Copied to
clipboard

## G = C32⋊3Dic10order 360 = 23·32·5

### 2nd semidirect product of C32 and Dic10 acting via Dic10/C10=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C30 — C32⋊3Dic10
 Chief series C1 — C5 — C15 — C3×C15 — C3×C30 — C3×Dic15 — C32⋊3Dic10
 Lower central C3×C15 — C3×C30 — C32⋊3Dic10
 Upper central C1 — C2

Generators and relations for C323Dic10
G = < a,b,c,d | a3=b3=c20=1, d2=c10, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 260 in 54 conjugacy classes, 23 normal (11 characteristic)
C1, C2, C3, C3, C4, C5, C6, C6, Q8, C32, C10, Dic3, C12, C15, C15, C3×C6, Dic5, C20, Dic6, C30, C30, C3×Dic3, C3⋊Dic3, Dic10, C3×C15, C5×Dic3, C3×Dic5, Dic15, C322Q8, C3×C30, C15⋊Q8, C3×Dic15, C5×C3⋊Dic3, C323Dic10
Quotients: C1, C2, C22, S3, Q8, D5, D6, D10, Dic6, S32, Dic10, S3×D5, C322Q8, C15⋊Q8, D15⋊S3, C323Dic10

Smallest permutation representation of C323Dic10
On 120 points
Generators in S120
(1 21 50)(2 51 22)(3 23 52)(4 53 24)(5 25 54)(6 55 26)(7 27 56)(8 57 28)(9 29 58)(10 59 30)(11 31 60)(12 41 32)(13 33 42)(14 43 34)(15 35 44)(16 45 36)(17 37 46)(18 47 38)(19 39 48)(20 49 40)(61 92 106)(62 107 93)(63 94 108)(64 109 95)(65 96 110)(66 111 97)(67 98 112)(68 113 99)(69 100 114)(70 115 81)(71 82 116)(72 117 83)(73 84 118)(74 119 85)(75 86 120)(76 101 87)(77 88 102)(78 103 89)(79 90 104)(80 105 91)
(1 50 21)(2 22 51)(3 52 23)(4 24 53)(5 54 25)(6 26 55)(7 56 27)(8 28 57)(9 58 29)(10 30 59)(11 60 31)(12 32 41)(13 42 33)(14 34 43)(15 44 35)(16 36 45)(17 46 37)(18 38 47)(19 48 39)(20 40 49)(61 92 106)(62 107 93)(63 94 108)(64 109 95)(65 96 110)(66 111 97)(67 98 112)(68 113 99)(69 100 114)(70 115 81)(71 82 116)(72 117 83)(73 84 118)(74 119 85)(75 86 120)(76 101 87)(77 88 102)(78 103 89)(79 90 104)(80 105 91)
(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 80 11 70)(2 79 12 69)(3 78 13 68)(4 77 14 67)(5 76 15 66)(6 75 16 65)(7 74 17 64)(8 73 18 63)(9 72 19 62)(10 71 20 61)(21 91 31 81)(22 90 32 100)(23 89 33 99)(24 88 34 98)(25 87 35 97)(26 86 36 96)(27 85 37 95)(28 84 38 94)(29 83 39 93)(30 82 40 92)(41 114 51 104)(42 113 52 103)(43 112 53 102)(44 111 54 101)(45 110 55 120)(46 109 56 119)(47 108 57 118)(48 107 58 117)(49 106 59 116)(50 105 60 115)

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

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

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

39 conjugacy classes

 class 1 2 3A 3B 3C 4A 4B 4C 5A 5B 6A 6B 6C 10A 10B 12A 12B 12C 12D 15A ··· 15H 20A 20B 20C 20D 30A ··· 30H order 1 2 3 3 3 4 4 4 5 5 6 6 6 10 10 12 12 12 12 15 ··· 15 20 20 20 20 30 ··· 30 size 1 1 2 2 4 18 30 30 2 2 2 2 4 2 2 30 30 30 30 4 ··· 4 18 18 18 18 4 ··· 4

39 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + - + + + - - + + - - image C1 C2 C2 S3 Q8 D5 D6 D10 Dic6 Dic10 S32 S3×D5 C32⋊2Q8 C15⋊Q8 D15⋊S3 C32⋊3Dic10 kernel C32⋊3Dic10 C3×Dic15 C5×C3⋊Dic3 Dic15 C3×C15 C3⋊Dic3 C30 C3×C6 C15 C32 C10 C6 C5 C3 C2 C1 # reps 1 2 1 2 1 2 2 2 4 4 1 4 1 4 4 4

Matrix representation of C323Dic10 in GL8(𝔽61)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 1 60
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 60 60 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 60 0 0 0 0 0 0 1 18 0 0 0 0 0 0 0 0 60 59 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 27 25 0 0 0 0 0 0 27 34 0 0 0 0 0 0 0 0 9 60 0 0 0 0 0 0 21 52 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,60,18,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,59,1,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[27,27,0,0,0,0,0,0,25,34,0,0,0,0,0,0,0,0,9,21,0,0,0,0,0,0,60,52,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C323Dic10 in GAP, Magma, Sage, TeX

C_3^2\rtimes_3{\rm Dic}_{10}
% in TeX

G:=Group("C3^2:3Dic10");
// GroupNames label

G:=SmallGroup(360,88);
// by ID

G=gap.SmallGroup(360,88);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,73,31,387,201,730,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^20=1,d^2=c^10,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽