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G = C23xDic3order 96 = 25·3

Direct product of C23 and Dic3

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C23xDic3, C24.3S3, C6.14C24, C23.41D6, C3:2(C23xC4), (C22xC6):5C4, C6:2(C22xC4), C2.2(S3xC23), (C23xC6).3C2, (C2xC6).69C23, C22.33(C22xS3), (C22xC6).47C22, (C2xC6):9(C2xC4), SmallGroup(96,218)

Series: Derived Chief Lower central Upper central

C1C3 — C23xDic3
C1C3C6Dic3C2xDic3C22xDic3 — C23xDic3
C3 — C23xDic3
C1C24

Generators and relations for C23xDic3
 G = < a,b,c,d,e | a2=b2=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 338 in 236 conjugacy classes, 185 normal (7 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C2xC4, C23, Dic3, C2xC6, C22xC4, C24, C2xDic3, C22xC6, C23xC4, C22xDic3, C23xC6, C23xDic3
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, Dic3, D6, C22xC4, C24, C2xDic3, C22xS3, C23xC4, C22xDic3, S3xC23, C23xDic3

Smallest permutation representation of C23xDic3
Regular action on 96 points
Generators in S96
(1 44)(2 45)(3 46)(4 47)(5 48)(6 43)(7 40)(8 41)(9 42)(10 37)(11 38)(12 39)(13 34)(14 35)(15 36)(16 31)(17 32)(18 33)(19 28)(20 29)(21 30)(22 25)(23 26)(24 27)(49 94)(50 95)(51 96)(52 91)(53 92)(54 93)(55 88)(56 89)(57 90)(58 85)(59 86)(60 87)(61 82)(62 83)(63 84)(64 79)(65 80)(66 81)(67 76)(68 77)(69 78)(70 73)(71 74)(72 75)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 19)(7 16)(8 17)(9 18)(10 13)(11 14)(12 15)(25 46)(26 47)(27 48)(28 43)(29 44)(30 45)(31 40)(32 41)(33 42)(34 37)(35 38)(36 39)(49 70)(50 71)(51 72)(52 67)(53 68)(54 69)(55 64)(56 65)(57 66)(58 61)(59 62)(60 63)(73 94)(74 95)(75 96)(76 91)(77 92)(78 93)(79 88)(80 89)(81 90)(82 85)(83 86)(84 87)
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 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 64 4 61)(2 63 5 66)(3 62 6 65)(7 68 10 71)(8 67 11 70)(9 72 12 69)(13 50 16 53)(14 49 17 52)(15 54 18 51)(19 56 22 59)(20 55 23 58)(21 60 24 57)(25 86 28 89)(26 85 29 88)(27 90 30 87)(31 92 34 95)(32 91 35 94)(33 96 36 93)(37 74 40 77)(38 73 41 76)(39 78 42 75)(43 80 46 83)(44 79 47 82)(45 84 48 81)

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

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

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

C23xDic3 is a maximal subgroup of
C24.55D6  C24.56D6  C24.57D6  C24.58D6  C24.60D6  C24.29D6  C24.67D6  S3xC23xC4
C23xDic3 is a maximal quotient of
C24.49D6  C6.422- 1+4  C12.76C24  C6.1442+ 1+4

48 conjugacy classes

class 1 2A···2O 3 4A···4P6A···6O
order12···234···46···6
size11···123···32···2

48 irreducible representations

dim1111222
type++++-+
imageC1C2C2C4S3Dic3D6
kernelC23xDic3C22xDic3C23xC6C22xC6C24C23C23
# reps114116187

Matrix representation of C23xDic3 in GL5(F13)

10000
01000
001200
00010
00001
,
10000
012000
001200
000120
000012
,
120000
01000
001200
00010
00001
,
10000
01000
00100
000112
00010
,
10000
01000
00100
00092
000114

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

C23xDic3 in GAP, Magma, Sage, TeX

C_2^3\times {\rm Dic}_3
% in TeX

G:=Group("C2^3xDic3");
// GroupNames label

G:=SmallGroup(96,218);
// by ID

G=gap.SmallGroup(96,218);
# by ID

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

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^6=1,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*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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