Copied to
clipboard

G = C12.28D10order 240 = 24·3·5

7th non-split extension by C12 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D609C2, Dic65D5, C20.18D6, D10.11D6, C12.28D10, C60.21C22, C30.10C23, Dic5.20D6, Dic3.4D10, D30.2C22, (C4×D5)⋊2S3, (D5×C12)⋊2C2, C157(C4○D4), C51(C4○D12), C3⋊D202C2, C4.14(S3×D5), D30.C22C2, (C5×Dic6)⋊3C2, C31(Q82D5), C6.10(C22×D5), C10.10(C22×S3), (C6×D5).12C22, (C5×Dic3).4C22, (C3×Dic5).14C22, C2.14(C2×S3×D5), SmallGroup(240,134)

Series: Derived Chief Lower central Upper central

C1C30 — C12.28D10
C1C5C15C30C6×D5C3⋊D20 — C12.28D10
C15C30 — C12.28D10
C1C2C4

Generators and relations for C12.28D10
 G = < a,b,c | a12=c2=1, b10=a6, bab-1=cac=a-1, cbc=a6b9 >

Subgroups: 384 in 80 conjugacy classes, 32 normal (22 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3 [×2], C6, C6, C2×C4 [×3], D4 [×3], Q8, D5 [×3], C10, Dic3 [×2], C12, C12, D6 [×2], C2×C6, C15, C4○D4, Dic5, C20, C20 [×2], D10, D10 [×2], Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12, C3×D5, D15 [×2], C30, C4×D5, C4×D5 [×2], D20 [×3], C5×Q8, C4○D12, C5×Dic3 [×2], C3×Dic5, C60, C6×D5, D30 [×2], Q82D5, D30.C2 [×2], C3⋊D20 [×2], D5×C12, C5×Dic6, D60, C12.28D10
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4, D10 [×3], C22×S3, C22×D5, C4○D12, S3×D5, Q82D5, C2×S3×D5, C12.28D10

Smallest permutation representation of C12.28D10
On 120 points
Generators in S120
(1 59 97 38 80 119 11 49 87 28 70 109)(2 110 71 29 88 50 12 120 61 39 98 60)(3 41 99 40 62 101 13 51 89 30 72 111)(4 112 73 31 90 52 14 102 63 21 100 42)(5 43 81 22 64 103 15 53 91 32 74 113)(6 114 75 33 92 54 16 104 65 23 82 44)(7 45 83 24 66 105 17 55 93 34 76 115)(8 116 77 35 94 56 18 106 67 25 84 46)(9 47 85 26 68 107 19 57 95 36 78 117)(10 118 79 37 96 58 20 108 69 27 86 48)
(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 5)(2 4)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(21 29)(22 28)(23 27)(24 26)(30 40)(31 39)(32 38)(33 37)(34 36)(41 111)(42 110)(43 109)(44 108)(45 107)(46 106)(47 105)(48 104)(49 103)(50 102)(51 101)(52 120)(53 119)(54 118)(55 117)(56 116)(57 115)(58 114)(59 113)(60 112)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 100)(72 99)(73 98)(74 97)(75 96)(76 95)(77 94)(78 93)(79 92)(80 91)

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

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

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

C12.28D10 is a maximal subgroup of
D602C4  D605C4  C24⋊D10  C24.2D10  C40.31D6  D120⋊C2  D20.9D6  C60.16C23  C60.39C23  D20.D6  D60.C4  Dic6.F5  D20.38D6  D5×C4○D12  D2029D6  D30.C23  D2014D6  C30.33C24  S3×Q82D5
C12.28D10 is a maximal quotient of
Dic5×Dic6  C4⋊Dic3⋊D5  Dic3.Dic10  Dic3⋊C4⋊D5  D30.D4  (C4×D5)⋊Dic3  C60.67D4  (C2×C60).C22  C60.70D4  (C4×Dic5)⋊S3  C20.Dic6  D30.C2⋊C4  D6017C4  D303Q8  D30.6D4  C1520(C4×D4)  C127D20  (C2×Dic6)⋊D5  D302D4

36 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B6A6B6C10A10B12A12B12C12D15A15B20A20B20C20D20E20F30A30B60A60B60C60D
order12222344444556661010121212121515202020202020303060606060
size11103030225566222101022221010444412121212444444

36 irreducible representations

dim1111112222222224444
type+++++++++++++++++
imageC1C2C2C2C2C2S3D5D6D6D6C4○D4D10D10C4○D12S3×D5Q82D5C2×S3×D5C12.28D10
kernelC12.28D10D30.C2C3⋊D20D5×C12C5×Dic6D60C4×D5Dic6Dic5C20D10C15Dic3C12C5C4C3C2C1
# reps1221111211124242224

Matrix representation of C12.28D10 in GL6(𝔽61)

60150000
1220000
001000
000100
0000603
0000401
,
100000
49600000
00444400
00176000
00005033
0000011
,
100000
49600000
00444400
00601700
000010
00002160

G:=sub<GL(6,GF(61))| [60,12,0,0,0,0,15,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,40,0,0,0,0,3,1],[1,49,0,0,0,0,0,60,0,0,0,0,0,0,44,17,0,0,0,0,44,60,0,0,0,0,0,0,50,0,0,0,0,0,33,11],[1,49,0,0,0,0,0,60,0,0,0,0,0,0,44,60,0,0,0,0,44,17,0,0,0,0,0,0,1,21,0,0,0,0,0,60] >;

C12.28D10 in GAP, Magma, Sage, TeX

C_{12}._{28}D_{10}
% in TeX

G:=Group("C12.28D10");
// GroupNames label

G:=SmallGroup(240,134);
// by ID

G=gap.SmallGroup(240,134);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,121,55,116,50,490,6917]);
// Polycyclic

G:=Group<a,b,c|a^12=c^2=1,b^10=a^6,b*a*b^-1=c*a*c=a^-1,c*b*c=a^6*b^9>;
// generators/relations

׿
×
𝔽