C3xC7^2:C3 - GroupNames

G = C₃×C₇²⋊C₃ order 441 = 3²·7²

Direct product of C₃ and C₇²⋊C₃

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C₃×C₇²⋊C₃, C₇²⋊₃C₃², C₂₁⋊₁(C₇⋊C₃), (C₇×C₂₁)⋊₂C₃, C₇⋊₂(C₃×C₇⋊C₃), SmallGroup(441,11)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₇² — C₃×C₇²⋊C₃

Chief series

C₁ — C₇ — C₇² — C₇²⋊C₃ — C₃×C₇²⋊C₃

Lower central

C₇² — C₃×C₇²⋊C₃

Upper central

C₁ — C₃

Generators and relations for C₃×C₇²⋊C₃
G = < a,b,c,d | a³=b⁷=c⁷=d³=1, ab=ba, ac=ca, ad=da, bc=cb, dbd^-1=b⁴, dcd^-1=c⁴ >

Subgroups: 444 in 60 conjugacy classes, 24 normal (5 characteristic)
C₁, C₃, C₃, C₇, C₃², C₇⋊C₃, C₂₁, C₇², C₃×C₇⋊C₃, C₇²⋊C₃, C₇×C₂₁, C₃×C₇²⋊C₃
Quotients: C₁, C₃, C₃², C₇⋊C₃, C₃×C₇⋊C₃, C₇²⋊C₃, C₃×C₇²⋊C₃

Smallest permutation representation of C₃×C₇²⋊C₃
►On 147 points

Generators in S₁₄₇

(1 145 96)(2 146 97)(3 147 98)(4 141 92)(5 142 93)(6 143 94)(7 144 95)(8 105 56)(9 99 50)(10 100 51)(11 101 52)(12 102 53)(13 103 54)(14 104 55)(15 112 63)(16 106 57)(17 107 58)(18 108 59)(19 109 60)(20 110 61)(21 111 62)(22 119 70)(23 113 64)(24 114 65)(25 115 66)(26 116 67)(27 117 68)(28 118 69)(29 126 77)(30 120 71)(31 121 72)(32 122 73)(33 123 74)(34 124 75)(35 125 76)(36 133 84)(37 127 78)(38 128 79)(39 129 80)(40 130 81)(41 131 82)(42 132 83)(43 140 91)(44 134 85)(45 135 86)(46 136 87)(47 137 88)(48 138 89)(49 139 90)

(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)(120 121 122 123 124 125 126)(127 128 129 130 131 132 133)(134 135 136 137 138 139 140)(141 142 143 144 145 146 147)

(1 90 82 75 68 61 54)(2 91 83 76 69 62 55)(3 85 84 77 70 63 56)(4 86 78 71 64 57 50)(5 87 79 72 65 58 51)(6 88 80 73 66 59 52)(7 89 81 74 67 60 53)(8 147 44 36 29 22 15)(9 141 45 37 30 23 16)(10 142 46 38 31 24 17)(11 143 47 39 32 25 18)(12 144 48 40 33 26 19)(13 145 49 41 34 27 20)(14 146 43 42 35 28 21)(92 135 127 120 113 106 99)(93 136 128 121 114 107 100)(94 137 129 122 115 108 101)(95 138 130 123 116 109 102)(96 139 131 124 117 110 103)(97 140 132 125 118 111 104)(98 134 133 126 119 112 105)

(1 145 96)(2 147 93)(3 142 97)(4 144 94)(5 146 98)(6 141 95)(7 143 92)(8 107 76)(9 109 73)(10 111 77)(11 106 74)(12 108 71)(13 110 75)(14 112 72)(15 121 55)(16 123 52)(17 125 56)(18 120 53)(19 122 50)(20 124 54)(21 126 51)(22 136 83)(23 138 80)(24 140 84)(25 135 81)(26 137 78)(27 139 82)(28 134 79)(29 100 62)(30 102 59)(31 104 63)(32 99 60)(33 101 57)(34 103 61)(35 105 58)(36 114 91)(37 116 88)(38 118 85)(39 113 89)(40 115 86)(41 117 90)(42 119 87)(43 133 65)(44 128 69)(45 130 66)(46 132 70)(47 127 67)(48 129 64)(49 131 68)

G:=sub<Sym(147)| (1,145,96)(2,146,97)(3,147,98)(4,141,92)(5,142,93)(6,143,94)(7,144,95)(8,105,56)(9,99,50)(10,100,51)(11,101,52)(12,102,53)(13,103,54)(14,104,55)(15,112,63)(16,106,57)(17,107,58)(18,108,59)(19,109,60)(20,110,61)(21,111,62)(22,119,70)(23,113,64)(24,114,65)(25,115,66)(26,116,67)(27,117,68)(28,118,69)(29,126,77)(30,120,71)(31,121,72)(32,122,73)(33,123,74)(34,124,75)(35,125,76)(36,133,84)(37,127,78)(38,128,79)(39,129,80)(40,130,81)(41,131,82)(42,132,83)(43,140,91)(44,134,85)(45,135,86)(46,136,87)(47,137,88)(48,138,89)(49,139,90), (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)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147), (1,90,82,75,68,61,54)(2,91,83,76,69,62,55)(3,85,84,77,70,63,56)(4,86,78,71,64,57,50)(5,87,79,72,65,58,51)(6,88,80,73,66,59,52)(7,89,81,74,67,60,53)(8,147,44,36,29,22,15)(9,141,45,37,30,23,16)(10,142,46,38,31,24,17)(11,143,47,39,32,25,18)(12,144,48,40,33,26,19)(13,145,49,41,34,27,20)(14,146,43,42,35,28,21)(92,135,127,120,113,106,99)(93,136,128,121,114,107,100)(94,137,129,122,115,108,101)(95,138,130,123,116,109,102)(96,139,131,124,117,110,103)(97,140,132,125,118,111,104)(98,134,133,126,119,112,105), (1,145,96)(2,147,93)(3,142,97)(4,144,94)(5,146,98)(6,141,95)(7,143,92)(8,107,76)(9,109,73)(10,111,77)(11,106,74)(12,108,71)(13,110,75)(14,112,72)(15,121,55)(16,123,52)(17,125,56)(18,120,53)(19,122,50)(20,124,54)(21,126,51)(22,136,83)(23,138,80)(24,140,84)(25,135,81)(26,137,78)(27,139,82)(28,134,79)(29,100,62)(30,102,59)(31,104,63)(32,99,60)(33,101,57)(34,103,61)(35,105,58)(36,114,91)(37,116,88)(38,118,85)(39,113,89)(40,115,86)(41,117,90)(42,119,87)(43,133,65)(44,128,69)(45,130,66)(46,132,70)(47,127,67)(48,129,64)(49,131,68)>;

G:=Group( (1,145,96)(2,146,97)(3,147,98)(4,141,92)(5,142,93)(6,143,94)(7,144,95)(8,105,56)(9,99,50)(10,100,51)(11,101,52)(12,102,53)(13,103,54)(14,104,55)(15,112,63)(16,106,57)(17,107,58)(18,108,59)(19,109,60)(20,110,61)(21,111,62)(22,119,70)(23,113,64)(24,114,65)(25,115,66)(26,116,67)(27,117,68)(28,118,69)(29,126,77)(30,120,71)(31,121,72)(32,122,73)(33,123,74)(34,124,75)(35,125,76)(36,133,84)(37,127,78)(38,128,79)(39,129,80)(40,130,81)(41,131,82)(42,132,83)(43,140,91)(44,134,85)(45,135,86)(46,136,87)(47,137,88)(48,138,89)(49,139,90), (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)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147), (1,90,82,75,68,61,54)(2,91,83,76,69,62,55)(3,85,84,77,70,63,56)(4,86,78,71,64,57,50)(5,87,79,72,65,58,51)(6,88,80,73,66,59,52)(7,89,81,74,67,60,53)(8,147,44,36,29,22,15)(9,141,45,37,30,23,16)(10,142,46,38,31,24,17)(11,143,47,39,32,25,18)(12,144,48,40,33,26,19)(13,145,49,41,34,27,20)(14,146,43,42,35,28,21)(92,135,127,120,113,106,99)(93,136,128,121,114,107,100)(94,137,129,122,115,108,101)(95,138,130,123,116,109,102)(96,139,131,124,117,110,103)(97,140,132,125,118,111,104)(98,134,133,126,119,112,105), (1,145,96)(2,147,93)(3,142,97)(4,144,94)(5,146,98)(6,141,95)(7,143,92)(8,107,76)(9,109,73)(10,111,77)(11,106,74)(12,108,71)(13,110,75)(14,112,72)(15,121,55)(16,123,52)(17,125,56)(18,120,53)(19,122,50)(20,124,54)(21,126,51)(22,136,83)(23,138,80)(24,140,84)(25,135,81)(26,137,78)(27,139,82)(28,134,79)(29,100,62)(30,102,59)(31,104,63)(32,99,60)(33,101,57)(34,103,61)(35,105,58)(36,114,91)(37,116,88)(38,118,85)(39,113,89)(40,115,86)(41,117,90)(42,119,87)(43,133,65)(44,128,69)(45,130,66)(46,132,70)(47,127,67)(48,129,64)(49,131,68) );

G=PermutationGroup([[(1,145,96),(2,146,97),(3,147,98),(4,141,92),(5,142,93),(6,143,94),(7,144,95),(8,105,56),(9,99,50),(10,100,51),(11,101,52),(12,102,53),(13,103,54),(14,104,55),(15,112,63),(16,106,57),(17,107,58),(18,108,59),(19,109,60),(20,110,61),(21,111,62),(22,119,70),(23,113,64),(24,114,65),(25,115,66),(26,116,67),(27,117,68),(28,118,69),(29,126,77),(30,120,71),(31,121,72),(32,122,73),(33,123,74),(34,124,75),(35,125,76),(36,133,84),(37,127,78),(38,128,79),(39,129,80),(40,130,81),(41,131,82),(42,132,83),(43,140,91),(44,134,85),(45,135,86),(46,136,87),(47,137,88),(48,138,89),(49,139,90)], [(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),(120,121,122,123,124,125,126),(127,128,129,130,131,132,133),(134,135,136,137,138,139,140),(141,142,143,144,145,146,147)], [(1,90,82,75,68,61,54),(2,91,83,76,69,62,55),(3,85,84,77,70,63,56),(4,86,78,71,64,57,50),(5,87,79,72,65,58,51),(6,88,80,73,66,59,52),(7,89,81,74,67,60,53),(8,147,44,36,29,22,15),(9,141,45,37,30,23,16),(10,142,46,38,31,24,17),(11,143,47,39,32,25,18),(12,144,48,40,33,26,19),(13,145,49,41,34,27,20),(14,146,43,42,35,28,21),(92,135,127,120,113,106,99),(93,136,128,121,114,107,100),(94,137,129,122,115,108,101),(95,138,130,123,116,109,102),(96,139,131,124,117,110,103),(97,140,132,125,118,111,104),(98,134,133,126,119,112,105)], [(1,145,96),(2,147,93),(3,142,97),(4,144,94),(5,146,98),(6,141,95),(7,143,92),(8,107,76),(9,109,73),(10,111,77),(11,106,74),(12,108,71),(13,110,75),(14,112,72),(15,121,55),(16,123,52),(17,125,56),(18,120,53),(19,122,50),(20,124,54),(21,126,51),(22,136,83),(23,138,80),(24,140,84),(25,135,81),(26,137,78),(27,139,82),(28,134,79),(29,100,62),(30,102,59),(31,104,63),(32,99,60),(33,101,57),(34,103,61),(35,105,58),(36,114,91),(37,116,88),(38,118,85),(39,113,89),(40,115,86),(41,117,90),(42,119,87),(43,133,65),(44,128,69),(45,130,66),(46,132,70),(47,127,67),(48,129,64),(49,131,68)]])

57 conjugacy classes

class	1	3A	3B	3C	···	3H	7A	···	7P	21A	···	21AF
order	1	3	3	3	···	3	7	···	7	21	···	21
size	1	1	1	49	···	49	3	···	3	3	···	3

57 irreducible representations

dim	1	1	1	3	3
type	+
image	C₁	C₃	C₃	C₇⋊C₃	C₃×C₇⋊C₃
kernel	C₃×C₇²⋊C₃	C₇²⋊C₃	C₇×C₂₁	C₂₁	C₇
# reps	1	6	2	16	32

Matrix representation of C₃×C₇²⋊C₃ ►in GL₆(𝔽₄₃)

36	0	0	0	0	0
0	36	0	0	0	0
0	0	36	0	0	0
0	0	0	36	0	0
0	0	0	0	36	0
0	0	0	0	0	36

0	1	0	0	0	0
0	0	1	0	0	0
1	25	24	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	1	0	0	0	0
0	0	1	0	0	0
1	25	24	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	25	24

3	0	36	0	0	0
32	36	33	0	0	0
36	0	4	0	0	0
0	0	0	36	36	0
0	0	0	11	14	10
0	0	0	0	36	36

G:=sub<GL(6,GF(43))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[0,0,1,0,0,0,1,0,25,0,0,0,0,1,24,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,0,1,0,0,0,1,0,25,0,0,0,0,1,24,0,0,0,0,0,0,0,0,1,0,0,0,1,0,25,0,0,0,0,1,24],[3,32,36,0,0,0,0,36,0,0,0,0,36,33,4,0,0,0,0,0,0,36,11,0,0,0,0,36,14,36,0,0,0,0,10,36] >;

C₃×C₇²⋊C₃ in GAP, Magma, Sage, TeX

C_3\times C_7^2\rtimes C_3

% in TeX

G:=Group("C3xC7^2:C3");

// GroupNames label

G:=SmallGroup(441,11);

// by ID

G=gap.SmallGroup(441,11);

# by ID

G:=PCGroup([4,-3,-3,-7,-7,218,2019]);

// Polycyclic

G:=Group<a,b,c,d|a^3=b^7=c^7=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^4,d*c*d^-1=c^4>;

// generators/relations

G = C3×C72⋊C3 order 441 = 32·72

Direct product of C3 and C72⋊C3

G = C₃×C₇²⋊C₃ order 441 = 3²·7²

Direct product of C₃ and C₇²⋊C₃