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## G = C2×C23.14D6order 192 = 26·3

### Direct product of C2 and C23.14D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C23.14D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C22×C3⋊D4 — C2×C23.14D6
 Lower central C3 — C2×C6 — C2×C23.14D6
 Upper central C1 — C23 — C22×D4

Generators and relations for C2×C23.14D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=1, f2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de-1 >

Subgroups: 1096 in 426 conjugacy classes, 135 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C23×C4, C22×D4, C22×D4, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C22×Dic3, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, S3×C23, C23×C6, C2×C4⋊D4, C2×Dic3⋊C4, C2×D6⋊C4, C23.14D6, C2×C6.D4, C23×Dic3, C22×C3⋊D4, D4×C2×C6, C2×C23.14D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C3⋊D4, C22×S3, C4⋊D4, C22×D4, C2×C4○D4, S3×D4, D42S3, C2×C3⋊D4, S3×C23, C2×C4⋊D4, C23.14D6, C2×S3×D4, C2×D42S3, C22×C3⋊D4, C2×C23.14D6

Smallest permutation representation of C2×C23.14D6
On 96 points
Generators in S96
(1 89)(2 90)(3 85)(4 86)(5 87)(6 88)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 94)(14 95)(15 96)(16 91)(17 92)(18 93)(19 84)(20 79)(21 80)(22 81)(23 82)(24 83)(31 68)(32 69)(33 70)(34 71)(35 72)(36 67)(37 58)(38 59)(39 60)(40 55)(41 56)(42 57)(43 64)(44 65)(45 66)(46 61)(47 62)(48 63)(49 75)(50 76)(51 77)(52 78)(53 73)(54 74)
(1 14)(2 48)(3 16)(4 44)(5 18)(6 46)(7 21)(8 33)(9 23)(10 35)(11 19)(12 31)(13 40)(15 42)(17 38)(20 50)(22 52)(24 54)(25 80)(26 70)(27 82)(28 72)(29 84)(30 68)(32 51)(34 53)(36 49)(37 43)(39 45)(41 47)(55 94)(56 62)(57 96)(58 64)(59 92)(60 66)(61 88)(63 90)(65 86)(67 75)(69 77)(71 73)(74 83)(76 79)(78 81)(85 91)(87 93)(89 95)
(1 33)(2 34)(3 35)(4 36)(5 31)(6 32)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(19 38)(20 39)(21 40)(22 41)(23 42)(24 37)(25 94)(26 95)(27 96)(28 91)(29 92)(30 93)(43 54)(44 49)(45 50)(46 51)(47 52)(48 53)(55 80)(56 81)(57 82)(58 83)(59 84)(60 79)(61 77)(62 78)(63 73)(64 74)(65 75)(66 76)(67 86)(68 87)(69 88)(70 89)(71 90)(72 85)
(1 41)(2 42)(3 37)(4 38)(5 39)(6 40)(7 51)(8 52)(9 53)(10 54)(11 49)(12 50)(13 46)(14 47)(15 48)(16 43)(17 44)(18 45)(19 36)(20 31)(21 32)(22 33)(23 34)(24 35)(25 77)(26 78)(27 73)(28 74)(29 75)(30 76)(55 88)(56 89)(57 90)(58 85)(59 86)(60 87)(61 94)(62 95)(63 96)(64 91)(65 92)(66 93)(67 84)(68 79)(69 80)(70 81)(71 82)(72 83)
(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 6 41 40)(2 39 42 5)(3 4 37 38)(7 47 51 14)(8 13 52 46)(9 45 53 18)(10 17 54 44)(11 43 49 16)(12 15 50 48)(19 35 36 24)(20 23 31 34)(21 33 32 22)(25 62 77 95)(26 94 78 61)(27 66 73 93)(28 92 74 65)(29 64 75 91)(30 96 76 63)(55 89 88 56)(57 87 90 60)(58 59 85 86)(67 83 84 72)(68 71 79 82)(69 81 80 70)

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A 4B 4C ··· 4J 4K 4L 6A ··· 6G 6H ··· 6O 12A 12B 12C 12D order 1 2 ··· 2 2 2 2 2 2 2 2 2 3 4 4 4 ··· 4 4 4 6 ··· 6 6 ··· 6 12 12 12 12 size 1 1 ··· 1 2 2 2 2 4 4 12 12 2 4 4 6 ··· 6 12 12 2 ··· 2 4 ··· 4 4 4 4 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 C4○D4 C3⋊D4 S3×D4 D4⋊2S3 kernel C2×C23.14D6 C2×Dic3⋊C4 C2×D6⋊C4 C23.14D6 C2×C6.D4 C23×Dic3 C22×C3⋊D4 D4×C2×C6 C22×D4 C2×Dic3 C22×C6 C22×C4 C2×D4 C24 C2×C6 C23 C22 C22 # reps 1 1 1 8 1 1 2 1 1 4 4 1 4 2 4 8 2 2

Matrix representation of C2×C23.14D6 in GL5(𝔽13)

 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 12 11 0 0 0 0 1 0 0 0 0 0 2 9 0 0 0 4 11
,
 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
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 8 3 0 0 0 5 5 0 0 0 0 0 1 12 0 0 0 1 0
,
 1 0 0 0 0 0 5 10 0 0 0 0 8 0 0 0 0 0 1 0 0 0 0 1 12

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,11,1,0,0,0,0,0,2,4,0,0,0,9,11],[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],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,8,5,0,0,0,3,5,0,0,0,0,0,1,1,0,0,0,12,0],[1,0,0,0,0,0,5,0,0,0,0,10,8,0,0,0,0,0,1,1,0,0,0,0,12] >;

C2×C23.14D6 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{14}D_6
% in TeX

G:=Group("C2xC2^3.14D6");
// GroupNames label

G:=SmallGroup(192,1361);
// by ID

G=gap.SmallGroup(192,1361);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,675,297,6278]);
// Polycyclic

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

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