direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C23.Q8, C24.8(C2×C6), (C2×C12).309D4, (C22×C6).3Q8, C23.3(C3×Q8), C22.72(C6×D4), C22.22(C6×Q8), C2.C42⋊7C6, C6.89(C22⋊Q8), (C23×C6).7C22, C6.139(C4⋊D4), C23.83(C22×C6), C6.34(C42⋊2C2), (C22×C12).34C22, (C22×C6).460C23, (C2×C4⋊C4)⋊6C6, (C6×C4⋊C4)⋊33C2, (C2×C4).16(C3×D4), C2.8(C3×C4⋊D4), C2.8(C3×C22⋊Q8), (C2×C6).612(C2×D4), (C2×C22⋊C4).9C6, (C2×C6).110(C2×Q8), (C6×C22⋊C4).28C2, (C22×C4).12(C2×C6), C2.4(C3×C42⋊2C2), C22.39(C3×C4○D4), (C2×C6).220(C4○D4), (C3×C2.C42)⋊6C2, SmallGroup(192,829)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.Q8
G = < a,b,c,d,e,f | a3=b2=c2=d2=e4=1, f2=ce2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
Subgroups: 330 in 186 conjugacy classes, 78 normal (14 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C23, C23, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C24, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C23×C6, C23.Q8, C3×C2.C42, C6×C22⋊C4, C6×C4⋊C4, C3×C23.Q8
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, C2×C6, C2×D4, C2×Q8, C4○D4, C3×D4, C3×Q8, C22×C6, C4⋊D4, C22⋊Q8, C42⋊2C2, C6×D4, C6×Q8, C3×C4○D4, C23.Q8, C3×C4⋊D4, C3×C22⋊Q8, C3×C42⋊2C2, C3×C23.Q8
►On 96 points
Generators in S96
(1 37 33)(2 38 34)(3 39 35)(4 40 36)(5 96 92)(6 93 89)(7 94 90)(8 95 91)(9 17 13)(10 18 14)(11 19 15)(12 20 16)(21 49 53)(22 50 54)(23 51 55)(24 52 56)(25 68 29)(26 65 30)(27 66 31)(28 67 32)(41 73 45)(42 74 46)(43 75 47)(44 76 48)(57 61 69)(58 62 70)(59 63 71)(60 64 72)(77 85 81)(78 86 82)(79 87 83)(80 88 84)
(1 11)(2 22)(3 9)(4 24)(5 70)(6 8)(7 72)(10 44)(12 42)(13 35)(14 48)(15 33)(16 46)(17 39)(18 76)(19 37)(20 74)(21 41)(23 43)(25 84)(26 28)(27 82)(29 88)(30 32)(31 86)(34 54)(36 56)(38 50)(40 52)(45 53)(47 55)(49 73)(51 75)(57 59)(58 96)(60 94)(61 63)(62 92)(64 90)(65 67)(66 78)(68 80)(69 71)(77 79)(81 83)(85 87)(89 91)(93 95)
(1 9)(2 10)(3 11)(4 12)(5 31)(6 32)(7 29)(8 30)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 43)(22 44)(23 41)(24 42)(25 94)(26 95)(27 96)(28 93)(45 55)(46 56)(47 53)(48 54)(49 75)(50 76)(51 73)(52 74)(57 81)(58 82)(59 83)(60 84)(61 77)(62 78)(63 79)(64 80)(65 91)(66 92)(67 89)(68 90)(69 85)(70 86)(71 87)(72 88)
(1 41)(2 42)(3 43)(4 44)(5 72)(6 69)(7 70)(8 71)(9 23)(10 24)(11 21)(12 22)(13 55)(14 56)(15 53)(16 54)(17 51)(18 52)(19 49)(20 50)(25 82)(26 83)(27 84)(28 81)(29 86)(30 87)(31 88)(32 85)(33 45)(34 46)(35 47)(36 48)(37 73)(38 74)(39 75)(40 76)(57 93)(58 94)(59 95)(60 96)(61 89)(62 90)(63 91)(64 92)(65 79)(66 80)(67 77)(68 78)
(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 77 11 63)(2 80 12 62)(3 79 9 61)(4 78 10 64)(5 76 29 52)(6 75 30 51)(7 74 31 50)(8 73 32 49)(13 57 35 83)(14 60 36 82)(15 59 33 81)(16 58 34 84)(17 69 39 87)(18 72 40 86)(19 71 37 85)(20 70 38 88)(21 91 41 67)(22 90 42 66)(23 89 43 65)(24 92 44 68)(25 56 96 48)(26 55 93 47)(27 54 94 46)(28 53 95 45)
G:=sub<Sym(96)| (1,37,33)(2,38,34)(3,39,35)(4,40,36)(5,96,92)(6,93,89)(7,94,90)(8,95,91)(9,17,13)(10,18,14)(11,19,15)(12,20,16)(21,49,53)(22,50,54)(23,51,55)(24,52,56)(25,68,29)(26,65,30)(27,66,31)(28,67,32)(41,73,45)(42,74,46)(43,75,47)(44,76,48)(57,61,69)(58,62,70)(59,63,71)(60,64,72)(77,85,81)(78,86,82)(79,87,83)(80,88,84), (1,11)(2,22)(3,9)(4,24)(5,70)(6,8)(7,72)(10,44)(12,42)(13,35)(14,48)(15,33)(16,46)(17,39)(18,76)(19,37)(20,74)(21,41)(23,43)(25,84)(26,28)(27,82)(29,88)(30,32)(31,86)(34,54)(36,56)(38,50)(40,52)(45,53)(47,55)(49,73)(51,75)(57,59)(58,96)(60,94)(61,63)(62,92)(64,90)(65,67)(66,78)(68,80)(69,71)(77,79)(81,83)(85,87)(89,91)(93,95), (1,9)(2,10)(3,11)(4,12)(5,31)(6,32)(7,29)(8,30)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,43)(22,44)(23,41)(24,42)(25,94)(26,95)(27,96)(28,93)(45,55)(46,56)(47,53)(48,54)(49,75)(50,76)(51,73)(52,74)(57,81)(58,82)(59,83)(60,84)(61,77)(62,78)(63,79)(64,80)(65,91)(66,92)(67,89)(68,90)(69,85)(70,86)(71,87)(72,88), (1,41)(2,42)(3,43)(4,44)(5,72)(6,69)(7,70)(8,71)(9,23)(10,24)(11,21)(12,22)(13,55)(14,56)(15,53)(16,54)(17,51)(18,52)(19,49)(20,50)(25,82)(26,83)(27,84)(28,81)(29,86)(30,87)(31,88)(32,85)(33,45)(34,46)(35,47)(36,48)(37,73)(38,74)(39,75)(40,76)(57,93)(58,94)(59,95)(60,96)(61,89)(62,90)(63,91)(64,92)(65,79)(66,80)(67,77)(68,78), (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), (1,77,11,63)(2,80,12,62)(3,79,9,61)(4,78,10,64)(5,76,29,52)(6,75,30,51)(7,74,31,50)(8,73,32,49)(13,57,35,83)(14,60,36,82)(15,59,33,81)(16,58,34,84)(17,69,39,87)(18,72,40,86)(19,71,37,85)(20,70,38,88)(21,91,41,67)(22,90,42,66)(23,89,43,65)(24,92,44,68)(25,56,96,48)(26,55,93,47)(27,54,94,46)(28,53,95,45)>;
G:=Group( (1,37,33)(2,38,34)(3,39,35)(4,40,36)(5,96,92)(6,93,89)(7,94,90)(8,95,91)(9,17,13)(10,18,14)(11,19,15)(12,20,16)(21,49,53)(22,50,54)(23,51,55)(24,52,56)(25,68,29)(26,65,30)(27,66,31)(28,67,32)(41,73,45)(42,74,46)(43,75,47)(44,76,48)(57,61,69)(58,62,70)(59,63,71)(60,64,72)(77,85,81)(78,86,82)(79,87,83)(80,88,84), (1,11)(2,22)(3,9)(4,24)(5,70)(6,8)(7,72)(10,44)(12,42)(13,35)(14,48)(15,33)(16,46)(17,39)(18,76)(19,37)(20,74)(21,41)(23,43)(25,84)(26,28)(27,82)(29,88)(30,32)(31,86)(34,54)(36,56)(38,50)(40,52)(45,53)(47,55)(49,73)(51,75)(57,59)(58,96)(60,94)(61,63)(62,92)(64,90)(65,67)(66,78)(68,80)(69,71)(77,79)(81,83)(85,87)(89,91)(93,95), (1,9)(2,10)(3,11)(4,12)(5,31)(6,32)(7,29)(8,30)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,43)(22,44)(23,41)(24,42)(25,94)(26,95)(27,96)(28,93)(45,55)(46,56)(47,53)(48,54)(49,75)(50,76)(51,73)(52,74)(57,81)(58,82)(59,83)(60,84)(61,77)(62,78)(63,79)(64,80)(65,91)(66,92)(67,89)(68,90)(69,85)(70,86)(71,87)(72,88), (1,41)(2,42)(3,43)(4,44)(5,72)(6,69)(7,70)(8,71)(9,23)(10,24)(11,21)(12,22)(13,55)(14,56)(15,53)(16,54)(17,51)(18,52)(19,49)(20,50)(25,82)(26,83)(27,84)(28,81)(29,86)(30,87)(31,88)(32,85)(33,45)(34,46)(35,47)(36,48)(37,73)(38,74)(39,75)(40,76)(57,93)(58,94)(59,95)(60,96)(61,89)(62,90)(63,91)(64,92)(65,79)(66,80)(67,77)(68,78), (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), (1,77,11,63)(2,80,12,62)(3,79,9,61)(4,78,10,64)(5,76,29,52)(6,75,30,51)(7,74,31,50)(8,73,32,49)(13,57,35,83)(14,60,36,82)(15,59,33,81)(16,58,34,84)(17,69,39,87)(18,72,40,86)(19,71,37,85)(20,70,38,88)(21,91,41,67)(22,90,42,66)(23,89,43,65)(24,92,44,68)(25,56,96,48)(26,55,93,47)(27,54,94,46)(28,53,95,45) );
G=PermutationGroup([[(1,37,33),(2,38,34),(3,39,35),(4,40,36),(5,96,92),(6,93,89),(7,94,90),(8,95,91),(9,17,13),(10,18,14),(11,19,15),(12,20,16),(21,49,53),(22,50,54),(23,51,55),(24,52,56),(25,68,29),(26,65,30),(27,66,31),(28,67,32),(41,73,45),(42,74,46),(43,75,47),(44,76,48),(57,61,69),(58,62,70),(59,63,71),(60,64,72),(77,85,81),(78,86,82),(79,87,83),(80,88,84)], [(1,11),(2,22),(3,9),(4,24),(5,70),(6,8),(7,72),(10,44),(12,42),(13,35),(14,48),(15,33),(16,46),(17,39),(18,76),(19,37),(20,74),(21,41),(23,43),(25,84),(26,28),(27,82),(29,88),(30,32),(31,86),(34,54),(36,56),(38,50),(40,52),(45,53),(47,55),(49,73),(51,75),(57,59),(58,96),(60,94),(61,63),(62,92),(64,90),(65,67),(66,78),(68,80),(69,71),(77,79),(81,83),(85,87),(89,91),(93,95)], [(1,9),(2,10),(3,11),(4,12),(5,31),(6,32),(7,29),(8,30),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,43),(22,44),(23,41),(24,42),(25,94),(26,95),(27,96),(28,93),(45,55),(46,56),(47,53),(48,54),(49,75),(50,76),(51,73),(52,74),(57,81),(58,82),(59,83),(60,84),(61,77),(62,78),(63,79),(64,80),(65,91),(66,92),(67,89),(68,90),(69,85),(70,86),(71,87),(72,88)], [(1,41),(2,42),(3,43),(4,44),(5,72),(6,69),(7,70),(8,71),(9,23),(10,24),(11,21),(12,22),(13,55),(14,56),(15,53),(16,54),(17,51),(18,52),(19,49),(20,50),(25,82),(26,83),(27,84),(28,81),(29,86),(30,87),(31,88),(32,85),(33,45),(34,46),(35,47),(36,48),(37,73),(38,74),(39,75),(40,76),(57,93),(58,94),(59,95),(60,96),(61,89),(62,90),(63,91),(64,92),(65,79),(66,80),(67,77),(68,78)], [(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)], [(1,77,11,63),(2,80,12,62),(3,79,9,61),(4,78,10,64),(5,76,29,52),(6,75,30,51),(7,74,31,50),(8,73,32,49),(13,57,35,83),(14,60,36,82),(15,59,33,81),(16,58,34,84),(17,69,39,87),(18,72,40,86),(19,71,37,85),(20,70,38,88),(21,91,41,67),(22,90,42,66),(23,89,43,65),(24,92,44,68),(25,56,96,48),(26,55,93,47),(27,54,94,46),(28,53,95,45)]])
66 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 3A | 3B | 4A | ··· | 4L | 6A | ··· | 6N | 6O | 6P | 6Q | 6R | 12A | ··· | 12X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | Q8 | C4○D4 | C3×D4 | C3×Q8 | C3×C4○D4 |
| kernel | C3×C23.Q8 | C3×C2.C42 | C6×C22⋊C4 | C6×C4⋊C4 | C23.Q8 | C2.C42 | C2×C22⋊C4 | C2×C4⋊C4 | C2×C12 | C22×C6 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 3 | 3 | 2 | 2 | 6 | 6 | 6 | 2 | 6 | 12 | 4 | 12 |
Matrix representation of C3×C23.Q8 ►in GL6(𝔽13)
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 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 |
| 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 11 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 5 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,1,0,0,0,0,11,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[5,5,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;
C3×C23.Q8 in GAP, Magma, Sage, TeX
C_3\times C_2^3.Q_8
% in TeX
G:=Group("C3xC2^3.Q8"); // GroupNames label
G:=SmallGroup(192,829);
// by ID
G=gap.SmallGroup(192,829);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,504,365,176,1094,1059]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^4=1,f^2=c*e^2,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*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations