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G = C62.30D6order 432 = 24·33

13rd non-split extension by C62 of D6 acting via D6/C2=S3

non-abelian, supersoluble, monomial

Aliases: C62.30D6, C4⋊(He33C4), He39(C4⋊C4), (C4×He3)⋊4C4, (C6×C12).11S3, (C3×C12)⋊2Dic3, (C3×C6).26D12, (C2×He3).7Q8, (C2×He3).25D4, (C3×C6).10Dic6, C2.1(He35D4), C325(C4⋊Dic3), C2.2(He34Q8), C6.19(C12⋊S3), C12.10(C3⋊Dic3), C6.10(C324Q8), C3.2(C12⋊Dic3), (C22×He3).23C22, (C2×C4×He3).7C2, C6.26(C2×C3⋊Dic3), C2.4(C2×He33C4), (C2×C12).21(C3⋊S3), (C2×He3).33(C2×C4), (C2×He33C4).2C2, (C3×C6).19(C2×Dic3), (C2×C4).3(He3⋊C2), C22.5(C2×He3⋊C2), (C2×C6).53(C2×C3⋊S3), SmallGroup(432,188)

Series: Derived Chief Lower central Upper central

C1C3C2×He3 — C62.30D6
C1C3C32He3C2×He3C22×He3C2×He33C4 — C62.30D6
He3C2×He3 — C62.30D6
C1C2×C6C2×C12

Generators and relations for C62.30D6
 G = < a,b,c,d | a6=b6=1, c6=b3, d2=a3, ab=ba, cac-1=ab2, dad-1=a-1b4, bc=cb, bd=db, dcd-1=b3c5 >

Subgroups: 453 in 143 conjugacy classes, 59 normal (19 characteristic)
C1, C2 [×3], C3, C3 [×4], C4 [×2], C4 [×2], C22, C6 [×3], C6 [×12], C2×C4, C2×C4 [×2], C32 [×4], Dic3 [×8], C12 [×2], C12 [×10], C2×C6, C2×C6 [×4], C4⋊C4, C3×C6 [×12], C2×Dic3 [×8], C2×C12, C2×C12 [×6], He3, C3×Dic3 [×8], C3×C12 [×8], C62 [×4], C4⋊Dic3 [×4], C3×C4⋊C4, C2×He3 [×3], C6×Dic3 [×8], C6×C12 [×4], He33C4 [×2], C4×He3 [×2], C22×He3, C3×C4⋊Dic3 [×4], C2×He33C4 [×2], C2×C4×He3, C62.30D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4, Q8, Dic3 [×8], D6 [×4], C4⋊C4, C3⋊S3, Dic6 [×4], D12 [×4], C2×Dic3 [×4], C3⋊Dic3 [×2], C2×C3⋊S3, C4⋊Dic3 [×4], He3⋊C2, C324Q8, C12⋊S3, C2×C3⋊Dic3, He33C4 [×2], C2×He3⋊C2, C12⋊Dic3, He34Q8, He35D4, C2×He33C4, C62.30D6

Smallest permutation representation of C62.30D6
On 144 points
Generators in S144
(1 75 67 36 112 121)(2 84 64 33 109 130)(3 81 61 34 118 127)(4 78 70 35 115 124)(5 105 86 28 133 49)(6 102 95 25 142 58)(7 99 92 26 139 55)(8 108 89 27 136 52)(9 74 62 30 111 128)(10 83 71 31 120 125)(11 80 68 32 117 122)(12 77 65 29 114 131)(13 141 53 42 101 90)(14 138 50 43 98 87)(15 135 59 44 107 96)(16 144 56 41 104 93)(17 137 57 46 97 94)(18 134 54 47 106 91)(19 143 51 48 103 88)(20 140 60 45 100 85)(21 110 123 37 73 69)(22 119 132 38 82 66)(23 116 129 39 79 63)(24 113 126 40 76 72)
(1 37 10 3 39 12)(2 38 11 4 40 9)(5 48 42 7 46 44)(6 45 43 8 47 41)(13 26 17 15 28 19)(14 27 18 16 25 20)(21 31 34 23 29 36)(22 32 35 24 30 33)(49 51 53 55 57 59)(50 52 54 56 58 60)(61 63 65 67 69 71)(62 64 66 68 70 72)(73 83 81 79 77 75)(74 84 82 80 78 76)(85 87 89 91 93 95)(86 88 90 92 94 96)(97 107 105 103 101 99)(98 108 106 104 102 100)(109 119 117 115 113 111)(110 120 118 116 114 112)(121 123 125 127 129 131)(122 124 126 128 130 132)(133 143 141 139 137 135)(134 144 142 140 138 136)
(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)
(1 43 36 14)(2 42 33 13)(3 41 34 16)(4 44 35 15)(5 24 28 40)(6 23 25 39)(7 22 26 38)(8 21 27 37)(9 48 30 19)(10 47 31 18)(11 46 32 17)(12 45 29 20)(49 109 86 84)(50 120 87 83)(51 119 88 82)(52 118 89 81)(53 117 90 80)(54 116 91 79)(55 115 92 78)(56 114 93 77)(57 113 94 76)(58 112 95 75)(59 111 96 74)(60 110 85 73)(61 140 127 100)(62 139 128 99)(63 138 129 98)(64 137 130 97)(65 136 131 108)(66 135 132 107)(67 134 121 106)(68 133 122 105)(69 144 123 104)(70 143 124 103)(71 142 125 102)(72 141 126 101)

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

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

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

62 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F4A4B4C4D4E4F6A···6F6G···6R12A12B12C12D12E···12T12U···12AB
order12223333334444446···66···61212121212···1212···12
size111111666622181818181···16···622226···618···18

62 irreducible representations

dim1111222222233366
type+++++--+-+
imageC1C2C2C4S3D4Q8Dic3D6Dic6D12He3⋊C2He33C4C2×He3⋊C2He34Q8He35D4
kernelC62.30D6C2×He33C4C2×C4×He3C4×He3C6×C12C2×He3C2×He3C3×C12C62C3×C6C3×C6C2×C4C4C22C2C2
# reps1214411848848422

Matrix representation of C62.30D6 in GL7(𝔽13)

0100000
121200000
00012000
0011000
0000010
0000001
0000100
,
12000000
01200000
00120000
00012000
0000900
0000090
0000009
,
6300000
10300000
00310000
0036000
0000100
0000030
0000009
,
71000000
3600000
00211000
00911000
00001200
0000004
00000100

G:=sub<GL(7,GF(13))| [0,12,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,9],[6,10,0,0,0,0,0,3,3,0,0,0,0,0,0,0,3,3,0,0,0,0,0,10,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,9],[7,3,0,0,0,0,0,10,6,0,0,0,0,0,0,0,2,9,0,0,0,0,0,11,11,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,10,0,0,0,0,0,4,0] >;

C62.30D6 in GAP, Magma, Sage, TeX

C_6^2._{30}D_6
% in TeX

G:=Group("C6^2.30D6");
// GroupNames label

G:=SmallGroup(432,188);
// by ID

G=gap.SmallGroup(432,188);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,64,1124,4037,537]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=1,c^6=b^3,d^2=a^3,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^4,b*c=c*b,b*d=d*b,d*c*d^-1=b^3*c^5>;
// generators/relations

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