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## G = C23⋊2Dic6order 192 = 26·3

### 1st semidirect product of C23 and Dic6 acting via Dic6/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — C23⋊2Dic6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C22×Dic6 — C23⋊2Dic6
 Lower central C3 — C22×C6 — C23⋊2Dic6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C232Dic6
G = < a,b,c,d,e | a2=b2=c2=d12=1, e2=d6, eae-1=ab=ba, dad-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 536 in 202 conjugacy classes, 63 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C3, C4 [×9], C22 [×3], C22 [×4], C22 [×10], C6 [×3], C6 [×4], C6 [×2], C2×C4 [×2], C2×C4 [×19], Q8 [×8], C23, C23 [×2], C23 [×6], Dic3 [×6], C12 [×3], C2×C6 [×3], C2×C6 [×4], C2×C6 [×10], C22⋊C4 [×6], C22×C4 [×2], C22×C4 [×4], C2×Q8 [×6], C24, Dic6 [×8], C2×Dic3 [×4], C2×Dic3 [×10], C2×C12 [×2], C2×C12 [×5], C22×C6, C22×C6 [×2], C22×C6 [×6], C2.C42 [×3], C2×C22⋊C4, C2×C22⋊C4 [×2], C22×Q8, C6.D4 [×4], C3×C22⋊C4 [×2], C2×Dic6 [×6], C22×Dic3 [×4], C22×C12 [×2], C23×C6, C23⋊Q8, C6.C42, C6.C42 [×2], C2×C6.D4 [×2], C6×C22⋊C4, C22×Dic6, C232Dic6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], Q8 [×2], C23, D6 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×3], Dic6 [×2], C3⋊D4 [×2], C22×S3, C22≀C2, C22⋊Q8 [×3], C4.4D4 [×3], C2×Dic6, C4○D12, S3×D4 [×2], D42S3 [×2], C2×C3⋊D4, C23⋊Q8, Dic3.D4 [×2], C23.11D6 [×2], C12.48D4, C23.12D6, C232D6, C232Dic6

Smallest permutation representation of C232Dic6
On 96 points
Generators in S96
(2 59)(4 49)(6 51)(8 53)(10 55)(12 57)(13 84)(14 87)(15 74)(16 89)(17 76)(18 91)(19 78)(20 93)(21 80)(22 95)(23 82)(24 85)(25 62)(27 64)(29 66)(31 68)(33 70)(35 72)(37 86)(38 73)(39 88)(40 75)(41 90)(42 77)(43 92)(44 79)(45 94)(46 81)(47 96)(48 83)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 25)(9 26)(10 27)(11 28)(12 29)(13 84)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(37 86)(38 87)(39 88)(40 89)(41 90)(42 91)(43 92)(44 93)(45 94)(46 95)(47 96)(48 85)(49 70)(50 71)(51 72)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)
(1 58)(2 59)(3 60)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 61)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 85)(84 86)
(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 41 7 47)(2 40 8 46)(3 39 9 45)(4 38 10 44)(5 37 11 43)(6 48 12 42)(13 56 19 50)(14 55 20 49)(15 54 21 60)(16 53 22 59)(17 52 23 58)(18 51 24 57)(25 95 31 89)(26 94 32 88)(27 93 33 87)(28 92 34 86)(29 91 35 85)(30 90 36 96)(61 82 67 76)(62 81 68 75)(63 80 69 74)(64 79 70 73)(65 78 71 84)(66 77 72 83)

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

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

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

42 conjugacy classes

 class 1 2A ··· 2G 2H 2I 3 4A 4B 4C 4D 4E ··· 4L 6A ··· 6G 6H 6I 6J 6K 12A ··· 12H order 1 2 ··· 2 2 2 3 4 4 4 4 4 ··· 4 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 ··· 1 4 4 2 4 4 4 4 12 ··· 12 2 ··· 2 4 4 4 4 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + - + + - + - image C1 C2 C2 C2 C2 S3 D4 D4 Q8 D6 D6 C4○D4 C3⋊D4 Dic6 C4○D12 S3×D4 D4⋊2S3 kernel C23⋊2Dic6 C6.C42 C2×C6.D4 C6×C22⋊C4 C22×Dic6 C2×C22⋊C4 C2×Dic3 C2×C12 C22×C6 C22×C4 C24 C2×C6 C2×C4 C23 C22 C22 C22 # reps 1 3 2 1 1 1 4 2 2 2 1 6 4 4 4 2 2

Matrix representation of C232Dic6 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 5 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 6 3 0 0 0 0 10 3 0 0 0 0 0 0 1 3 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 2 4 0 0 0 0 2 11 0 0 0 0 0 0 12 10 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,5,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[6,10,0,0,0,0,3,3,0,0,0,0,0,0,1,0,0,0,0,0,3,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[2,2,0,0,0,0,4,11,0,0,0,0,0,0,12,0,0,0,0,0,10,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;

C232Dic6 in GAP, Magma, Sage, TeX

C_2^3\rtimes_2{\rm Dic}_6
% in TeX

G:=Group("C2^3:2Dic6");
// GroupNames label

G:=SmallGroup(192,506);
// by ID

G=gap.SmallGroup(192,506);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,254,387,6278]);
// Polycyclic

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

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