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## G = Dic6.32D4order 192 = 26·3

### 2nd non-split extension by Dic6 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Dic6.32D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C22×Dic6 — Dic6.32D4
 Lower central C3 — C6 — C2×C12 — Dic6.32D4
 Upper central C1 — C22 — C22×C4 — C22⋊C8

Generators and relations for Dic6.32D4
G = < a,b,c,d | a12=d2=1, b2=c4=a6, bab-1=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd=a9c3 >

Subgroups: 416 in 148 conjugacy classes, 47 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C24, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C22×C6, C22⋊C8, Q8⋊C4, C22⋊Q8, C2×Q16, C22×Q8, Dic12, Dic3⋊C4, C4⋊Dic3, C6.D4, C2×C24, C2×Dic6, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C22⋊Q16, C2.Dic12, C3×C22⋊C8, C2×Dic12, C12.48D4, C22×Dic6, Dic6.32D4
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, D12, C22×S3, C22≀C2, C2×Q16, C8.C22, Dic12, C2×D12, S3×D4, C22⋊Q16, D6⋊D4, C2×Dic12, C8.D6, Dic6.32D4

Smallest permutation representation of Dic6.32D4
On 96 points
Generators in S96
(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)
(1 52 7 58)(2 51 8 57)(3 50 9 56)(4 49 10 55)(5 60 11 54)(6 59 12 53)(13 43 19 37)(14 42 20 48)(15 41 21 47)(16 40 22 46)(17 39 23 45)(18 38 24 44)(25 90 31 96)(26 89 32 95)(27 88 33 94)(28 87 34 93)(29 86 35 92)(30 85 36 91)(61 76 67 82)(62 75 68 81)(63 74 69 80)(64 73 70 79)(65 84 71 78)(66 83 72 77)
(1 89 73 39 7 95 79 45)(2 90 74 40 8 96 80 46)(3 91 75 41 9 85 81 47)(4 92 76 42 10 86 82 48)(5 93 77 43 11 87 83 37)(6 94 78 44 12 88 84 38)(13 51 34 63 19 57 28 69)(14 52 35 64 20 58 29 70)(15 53 36 65 21 59 30 71)(16 54 25 66 22 60 31 72)(17 55 26 67 23 49 32 61)(18 56 27 68 24 50 33 62)
(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 96)(44 85)(45 86)(46 87)(47 88)(48 89)

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 24A ··· 24H order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 2 2 2 4 12 12 12 12 24 24 2 2 2 4 4 4 4 4 4 2 2 2 2 4 4 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + - + + - - + - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 Q16 D12 D12 Dic12 C8.C22 S3×D4 C8.D6 kernel Dic6.32D4 C2.Dic12 C3×C22⋊C8 C2×Dic12 C12.48D4 C22×Dic6 C22⋊C8 Dic6 C2×C12 C22×C6 C2×C8 C22×C4 C2×C6 C2×C4 C23 C22 C6 C4 C2 # reps 1 2 1 2 1 1 1 4 1 1 2 1 4 2 2 8 1 2 2

Matrix representation of Dic6.32D4 in GL6(𝔽73)

 0 1 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 1 0
,
 56 41 0 0 0 0 41 17 0 0 0 0 0 0 1 0 0 0 0 0 41 72 0 0 0 0 0 0 1 0 0 0 0 0 1 72
,
 16 57 0 0 0 0 16 16 0 0 0 0 0 0 41 71 0 0 0 0 37 32 0 0 0 0 0 0 7 59 0 0 0 0 14 66
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 41 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[56,41,0,0,0,0,41,17,0,0,0,0,0,0,1,41,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,72],[16,16,0,0,0,0,57,16,0,0,0,0,0,0,41,37,0,0,0,0,71,32,0,0,0,0,0,0,7,14,0,0,0,0,59,66],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,41,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic6.32D4 in GAP, Magma, Sage, TeX

{\rm Dic}_6._{32}D_4
% in TeX

G:=Group("Dic6.32D4");
// GroupNames label

G:=SmallGroup(192,298);
// by ID

G=gap.SmallGroup(192,298);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,254,219,226,1123,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=d^2=1,b^2=c^4=a^6,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=a^9*c^3>;
// generators/relations

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