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## G = C2×C23.28D6order 192 = 26·3

### Direct product of C2 and C23.28D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C23.28D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C22×C3⋊D4 — C2×C23.28D6
 Lower central C3 — C2×C6 — C2×C23.28D6
 Upper central C1 — C23 — C23×C4

Generators and relations for C2×C23.28D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=d, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce5 >

Subgroups: 824 in 342 conjugacy classes, 127 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×6], C3, C4 [×10], C22, C22 [×10], C22 [×22], S3 [×2], C6, C6 [×6], C6 [×4], C2×C4 [×4], C2×C4 [×24], D4 [×8], C23, C23 [×6], C23 [×12], Dic3 [×6], C12 [×4], D6 [×10], C2×C6, C2×C6 [×10], C2×C6 [×12], C22⋊C4 [×12], C4⋊C4 [×8], C22×C4 [×6], C22×C4 [×7], C2×D4 [×8], C24, C24, C2×Dic3 [×6], C2×Dic3 [×6], C3⋊D4 [×8], C2×C12 [×4], C2×C12 [×12], C22×S3 [×2], C22×S3 [×6], C22×C6, C22×C6 [×6], C22×C6 [×4], C2×C22⋊C4 [×3], C2×C4⋊C4 [×2], C22.D4 [×8], C23×C4, C22×D4, Dic3⋊C4 [×8], D6⋊C4 [×8], C6.D4 [×4], C22×Dic3, C22×Dic3 [×2], C2×C3⋊D4 [×4], C2×C3⋊D4 [×4], C22×C12 [×6], C22×C12 [×4], S3×C23, C23×C6, C2×C22.D4, C2×Dic3⋊C4 [×2], C2×D6⋊C4 [×2], C23.28D6 [×8], C2×C6.D4, C22×C3⋊D4, C23×C12, C2×C23.28D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, C3⋊D4 [×4], C22×S3 [×7], C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C4○D12 [×4], C2×C3⋊D4 [×6], S3×C23, C2×C22.D4, C23.28D6 [×4], C2×C4○D12 [×2], C22×C3⋊D4, C2×C23.28D6

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

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 C4○D4 C3⋊D4 C4○D12 kernel C2×C23.28D6 C2×Dic3⋊C4 C2×D6⋊C4 C23.28D6 C2×C6.D4 C22×C3⋊D4 C23×C12 C23×C4 C22×C6 C22×C4 C24 C2×C6 C23 C22 # reps 1 2 2 8 1 1 1 1 4 6 1 8 8 16

Matrix representation of C2×C23.28D6 in GL7(𝔽13)

 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 0 0 5 0 0 0 0 0 8 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 2 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 12 1 0 0 0 0 0 12 0 0 0 0 0 0 0 0 8 10 0 0 0 0 0 0 5
,
 1 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 5 3 0 0 0 0 0 5 8

G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,8,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,2,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,10,5],[1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,5,0,0,0,0,0,3,8] >;

C2×C23.28D6 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{28}D_6
% in TeX

G:=Group("C2xC2^3.28D6");
// GroupNames label

G:=SmallGroup(192,1348);
// by ID

G=gap.SmallGroup(192,1348);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,675,136,6278]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=d,f^2=d*c=c*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,f*b*f^-1=b*d=d*b,b*e=e*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^5>;
// generators/relations

׿
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