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G = C2×C23.23D6order 192 = 26·3

Direct product of C2 and C23.23D6

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

Aliases: C2×C23.23D6, C24.69D6, (C2×D4).228D6, (C2×C6).291C24, (C23×Dic3)⋊8C2, (C22×D4).11S3, (C22×C6).121D4, (C22×C4).285D6, C6.139(C22×D4), (C2×C12).642C23, Dic3⋊C472C22, (C6×D4).311C22, C65(C22.D4), C23.74(C3⋊D4), (C23×C6).73C22, C6.D457C22, (C22×C6).227C23, C23.143(C22×S3), C22.305(S3×C23), C22.77(D42S3), (C22×C12).437C22, (C2×Dic3).281C23, (C22×Dic3)⋊48C22, (D4×C2×C6).21C2, (C2×C6).73(C2×D4), C6.103(C2×C4○D4), C36(C2×C22.D4), (C2×Dic3⋊C4)⋊47C2, C2.67(C2×D42S3), C2.12(C22×C3⋊D4), (C2×C6).175(C4○D4), (C2×C6.D4)⋊24C2, (C2×C4).236(C22×S3), C22.108(C2×C3⋊D4), SmallGroup(192,1355)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C2×C23.23D6
C1C3C6C2×C6C2×Dic3C22×Dic3C23×Dic3 — C2×C23.23D6
C3C2×C6 — C2×C23.23D6
C1C23C22×D4

Generators and relations for C2×C23.23D6
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=1, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce-1 >

Subgroups: 744 in 342 conjugacy classes, 127 normal (19 characteristic)
C1, C2, C2 [×6], C2 [×6], C3, C4 [×10], C22, C22 [×10], C22 [×22], C6, C6 [×6], C6 [×6], C2×C4 [×2], C2×C4 [×26], D4 [×8], C23, C23 [×8], C23 [×10], Dic3 [×8], C12 [×2], C2×C6, C2×C6 [×10], C2×C6 [×22], C22⋊C4 [×12], C4⋊C4 [×8], C22×C4, C22×C4 [×12], C2×D4 [×4], C2×D4 [×4], C24 [×2], C2×Dic3 [×8], C2×Dic3 [×16], C2×C12 [×2], C2×C12 [×2], C3×D4 [×8], C22×C6, C22×C6 [×8], C22×C6 [×10], C2×C22⋊C4 [×3], C2×C4⋊C4 [×2], C22.D4 [×8], C23×C4, C22×D4, Dic3⋊C4 [×8], C6.D4 [×12], C22×Dic3 [×8], C22×Dic3 [×4], C22×C12, C6×D4 [×4], C6×D4 [×4], C23×C6 [×2], C2×C22.D4, C2×Dic3⋊C4 [×2], C23.23D6 [×8], C2×C6.D4, C2×C6.D4 [×2], C23×Dic3, D4×C2×C6, C2×C23.23D6
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], D42S3 [×4], C2×C3⋊D4 [×6], S3×C23, C2×C22.D4, C23.23D6 [×4], C2×D42S3 [×2], C22×C3⋊D4, C2×C23.23D6

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

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

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

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

48 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M 3 4A4B4C···4J4K4L4M4N6A···6G6H···6O12A12B12C12D
order12···22222223444···444446···66···612121212
size11···12222442446···6121212122···24···44444

48 irreducible representations

dim11111122222224
type+++++++++++-
imageC1C2C2C2C2C2S3D4D6D6D6C4○D4C3⋊D4D42S3
kernelC2×C23.23D6C2×Dic3⋊C4C23.23D6C2×C6.D4C23×Dic3D4×C2×C6C22×D4C22×C6C22×C4C2×D4C24C2×C6C23C22
# reps12831114142884

Matrix representation of C2×C23.23D6 in GL5(𝔽13)

120000
01000
00100
000120
000012
,
120000
012000
001200
00001
00010
,
10000
012000
001200
00010
00001
,
10000
01000
00100
000120
000012
,
120000
03000
00400
00010
000012
,
120000
00400
03000
00080
00005

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

C2×C23.23D6 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{23}D_6
% in TeX

G:=Group("C2xC2^3.23D6");
// GroupNames label

G:=SmallGroup(192,1355);
// by ID

G=gap.SmallGroup(192,1355);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,675,297,6278]);
// Polycyclic

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

׿
×
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