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G = C2×C23.12D6order 192 = 26·3

Direct product of C2 and C23.12D6

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

Aliases: C2×C23.12D6, C24.48D6, (C2×D4).229D6, C12.251(C2×D4), (C2×C12).209D4, C63(C4.4D4), (C2×C6).292C24, (C22×D4).12S3, (C22×C4).394D6, C6.140(C22×D4), (C2×C12).540C23, (C2×Dic6)⋊67C22, (C22×Dic6)⋊20C2, (C4×Dic3)⋊67C22, (C6×D4).269C22, (C23×C6).74C22, C6.D458C22, (C22×C6).228C23, C22.306(S3×C23), C23.144(C22×S3), C22.78(D42S3), (C22×C12).273C22, (C2×Dic3).282C23, (C22×Dic3).231C22, (D4×C2×C6).8C2, C34(C2×C4.4D4), (C2×C4×Dic3)⋊11C2, C4.23(C2×C3⋊D4), C6.104(C2×C4○D4), (C2×C6).579(C2×D4), C2.68(C2×D42S3), C2.13(C22×C3⋊D4), (C2×C6).176(C4○D4), (C2×C6.D4)⋊25C2, (C2×C4).153(C3⋊D4), (C2×C4).623(C22×S3), C22.109(C2×C3⋊D4), SmallGroup(192,1356)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C2×C23.12D6
C1C3C6C2×C6C2×Dic3C22×Dic3C2×C4×Dic3 — C2×C23.12D6
C3C2×C6 — C2×C23.12D6
C1C23C22×D4

Generators and relations for C2×C23.12D6
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=c, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=bc=cb, fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 744 in 330 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×4], C3, C4 [×4], C4 [×8], C22, C22 [×6], C22 [×20], C6, C6 [×6], C6 [×4], C2×C4 [×6], C2×C4 [×16], D4 [×8], Q8 [×8], C23, C23 [×4], C23 [×12], Dic3 [×8], C12 [×4], C2×C6, C2×C6 [×6], C2×C6 [×20], C42 [×4], C22⋊C4 [×16], C22×C4, C22×C4 [×4], C2×D4 [×4], C2×D4 [×4], C2×Q8 [×8], C24 [×2], Dic6 [×8], C2×Dic3 [×8], C2×Dic3 [×8], C2×C12 [×6], C3×D4 [×8], C22×C6, C22×C6 [×4], C22×C6 [×12], C2×C42, C2×C22⋊C4 [×4], C4.4D4 [×8], C22×D4, C22×Q8, C4×Dic3 [×4], C6.D4 [×16], C2×Dic6 [×4], C2×Dic6 [×4], C22×Dic3 [×4], C22×C12, C6×D4 [×4], C6×D4 [×4], C23×C6 [×2], C2×C4.4D4, C2×C4×Dic3, C23.12D6 [×8], C2×C6.D4 [×4], C22×Dic6, D4×C2×C6, C2×C23.12D6
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], C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], D42S3 [×4], C2×C3⋊D4 [×6], S3×C23, C2×C4.4D4, C23.12D6 [×4], C2×D42S3 [×2], C22×C3⋊D4, C2×C23.12D6

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

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

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

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

48 conjugacy classes

class 1 2A···2G2H2I2J2K 3 4A4B4C4D4E···4L4M4N4O4P6A···6G6H···6O12A12B12C12D
order12···22222344444···444446···66···612121212
size11···14444222226···6121212122···24···44444

48 irreducible representations

dim11111122222224
type+++++++++++-
imageC1C2C2C2C2C2S3D4D6D6D6C4○D4C3⋊D4D42S3
kernelC2×C23.12D6C2×C4×Dic3C23.12D6C2×C6.D4C22×Dic6D4×C2×C6C22×D4C2×C12C22×C4C2×D4C24C2×C6C2×C4C22
# reps11841114142884

Matrix representation of C2×C23.12D6 in GL5(𝔽13)

120000
012000
001200
000120
000012
,
10000
012000
00100
00001
00010
,
10000
01000
00100
000120
000012
,
10000
012000
001200
000120
000012
,
10000
04000
001000
000012
00010
,
120000
001000
09000
00005
00080

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

C2×C23.12D6 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{12}D_6
% in TeX

G:=Group("C2xC2^3.12D6");
// GroupNames label

G:=SmallGroup(192,1356);
// by ID

G=gap.SmallGroup(192,1356);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,100,1571,185,6278]);
// Polycyclic

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

׿
×
𝔽