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G = C2×C23.8D6  order 192 = 26·3

Direct product of C2 and C23.8D6

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

Aliases: C2×C23.8D6, C24.33D6, (C2×C6).26C24, C22⋊C4.85D6, C6⋊2(C42⋊2C2), C4⋊Dic3⋊50C22, (C22×C4).184D6, (C2×C12).126C23, Dic3⋊C4⋊47C22, (C4×Dic3)⋊72C22, (C23×C6).52C22, C23.88(C22×S3), C22.68(S3×C23), C22.71(C4○D12), (C22×C6).118C23, C22.65(D4⋊2S3), (C22×C12).350C22, (C2×Dic3).176C23, C6.D4.84C22, (C22×Dic3).203C22, C3⋊2(C2×C42⋊2C2), (C2×C4×Dic3)⋊29C2, C6.11(C2×C4○D4), (C2×C4⋊Dic3)⋊18C2, C2.13(C2×C4○D12), C2.8(C2×D4⋊2S3), (C2×Dic3⋊C4)⋊34C2, (C2×C22⋊C4).17S3, (C6×C22⋊C4).20C2, (C2×C6).100(C4○D4), (C2×C4).255(C22×S3), (C2×C6.D4).21C2, (C3×C22⋊C4).108C22, SmallGroup(192,1041)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C2×C23.8D6
Chief series
C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C2×C4×Dic3 — C2×C23.8D6
Lower central
C3 — C2×C6 — C2×C23.8D6

Subgroups: 520 in 246 conjugacy classes, 111 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C3, C4 [×12], C22, C22 [×6], C22 [×10], C6 [×3], C6 [×4], C6 [×2], C2×C4 [×4], C2×C4 [×20], C23, C23 [×2], C23 [×6], Dic3 [×8], C12 [×4], C2×C6, C2×C6 [×6], C2×C6 [×10], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×4], C24, C2×Dic3 [×8], C2×Dic3 [×8], C2×C12 [×4], C2×C12 [×4], C22×C6, C22×C6 [×2], C22×C6 [×6], C2×C42, C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C42⋊2C2 [×8], C4×Dic3 [×4], Dic3⋊C4 [×8], C4⋊Dic3 [×4], C6.D4 [×8], C3×C22⋊C4 [×4], C22×Dic3 [×4], C22×C12 [×2], C23×C6, C2×C42⋊2C2, C23.8D6 [×8], C2×C4×Dic3, C2×Dic3⋊C4 [×2], C2×C4⋊Dic3, C2×C6.D4 [×2], C6×C22⋊C4, C2×C23.8D6

Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×6], C24, C22×S3 [×7], C42⋊2C2 [×4], C2×C4○D4 [×3], C4○D12 [×2], D4⋊2S3 [×4], S3×C23, C2×C42⋊2C2, C23.8D6 [×4], C2×C4○D12, C2×D4⋊2S3 [×2], C2×C23.8D6

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

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

(2 75)(4 77)(6 79)(8 81)(10 83)(12 73)(14 56)(16 58)(18 60)(20 50)(22 52)(24 54)(25 68)(26 32)(27 70)(28 34)(29 72)(30 36)(31 62)(33 64)(35 66)(37 90)(38 44)(39 92)(40 46)(41 94)(42 48)(43 96)(45 86)(47 88)(61 67)(63 69)(65 71)(85 91)(87 93)(89 95)

(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 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 61)(37 96)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)

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

Matrix representation ►G ⊆ GL6(𝔽13)

1200000
0120000
0012000
0001200
0000120
0000012
,
100000
010000
001000
00121200
000010
00001212
,
100000
010000
0012000
0001200
000010
000001
,
100000
010000
001000
000100
0000120
0000012
,
1210000
1200000
005000
000500
00001211
000001
,
330000
6100000
008300
000500
000050
000005

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,11,1],[3,6,0,0,0,0,3,10,0,0,0,0,0,0,8,0,0,0,0,0,3,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;
 

48 conjugacy classes

class 1 2A···2G2H2I 3 4A4B4C4D4E4F4G···4N4O4P4Q4R6A···6G6H6I6J6K12A···12H
order12···22234444444···444446···6666612···12
size11···14422222446···6121212122···244444···4

48 irreducible representations

dim11111112222224
type+++++++++++-
imageC1C2C2C2C2C2C2S3D6D6D6C4○D4C4○D12D4⋊2S3
kernelC2×C23.8D6C23.8D6C2×C4×Dic3C2×Dic3⋊C4C2×C4⋊Dic3C2×C6.D4C6×C22⋊C4C2×C22⋊C4C22⋊C4C22×C4C24C2×C6C22C22
# reps181212114211284

In GAP, Magma, Sage, TeX 

C_2\times C_2^3._8D_6
 
% in TeX
 
G:=Group("C2xC2^3.8D6");
 
// GroupNames label
 
G:=SmallGroup(192,1041);
 
// by ID
 
G=gap.SmallGroup(192,1041);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,100,1571,297,6278]);
 
// 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,f*b*f^-1=b*c=c*b,e*b*e^-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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