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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, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C23, C23, C23, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C22×C4, C2×Q8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C22×Q8, C6.D4, C3×C22⋊C4, C2×Dic6, C22×Dic3, C22×C12, C23×C6, C23⋊Q8, C6.C42, C6.C42, C2×C6.D4, C6×C22⋊C4, C22×Dic6, C232Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, Dic6, C3⋊D4, C22×S3, C22≀C2, C22⋊Q8, C4.4D4, C2×Dic6, C4○D12, S3×D4, D42S3, C2×C3⋊D4, C23⋊Q8, Dic3.D4, C23.11D6, C12.48D4, C23.12D6, C232D6, C232Dic6

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

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

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

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

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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