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## G = C23×Dic3order 96 = 25·3

### Direct product of C23 and Dic3

Aliases: C23×Dic3, C24.3S3, C6.14C24, C23.41D6, C32(C23×C4), (C22×C6)⋊5C4, C62(C22×C4), C2.2(S3×C23), (C23×C6).3C2, (C2×C6).69C23, C22.33(C22×S3), (C22×C6).47C22, (C2×C6)⋊9(C2×C4), SmallGroup(96,218)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C23×Dic3
 Chief series C1 — C3 — C6 — Dic3 — C2×Dic3 — C22×Dic3 — C23×Dic3
 Lower central C3 — C23×Dic3
 Upper central C1 — C24

Generators and relations for C23×Dic3
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 [×14], C3, C4 [×8], C22 [×35], C6, C6 [×14], C2×C4 [×28], C23 [×15], Dic3 [×8], C2×C6 [×35], C22×C4 [×14], C24, C2×Dic3 [×28], C22×C6 [×15], C23×C4, C22×Dic3 [×14], C23×C6, C23×Dic3
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], Dic3 [×8], D6 [×7], C22×C4 [×14], C24, C2×Dic3 [×28], C22×S3 [×7], C23×C4, C22×Dic3 [×14], S3×C23, C23×Dic3

Smallest permutation representation of C23×Dic3
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)])

C23×Dic3 is a maximal subgroup of
C24.55D6  C24.56D6  C24.57D6  C24.58D6  C24.60D6  C24.29D6  C24.67D6  S3×C23×C4
C23×Dic3 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 ··· 4P 6A ··· 6O order 1 2 ··· 2 3 4 ··· 4 6 ··· 6 size 1 1 ··· 1 2 3 ··· 3 2 ··· 2

48 irreducible representations

 dim 1 1 1 1 2 2 2 type + + + + - + image C1 C2 C2 C4 S3 Dic3 D6 kernel C23×Dic3 C22×Dic3 C23×C6 C22×C6 C24 C23 C23 # reps 1 14 1 16 1 8 7

Matrix representation of C23×Dic3 in GL5(𝔽13)

 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 12 0 0 0 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 9 2 0 0 0 11 4

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] >;

C23×Dic3 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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