direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C23.8Q8, C2.4(D4×C12), C22⋊C4⋊4C12, C6.105(C4×D4), (C23×C4).9C6, C6.88C22≀C2, (C2×C12).454D4, C23.9(C3×Q8), C24.32(C2×C6), (C23×C12).7C2, C23.40(C3×D4), C22.34(C6×D4), (C22×C6).20Q8, C22.12(C6×Q8), C23.20(C2×C12), (C22×C6).155D4, C6.83(C22⋊Q8), C2.C42⋊10C6, (C23×C6).86C22, C23.62(C22×C6), C22.34(C22×C12), (C22×C6).449C23, (C22×C12).493C22, C6.86(C22.D4), (C2×C4⋊C4)⋊2C6, C2.7(C6×C4⋊C4), (C6×C4⋊C4)⋊29C2, (C2×C6)⋊5(C4⋊C4), (C2×C4)⋊2(C2×C12), C6.62(C2×C4⋊C4), C22⋊3(C3×C4⋊C4), (C2×C12)⋊23(C2×C4), (C2×C4).99(C3×D4), (C3×C22⋊C4)⋊10C4, C2.2(C3×C22⋊Q8), C2.2(C3×C22≀C2), (C2×C6).601(C2×D4), (C2×C22⋊C4).5C6, (C2×C6).104(C2×Q8), (C6×C22⋊C4).11C2, (C22×C6).82(C2×C4), (C22×C4).93(C2×C6), C22.19(C3×C4○D4), (C2×C6).209(C4○D4), (C2×C6).221(C22×C4), C2.2(C3×C22.D4), (C3×C2.C42)⋊23C2, SmallGroup(192,818)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.8Q8
G = < a,b,c,d,e,f | a3=b2=c2=d2=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce-1 >
Subgroups: 370 in 234 conjugacy classes, 106 normal (30 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, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C22×C12, C23×C6, C23.8Q8, C3×C2.C42, C6×C22⋊C4, C6×C4⋊C4, C23×C12, C3×C23.8Q8
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C23, C12, C2×C6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C23.8Q8, C6×C4⋊C4, D4×C12, C3×C22≀C2, C3×C22⋊Q8, C3×C22.D4, C3×C23.8Q8
►On 96 points
Generators in S96
(1 6 35)(2 7 36)(3 8 33)(4 5 34)(9 29 13)(10 30 14)(11 31 15)(12 32 16)(17 53 21)(18 54 22)(19 55 23)(20 56 24)(25 67 71)(26 68 72)(27 65 69)(28 66 70)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 59 63)(50 60 64)(51 57 61)(52 58 62)(73 84 80)(74 81 77)(75 82 78)(76 83 79)(85 93 89)(86 94 90)(87 95 91)(88 96 92)
(1 9)(2 10)(3 11)(4 12)(5 32)(6 29)(7 30)(8 31)(13 35)(14 36)(15 33)(16 34)(17 39)(18 40)(19 37)(20 38)(21 43)(22 44)(23 41)(24 42)(25 51)(26 52)(27 49)(28 50)(45 55)(46 56)(47 53)(48 54)(57 67)(58 68)(59 65)(60 66)(61 71)(62 72)(63 69)(64 70)(73 94)(74 95)(75 96)(76 93)(77 87)(78 88)(79 85)(80 86)(81 91)(82 92)(83 89)(84 90)
(1 9)(2 10)(3 11)(4 12)(5 32)(6 29)(7 30)(8 31)(13 35)(14 36)(15 33)(16 34)(17 39)(18 40)(19 37)(20 38)(21 43)(22 44)(23 41)(24 42)(25 93)(26 94)(27 95)(28 96)(45 55)(46 56)(47 53)(48 54)(49 74)(50 75)(51 76)(52 73)(57 83)(58 84)(59 81)(60 82)(61 79)(62 80)(63 77)(64 78)(65 91)(66 92)(67 89)(68 90)(69 87)(70 88)(71 85)(72 86)
(1 39)(2 40)(3 37)(4 38)(5 46)(6 47)(7 48)(8 45)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(25 76)(26 73)(27 74)(28 75)(29 53)(30 54)(31 55)(32 56)(33 41)(34 42)(35 43)(36 44)(49 95)(50 96)(51 93)(52 94)(57 89)(58 90)(59 91)(60 92)(61 85)(62 86)(63 87)(64 88)(65 81)(66 82)(67 83)(68 84)(69 77)(70 78)(71 79)(72 80)
(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 3 79)(2 62 4 64)(5 50 7 52)(6 74 8 76)(9 63 11 61)(10 80 12 78)(13 59 15 57)(14 84 16 82)(17 87 19 85)(18 72 20 70)(21 91 23 89)(22 68 24 66)(25 47 27 45)(26 56 28 54)(29 49 31 51)(30 73 32 75)(33 83 35 81)(34 60 36 58)(37 71 39 69)(38 88 40 86)(41 67 43 65)(42 92 44 90)(46 96 48 94)(53 95 55 93)
G:=sub<Sym(96)| (1,6,35)(2,7,36)(3,8,33)(4,5,34)(9,29,13)(10,30,14)(11,31,15)(12,32,16)(17,53,21)(18,54,22)(19,55,23)(20,56,24)(25,67,71)(26,68,72)(27,65,69)(28,66,70)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,59,63)(50,60,64)(51,57,61)(52,58,62)(73,84,80)(74,81,77)(75,82,78)(76,83,79)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (1,9)(2,10)(3,11)(4,12)(5,32)(6,29)(7,30)(8,31)(13,35)(14,36)(15,33)(16,34)(17,39)(18,40)(19,37)(20,38)(21,43)(22,44)(23,41)(24,42)(25,51)(26,52)(27,49)(28,50)(45,55)(46,56)(47,53)(48,54)(57,67)(58,68)(59,65)(60,66)(61,71)(62,72)(63,69)(64,70)(73,94)(74,95)(75,96)(76,93)(77,87)(78,88)(79,85)(80,86)(81,91)(82,92)(83,89)(84,90), (1,9)(2,10)(3,11)(4,12)(5,32)(6,29)(7,30)(8,31)(13,35)(14,36)(15,33)(16,34)(17,39)(18,40)(19,37)(20,38)(21,43)(22,44)(23,41)(24,42)(25,93)(26,94)(27,95)(28,96)(45,55)(46,56)(47,53)(48,54)(49,74)(50,75)(51,76)(52,73)(57,83)(58,84)(59,81)(60,82)(61,79)(62,80)(63,77)(64,78)(65,91)(66,92)(67,89)(68,90)(69,87)(70,88)(71,85)(72,86), (1,39)(2,40)(3,37)(4,38)(5,46)(6,47)(7,48)(8,45)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(25,76)(26,73)(27,74)(28,75)(29,53)(30,54)(31,55)(32,56)(33,41)(34,42)(35,43)(36,44)(49,95)(50,96)(51,93)(52,94)(57,89)(58,90)(59,91)(60,92)(61,85)(62,86)(63,87)(64,88)(65,81)(66,82)(67,83)(68,84)(69,77)(70,78)(71,79)(72,80), (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), (1,77,3,79)(2,62,4,64)(5,50,7,52)(6,74,8,76)(9,63,11,61)(10,80,12,78)(13,59,15,57)(14,84,16,82)(17,87,19,85)(18,72,20,70)(21,91,23,89)(22,68,24,66)(25,47,27,45)(26,56,28,54)(29,49,31,51)(30,73,32,75)(33,83,35,81)(34,60,36,58)(37,71,39,69)(38,88,40,86)(41,67,43,65)(42,92,44,90)(46,96,48,94)(53,95,55,93)>;
G:=Group( (1,6,35)(2,7,36)(3,8,33)(4,5,34)(9,29,13)(10,30,14)(11,31,15)(12,32,16)(17,53,21)(18,54,22)(19,55,23)(20,56,24)(25,67,71)(26,68,72)(27,65,69)(28,66,70)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,59,63)(50,60,64)(51,57,61)(52,58,62)(73,84,80)(74,81,77)(75,82,78)(76,83,79)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (1,9)(2,10)(3,11)(4,12)(5,32)(6,29)(7,30)(8,31)(13,35)(14,36)(15,33)(16,34)(17,39)(18,40)(19,37)(20,38)(21,43)(22,44)(23,41)(24,42)(25,51)(26,52)(27,49)(28,50)(45,55)(46,56)(47,53)(48,54)(57,67)(58,68)(59,65)(60,66)(61,71)(62,72)(63,69)(64,70)(73,94)(74,95)(75,96)(76,93)(77,87)(78,88)(79,85)(80,86)(81,91)(82,92)(83,89)(84,90), (1,9)(2,10)(3,11)(4,12)(5,32)(6,29)(7,30)(8,31)(13,35)(14,36)(15,33)(16,34)(17,39)(18,40)(19,37)(20,38)(21,43)(22,44)(23,41)(24,42)(25,93)(26,94)(27,95)(28,96)(45,55)(46,56)(47,53)(48,54)(49,74)(50,75)(51,76)(52,73)(57,83)(58,84)(59,81)(60,82)(61,79)(62,80)(63,77)(64,78)(65,91)(66,92)(67,89)(68,90)(69,87)(70,88)(71,85)(72,86), (1,39)(2,40)(3,37)(4,38)(5,46)(6,47)(7,48)(8,45)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(25,76)(26,73)(27,74)(28,75)(29,53)(30,54)(31,55)(32,56)(33,41)(34,42)(35,43)(36,44)(49,95)(50,96)(51,93)(52,94)(57,89)(58,90)(59,91)(60,92)(61,85)(62,86)(63,87)(64,88)(65,81)(66,82)(67,83)(68,84)(69,77)(70,78)(71,79)(72,80), (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), (1,77,3,79)(2,62,4,64)(5,50,7,52)(6,74,8,76)(9,63,11,61)(10,80,12,78)(13,59,15,57)(14,84,16,82)(17,87,19,85)(18,72,20,70)(21,91,23,89)(22,68,24,66)(25,47,27,45)(26,56,28,54)(29,49,31,51)(30,73,32,75)(33,83,35,81)(34,60,36,58)(37,71,39,69)(38,88,40,86)(41,67,43,65)(42,92,44,90)(46,96,48,94)(53,95,55,93) );
G=PermutationGroup([[(1,6,35),(2,7,36),(3,8,33),(4,5,34),(9,29,13),(10,30,14),(11,31,15),(12,32,16),(17,53,21),(18,54,22),(19,55,23),(20,56,24),(25,67,71),(26,68,72),(27,65,69),(28,66,70),(37,45,41),(38,46,42),(39,47,43),(40,48,44),(49,59,63),(50,60,64),(51,57,61),(52,58,62),(73,84,80),(74,81,77),(75,82,78),(76,83,79),(85,93,89),(86,94,90),(87,95,91),(88,96,92)], [(1,9),(2,10),(3,11),(4,12),(5,32),(6,29),(7,30),(8,31),(13,35),(14,36),(15,33),(16,34),(17,39),(18,40),(19,37),(20,38),(21,43),(22,44),(23,41),(24,42),(25,51),(26,52),(27,49),(28,50),(45,55),(46,56),(47,53),(48,54),(57,67),(58,68),(59,65),(60,66),(61,71),(62,72),(63,69),(64,70),(73,94),(74,95),(75,96),(76,93),(77,87),(78,88),(79,85),(80,86),(81,91),(82,92),(83,89),(84,90)], [(1,9),(2,10),(3,11),(4,12),(5,32),(6,29),(7,30),(8,31),(13,35),(14,36),(15,33),(16,34),(17,39),(18,40),(19,37),(20,38),(21,43),(22,44),(23,41),(24,42),(25,93),(26,94),(27,95),(28,96),(45,55),(46,56),(47,53),(48,54),(49,74),(50,75),(51,76),(52,73),(57,83),(58,84),(59,81),(60,82),(61,79),(62,80),(63,77),(64,78),(65,91),(66,92),(67,89),(68,90),(69,87),(70,88),(71,85),(72,86)], [(1,39),(2,40),(3,37),(4,38),(5,46),(6,47),(7,48),(8,45),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24),(25,76),(26,73),(27,74),(28,75),(29,53),(30,54),(31,55),(32,56),(33,41),(34,42),(35,43),(36,44),(49,95),(50,96),(51,93),(52,94),(57,89),(58,90),(59,91),(60,92),(61,85),(62,86),(63,87),(64,88),(65,81),(66,82),(67,83),(68,84),(69,77),(70,78),(71,79),(72,80)], [(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)], [(1,77,3,79),(2,62,4,64),(5,50,7,52),(6,74,8,76),(9,63,11,61),(10,80,12,78),(13,59,15,57),(14,84,16,82),(17,87,19,85),(18,72,20,70),(21,91,23,89),(22,68,24,66),(25,47,27,45),(26,56,28,54),(29,49,31,51),(30,73,32,75),(33,83,35,81),(34,60,36,58),(37,71,39,69),(38,88,40,86),(41,67,43,65),(42,92,44,90),(46,96,48,94),(53,95,55,93)]])
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 4A | ··· | 4H | 4I | ··· | 4P | 6A | ··· | 6N | 6O | ··· | 6V | 12A | ··· | 12P | 12Q | ··· | 12AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C12 | D4 | D4 | Q8 | C4○D4 | C3×D4 | C3×D4 | C3×Q8 | C3×C4○D4 |
| kernel | C3×C23.8Q8 | C3×C2.C42 | C6×C22⋊C4 | C6×C4⋊C4 | C23×C12 | C23.8Q8 | C3×C22⋊C4 | C2.C42 | C2×C22⋊C4 | C2×C4⋊C4 | C23×C4 | C22⋊C4 | C2×C12 | C22×C6 | C22×C6 | C2×C6 | C2×C4 | C23 | C23 | C22 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 8 | 4 | 4 | 4 | 2 | 16 | 4 | 2 | 2 | 4 | 8 | 4 | 4 | 8 |
Matrix representation of C3×C23.8Q8 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 7 | 12 |
| 12 | 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 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 5 | 1 | 0 | 0 |
| 0 | 2 | 8 | 0 | 0 |
| 0 | 0 | 0 | 3 | 1 |
| 0 | 0 | 0 | 5 | 10 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[1,0,0,0,0,0,12,0,0,0,0,5,1,0,0,0,0,0,1,7,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[8,0,0,0,0,0,12,0,0,0,0,5,1,0,0,0,0,0,12,0,0,0,0,0,12],[8,0,0,0,0,0,5,2,0,0,0,1,8,0,0,0,0,0,3,5,0,0,0,1,10] >;
C3×C23.8Q8 in GAP, Magma, Sage, TeX
C_3\times C_2^3._8Q_8
% in TeX
G:=Group("C3xC2^3.8Q8"); // GroupNames label
G:=SmallGroup(192,818);
// by ID
G=gap.SmallGroup(192,818);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,680,1094]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^4=1,f^2=e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*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=c*e^-1>;
// generators/relations