5

2nd non-split extension by C2×C6 of D20 acting via D20/C10=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C6).2D20, C157(C23⋊C4), (C6×Dic5)⋊1C4, (C2×C30).36D4, C23.8(S3×D5), C6.D41D5, (C2×Dic5)⋊1Dic3, (C22×D5)⋊3Dic3, (C22×C10).27D6, (C22×C6).12D10, C53(C23.7D6), C30.38D414C2, C22.5(D5×Dic3), C30.66(C22⋊C4), C22.8(C15⋊D4), C22.8(C3⋊D20), C33(C23.1D10), C6.32(D10⋊C4), (C22×C30).26C22, C10.22(C6.D4), C2.11(D10⋊Dic3), (D5×C2×C6)⋊1C4, (C2×C6).49(C4×D5), (C6×C5⋊D4).1C2, (C2×C5⋊D4).1S3, (C2×C30).90(C2×C4), (C2×C6).4(C5⋊D4), (C5×C6.D4)⋊1C2, (C2×C10).50(C3⋊D4), (C2×C10).23(C2×Dic3), SmallGroup(480,71)

Series: Derived Chief Lower central Upper central

C1C2×C30 — (C2×C6).D20
C1C5C15C30C2×C30C22×C30C6×C5⋊D4 — (C2×C6).D20
C15C30C2×C30 — (C2×C6).D20
C1C2C23

Generators and relations for (C2×C6).D20
 G = < a,b,c,d | a2=b6=c20=1, d2=a, ab=ba, cac-1=ab3, ad=da, cbc-1=dbd-1=b-1, dcd-1=ac-1 >

Subgroups: 540 in 104 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2 [×4], C3, C4 [×3], C22 [×3], C22 [×3], C5, C6, C6 [×4], C2×C4 [×3], D4 [×2], C23, C23, D5, C10, C10 [×3], Dic3 [×2], C12, C2×C6 [×3], C2×C6 [×3], C15, C22⋊C4 [×2], C2×D4, Dic5 [×2], C20, D10 [×2], C2×C10 [×3], C2×C10, C2×Dic3 [×2], C2×C12, C3×D4 [×2], C22×C6, C22×C6, C3×D5, C30, C30 [×3], C23⋊C4, C2×Dic5, C2×Dic5, C5⋊D4 [×2], C2×C20, C22×D5, C22×C10, C6.D4, C6.D4, C6×D4, C5×Dic3, C3×Dic5, Dic15, C6×D5 [×2], C2×C30 [×3], C2×C30, C23.D5, C5×C22⋊C4, C2×C5⋊D4, C23.7D6, C6×Dic5, C3×C5⋊D4 [×2], C10×Dic3, C2×Dic15, D5×C2×C6, C22×C30, C23.1D10, C5×C6.D4, C30.38D4, C6×C5⋊D4, (C2×C6).D20
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, Dic3 [×2], D6, C22⋊C4, D10, C2×Dic3, C3⋊D4 [×2], C23⋊C4, C4×D5, D20, C5⋊D4, C6.D4, S3×D5, D10⋊C4, C23.7D6, D5×Dic3, C15⋊D4, C3⋊D20, C23.1D10, D10⋊Dic3, (C2×C6).D20

Smallest permutation representation of (C2×C6).D20
On 120 points
Generators in S120
(1 27)(3 29)(5 31)(7 33)(9 35)(11 37)(13 39)(15 21)(17 23)(19 25)(42 75)(44 77)(46 79)(48 61)(50 63)(52 65)(54 67)(56 69)(58 71)(60 73)(81 102)(83 104)(85 106)(87 108)(89 110)(91 112)(93 114)(95 116)(97 118)(99 120)
(1 58 97 27 71 118)(2 119 72 28 98 59)(3 60 99 29 73 120)(4 101 74 30 100 41)(5 42 81 31 75 102)(6 103 76 32 82 43)(7 44 83 33 77 104)(8 105 78 34 84 45)(9 46 85 35 79 106)(10 107 80 36 86 47)(11 48 87 37 61 108)(12 109 62 38 88 49)(13 50 89 39 63 110)(14 111 64 40 90 51)(15 52 91 21 65 112)(16 113 66 22 92 53)(17 54 93 23 67 114)(18 115 68 24 94 55)(19 56 95 25 69 116)(20 117 70 26 96 57)
(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 21 27 15)(2 40)(3 13 29 39)(4 12)(5 37 31 11)(6 36)(7 9 33 35)(10 32)(14 28)(16 20)(17 25 23 19)(18 24)(22 26)(30 38)(41 109)(42 87 75 108)(43 86)(44 106 77 85)(45 105)(46 83 79 104)(47 82)(48 102 61 81)(49 101)(50 99 63 120)(51 98)(52 118 65 97)(53 117)(54 95 67 116)(55 94)(56 114 69 93)(57 113)(58 91 71 112)(59 90)(60 110 73 89)(62 100)(64 119)(66 96)(68 115)(70 92)(72 111)(74 88)(76 107)(78 84)(80 103)

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

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

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

57 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E5A5B6A6B6C6D6E6F6G10A···10F10G10H10I10J12A12B15A15B20A···20H30A···30N
order12222234444455666666610···10101010101212151520···2030···30
size112222021212206060222224420202···2444420204412···124···4

57 irreducible representations

dim1111112222222222244444444
type+++++++--+++++--+
imageC1C2C2C2C4C4S3D4D5Dic3Dic3D6D10C3⋊D4C4×D5D20C5⋊D4C23⋊C4S3×D5C23.7D6D5×Dic3C15⋊D4C3⋊D20C23.1D10(C2×C6).D20
kernel(C2×C6).D20C5×C6.D4C30.38D4C6×C5⋊D4C6×Dic5D5×C2×C6C2×C5⋊D4C2×C30C6.D4C2×Dic5C22×D5C22×C10C22×C6C2×C10C2×C6C2×C6C2×C6C15C23C5C22C22C22C3C1
# reps1111221221112444412222248

Matrix representation of (C2×C6).D20 in GL4(𝔽61) generated by

60000
06000
334410
334401
,
254500
493700
14535316
4128459
,
36486042
59381817
60581731
13271731
,
474400
81400
412501
412510
G:=sub<GL(4,GF(61))| [60,0,33,33,0,60,44,44,0,0,1,0,0,0,0,1],[25,49,14,41,45,37,53,28,0,0,53,45,0,0,16,9],[36,59,60,13,48,38,58,27,60,18,17,17,42,17,31,31],[47,8,41,41,44,14,25,25,0,0,0,1,0,0,1,0] >;

(C2×C6).D20 in GAP, Magma, Sage, TeX

(C_2\times C_6).D_{20}
% in TeX

G:=Group("(C2xC6).D20");
// GroupNames label

G:=SmallGroup(480,71);
// by ID

G=gap.SmallGroup(480,71);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,422,346,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^20=1,d^2=a,a*b=b*a,c*a*c^-1=a*b^3,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a*c^-1>;
// generators/relations

׿
×
𝔽