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## G = C22×Dic6order 96 = 25·3

### Direct product of C22 and Dic6

Series: Derived Chief Lower central Upper central

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

Generators and relations for C22×Dic6
G = < a,b,c,d | a2=b2=c12=1, d2=c6, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 258 in 156 conjugacy classes, 105 normal (9 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, Q8, C23, Dic3, C12, C2×C6, C22×C4, C22×C4, C2×Q8, Dic6, C2×Dic3, C2×C12, C22×C6, C22×Q8, C2×Dic6, C22×Dic3, C22×C12, C22×Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C24, Dic6, C22×S3, C22×Q8, C2×Dic6, S3×C23, C22×Dic6

Smallest permutation representation of C22×Dic6
Regular action on 96 points
Generators in S96
(1 56)(2 57)(3 58)(4 59)(5 60)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 70)(14 71)(15 72)(16 61)(17 62)(18 63)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 93)(26 94)(27 95)(28 96)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)
(1 34)(2 35)(3 36)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 37)(22 38)(23 39)(24 40)(49 95)(50 96)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)(57 91)(58 92)(59 93)(60 94)(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 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 24 7 18)(2 23 8 17)(3 22 9 16)(4 21 10 15)(5 20 11 14)(6 19 12 13)(25 37 31 43)(26 48 32 42)(27 47 33 41)(28 46 34 40)(29 45 35 39)(30 44 36 38)(49 64 55 70)(50 63 56 69)(51 62 57 68)(52 61 58 67)(53 72 59 66)(54 71 60 65)(73 93 79 87)(74 92 80 86)(75 91 81 85)(76 90 82 96)(77 89 83 95)(78 88 84 94)

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + - + + - image C1 C2 C2 C2 S3 Q8 D6 D6 Dic6 kernel C22×Dic6 C2×Dic6 C22×Dic3 C22×C12 C22×C4 C2×C6 C2×C4 C23 C22 # reps 1 12 2 1 1 4 6 1 8

Matrix representation of C22×Dic6 in GL5(𝔽13)

 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 1 12 0 0 0 1 0 0 0 0 0 0 6 3 0 0 0 10 3
,
 12 0 0 0 0 0 1 12 0 0 0 0 12 0 0 0 0 0 4 2 0 0 0 11 9

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,1,1,0,0,0,12,0,0,0,0,0,0,6,10,0,0,0,3,3],[12,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,4,11,0,0,0,2,9] >;

C22×Dic6 in GAP, Magma, Sage, TeX

C_2^2\times {\rm Dic}_6
% in TeX

G:=Group("C2^2xDic6");
// GroupNames label

G:=SmallGroup(96,205);
// by ID

G=gap.SmallGroup(96,205);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,579,69,2309]);
// Polycyclic

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

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