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## G = C62.10Dic3order 432 = 24·33

### 10th non-split extension by C62 of Dic3 acting via Dic3/C2=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C3.A4 — C62.10Dic3
 Chief series C1 — C22 — C2×C6 — C62 — C3×C3.A4 — C6×C3.A4 — C62.10Dic3
 Lower central C3×C3.A4 — C62.10Dic3
 Upper central C1 — C2

Generators and relations for C62.10Dic3
G = < a,b,c,d | a6=b6=1, c6=b2, d2=b2c3, ab=ba, cac-1=ab3, dad-1=a2b3, cbc-1=a3b4, dbd-1=a3b2, dcd-1=b4c5 >

Subgroups: 648 in 102 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C6, C2×C4, C23, C9, C32, Dic3, C2×C6, C2×C6, C2×C6, C22⋊C4, C18, C3×C6, C3×C6, C2×Dic3, C22×C6, C22×C6, C3×C9, Dic9, C3.A4, C3⋊Dic3, C62, C62, C6.D4, C3×C18, C2×C3.A4, C2×C3⋊Dic3, C2×C62, C9⋊Dic3, C3×C3.A4, C6.S4, C625C4, C6×C3.A4, C62.10Dic3
Quotients: C1, C2, C4, S3, Dic3, D9, C3⋊S3, S4, Dic9, C3⋊Dic3, A4⋊C4, C9⋊S3, C3.S4, C3⋊S4, C9⋊Dic3, C6.S4, C6.7S4, C32.3S4, C62.10Dic3

Smallest permutation representation of C62.10Dic3
On 108 points
Generators in S108
(1 78 21)(2 88 22 11 79 31)(3 89 23 12 80 32)(4 81 24)(5 73 25 14 82 34)(6 74 26 15 83 35)(7 84 27)(8 76 28 17 85 19)(9 77 29 18 86 20)(10 87 30)(13 90 33)(16 75 36)(37 101 60 46 92 69)(38 93 61)(39 103 62 48 94 71)(40 104 63 49 95 72)(41 96 64)(42 106 65 51 97 56)(43 107 66 52 98 57)(44 99 67)(45 91 68 54 100 59)(47 102 70)(50 105 55)(53 108 58)
(1 4 7 10 13 16)(2 14 8)(3 6 9 12 15 18)(5 17 11)(19 31 25)(20 23 26 29 32 35)(21 24 27 30 33 36)(22 34 28)(37 40 43 46 49 52)(38 41 44 47 50 53)(39 51 45)(42 54 48)(55 58 61 64 67 70)(56 68 62)(57 60 63 66 69 72)(59 71 65)(73 85 79)(74 77 80 83 86 89)(75 78 81 84 87 90)(76 88 82)(91 103 97)(92 95 98 101 104 107)(93 96 99 102 105 108)(94 106 100)
(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)
(1 51 10 42)(2 50 11 41)(3 49 12 40)(4 48 13 39)(5 47 14 38)(6 46 15 37)(7 45 16 54)(8 44 17 53)(9 43 18 52)(19 108 28 99)(20 107 29 98)(21 106 30 97)(22 105 31 96)(23 104 32 95)(24 103 33 94)(25 102 34 93)(26 101 35 92)(27 100 36 91)(55 88 64 79)(56 87 65 78)(57 86 66 77)(58 85 67 76)(59 84 68 75)(60 83 69 74)(61 82 70 73)(62 81 71 90)(63 80 72 89)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 6A 6B 6C 6D 6E ··· 6L 9A ··· 9I 18A ··· 18I order 1 2 2 2 3 3 3 3 4 4 4 4 6 6 6 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 3 3 2 2 2 2 54 54 54 54 2 2 2 2 6 ··· 6 8 ··· 8 8 ··· 8

42 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 3 3 6 6 6 6 type + + + + - - + - + + + - - image C1 C2 C4 S3 S3 Dic3 Dic3 D9 Dic9 S4 A4⋊C4 C3.S4 C3⋊S4 C6.S4 C6.7S4 kernel C62.10Dic3 C6×C3.A4 C3×C3.A4 C2×C3.A4 C2×C62 C3.A4 C62 C22×C6 C2×C6 C3×C6 C32 C6 C6 C3 C3 # reps 1 1 2 3 1 3 1 9 9 2 2 3 1 3 1

Matrix representation of C62.10Dic3 in GL7(𝔽37)

 36 1 0 0 0 0 0 36 0 0 0 0 0 0 0 0 35 34 0 0 0 0 0 1 1 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 1
,
 0 36 0 0 0 0 0 1 36 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36
,
 26 17 0 0 0 0 0 20 6 0 0 0 0 0 0 0 2 3 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 1 0 0 0 0 36 0 0 0 0 0 0 0 36 0
,
 8 28 0 0 0 0 0 36 29 0 0 0 0 0 0 0 31 19 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0

G:=sub<GL(7,GF(37))| [36,36,0,0,0,0,0,1,0,0,0,0,0,0,0,0,35,1,0,0,0,0,0,34,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,36,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36],[26,20,0,0,0,0,0,17,6,0,0,0,0,0,0,0,2,36,0,0,0,0,0,3,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,1,0,0],[8,36,0,0,0,0,0,28,29,0,0,0,0,0,0,0,31,0,0,0,0,0,0,19,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0] >;

C62.10Dic3 in GAP, Magma, Sage, TeX

C_6^2._{10}{\rm Dic}_3
% in TeX

G:=Group("C6^2.10Dic3");
// GroupNames label

G:=SmallGroup(432,259);
// by ID

G=gap.SmallGroup(432,259);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,14,926,394,675,2524,9077,2287,5298,3989]);
// Polycyclic

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

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