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G = C2×C23.16D6  order 192 = 26·3

Direct product of C2 and C23.16D6

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

Aliases: C2×C23.16D6, C24.61D6, C6.6(C23×C4), C23.62(C4×S3), (C2×C6).24C24, C6⋊2(C42⋊C2), C22⋊C4.122D6, (C22×C4).327D6, (C2×C12).569C23, Dic3⋊C4⋊56C22, (C22×Dic3)⋊10C4, (C4×Dic3)⋊71C22, (C23×C6).50C22, C22.16(S3×C23), (C23×Dic3).7C2, C23.153(C22×S3), (C22×C6).386C23, Dic3.22(C22×C4), C22.63(D4⋊2S3), (C22×C12).349C22, (C2×Dic3).302C23, C6.D4.82C22, (C22×Dic3).243C22, C2.8(S3×C22×C4), C3⋊2(C2×C42⋊C2), (C2×C4×Dic3)⋊28C2, C6.65(C2×C4○D4), C22.23(S3×C2×C4), C2.1(C2×D4⋊2S3), (C2×Dic3⋊C4)⋊33C2, (C2×Dic3)⋊21(C2×C4), (C6×C22⋊C4).25C2, (C2×C22⋊C4).20S3, (C22×C6).76(C2×C4), (C2×C6).17(C22×C4), (C2×C6).165(C4○D4), (C2×C4).254(C22×S3), (C2×C6.D4).19C2, (C3×C22⋊C4).132C22, SmallGroup(192,1039)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C2×C23.16D6
Chief series
C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C23×Dic3 — C2×C23.16D6
Lower central
C3 — C6 — C2×C23.16D6

Subgroups: 616 in 330 conjugacy classes, 167 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×4], C3, C4 [×16], C22, C22 [×10], C22 [×12], C6, C6 [×6], C6 [×4], C2×C4 [×4], C2×C4 [×40], C23, C23 [×6], C23 [×4], Dic3 [×8], Dic3 [×4], C12 [×4], C2×C6, C2×C6 [×10], C2×C6 [×12], C42 [×8], C22⋊C4 [×4], C22⋊C4 [×4], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×16], C24, C2×Dic3 [×32], C2×Dic3 [×4], C2×C12 [×4], C2×C12 [×4], C22×C6, C22×C6 [×6], C22×C6 [×4], C2×C42 [×2], C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4 [×2], C42⋊C2 [×8], C23×C4, C4×Dic3 [×8], Dic3⋊C4 [×8], C6.D4 [×4], C3×C22⋊C4 [×4], C22×Dic3 [×2], C22×Dic3 [×14], C22×C12 [×2], C23×C6, C2×C42⋊C2, C23.16D6 [×8], C2×C4×Dic3 [×2], C2×Dic3⋊C4 [×2], C2×C6.D4, C6×C22⋊C4, C23×Dic3, C2×C23.16D6

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], C22×C4 [×14], C4○D4 [×4], C24, C4×S3 [×4], C22×S3 [×7], C42⋊C2 [×4], C23×C4, C2×C4○D4 [×2], S3×C2×C4 [×6], D4⋊2S3 [×4], S3×C23, C2×C42⋊C2, C23.16D6 [×4], S3×C22×C4, C2×D4⋊2S3 [×2], C2×C23.16D6

Generators and relations
 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, bc=cb, ebe-1=fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽13)

1200000
0120000
001000
000100
0000120
0000012
,
100000
010000
0012000
0001200
0000120
0000121
,
1200000
0120000
0012000
0001200
000010
000001
,
100000
010000
001000
000100
0000120
0000012
,
050000
880000
000800
005500
0000810
000085
,
050000
500000
008800
000500
0000111
0000112

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,8,0,0,0,0,5,8,0,0,0,0,0,0,0,5,0,0,0,0,8,5,0,0,0,0,0,0,8,8,0,0,0,0,10,5],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,8,0,0,0,0,0,8,5,0,0,0,0,0,0,1,1,0,0,0,0,11,12] >;
 

60 conjugacy classes

class 1 2A···2G2H2I2J2K 3 4A···4H4I···4P4Q···4AB6A···6G6H6I6J6K12A···12H
order12···2222234···44···44···46···6666612···12
size11···1222222···23···36···62···244444···4

60 irreducible representations

dim111111112222224
type+++++++++++-
imageC1C2C2C2C2C2C2C4S3D6D6D6C4○D4C4×S3D4⋊2S3
kernelC2×C23.16D6C23.16D6C2×C4×Dic3C2×Dic3⋊C4C2×C6.D4C6×C22⋊C4C23×Dic3C22×Dic3C2×C22⋊C4C22⋊C4C22×C4C24C2×C6C23C22
# reps1822111161421884

In GAP, Magma, Sage, TeX 

C_2\times C_2^3._{16}D_6
 
% in TeX
 
G:=Group("C2xC2^3.16D6");
 
// GroupNames label
 
G:=SmallGroup(192,1039);
 
// by ID
 
G=gap.SmallGroup(192,1039);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,297,80,6278]);
 
// 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,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=e^5>;
 
// generators/relations
 

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