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## G = C23×D12order 192 = 26·3

### Direct product of C23 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C23×D12
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C23 — S3×C24 — C23×D12
 Lower central C3 — C6 — C23×D12
 Upper central C1 — C24 — C23×C4

Generators and relations for C23×D12
G = < a,b,c,d,e | a2=b2=c2=d12=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 3512 in 1362 conjugacy classes, 543 normal (9 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C12, D6, D6, C2×C6, C22×C4, C2×D4, C24, C24, D12, C2×C12, C22×S3, C22×S3, C22×C6, C23×C4, C22×D4, C25, C2×D12, C22×C12, S3×C23, S3×C23, C23×C6, D4×C23, C22×D12, C23×C12, S3×C24, C23×D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, D12, C22×S3, C22×D4, C25, C2×D12, S3×C23, D4×C23, C22×D12, S3×C24, C23×D12

Smallest permutation representation of C23×D12
On 96 points
Generators in S96
(1 32)(2 33)(3 34)(4 35)(5 36)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 49)(22 50)(23 51)(24 52)(37 92)(38 93)(39 94)(40 95)(41 96)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(61 79)(62 80)(63 81)(64 82)(65 83)(66 84)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 72)(14 61)(15 62)(16 63)(17 64)(18 65)(19 66)(20 67)(21 68)(22 69)(23 70)(24 71)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 81)(57 82)(58 83)(59 84)(60 73)
(1 77)(2 78)(3 79)(4 80)(5 81)(6 82)(7 83)(8 84)(9 73)(10 74)(11 75)(12 76)(13 87)(14 88)(15 89)(16 90)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 85)(24 86)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(33 72)(34 61)(35 62)(36 63)(37 58)(38 59)(39 60)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 57)
(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 39)(2 38)(3 37)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 66)(14 65)(15 64)(16 63)(17 62)(18 61)(19 72)(20 71)(21 70)(22 69)(23 68)(24 67)(25 89)(26 88)(27 87)(28 86)(29 85)(30 96)(31 95)(32 94)(33 93)(34 92)(35 91)(36 90)(49 76)(50 75)(51 74)(52 73)(53 84)(54 83)(55 82)(56 81)(57 80)(58 79)(59 78)(60 77)

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

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

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

72 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2AE 3 4A ··· 4H 6A ··· 6O 12A ··· 12P order 1 2 ··· 2 2 ··· 2 3 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 6 ··· 6 2 2 ··· 2 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 S3 D4 D6 D6 D12 kernel C23×D12 C22×D12 C23×C12 S3×C24 C23×C4 C22×C6 C22×C4 C24 C23 # reps 1 28 1 2 1 8 14 1 16

Matrix representation of C23×D12 in GL6(𝔽13)

 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 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 12 0 0 0 0 0 0 0 10 3 0 0 0 0 10 7
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 12 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 0 1

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,10,10,0,0,0,0,3,7],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,12,1] >;

C23×D12 in GAP, Magma, Sage, TeX

C_2^3\times D_{12}
% in TeX

G:=Group("C2^3xD12");
// GroupNames label

G:=SmallGroup(192,1512);
// by ID

G=gap.SmallGroup(192,1512);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,1684,102,6278]);
// Polycyclic

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

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