direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C6xC22:C8, C23:3C24, C24.5C12, (C22xC8):5C6, (C22xC6):3C8, C4.69(C6xD4), C22:3(C2xC24), (C22xC24):5C2, (C23xC6).5C4, (C2xC24):42C22, (C2xC12).535D4, C12.474(C2xD4), (C23xC12).8C2, C6.30(C22xC8), (C23xC4).10C6, C2.1(C22xC24), C2.3(C6xM4(2)), C23.33(C2xC12), (C22xC12).19C4, (C22xC4).14C12, C6.47(C2xM4(2)), (C2xC6).30M4(2), (C2xC12).981C23, C12.114(C22:C4), C22.9(C3xM4(2)), C22.19(C22xC12), (C22xC12).495C22, (C2xC6):7(C2xC8), (C2xC8):10(C2xC6), C2.3(C6xC22:C4), (C2xC4).59(C2xC12), (C2xC4).145(C3xD4), C4.31(C3xC22:C4), C6.97(C2xC22:C4), (C2xC12).288(C2xC4), (C2xC6).231(C22xC4), (C22xC4).142(C2xC6), (C2xC4).149(C22xC6), (C22xC6).114(C2xC4), C22.32(C3xC22:C4), (C2xC6).134(C22:C4), SmallGroup(192,839)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6xC22:C8
G = < a,b,c,d | a6=b2=c2=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Subgroups: 290 in 202 conjugacy classes, 114 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2xC4, C2xC4, C2xC4, C23, C23, C23, C12, C12, C2xC6, C2xC6, C2xC6, C2xC8, C2xC8, C22xC4, C22xC4, C22xC4, C24, C24, C2xC12, C2xC12, C2xC12, C22xC6, C22xC6, C22xC6, C22:C8, C22xC8, C23xC4, C2xC24, C2xC24, C22xC12, C22xC12, C22xC12, C23xC6, C2xC22:C8, C3xC22:C8, C22xC24, C23xC12, C6xC22:C8
Quotients: C1, C2, C3, C4, C22, C6, C8, C2xC4, D4, C23, C12, C2xC6, C22:C4, C2xC8, M4(2), C22xC4, C2xD4, C24, C2xC12, C3xD4, C22xC6, C22:C8, C2xC22:C4, C22xC8, C2xM4(2), C3xC22:C4, C2xC24, C3xM4(2), C22xC12, C6xD4, C2xC22:C8, C3xC22:C8, C6xC22:C4, C22xC24, C6xM4(2), C6xC22:C8
►On 96 points
(1 11 25 67 55 83)(2 12 26 68 56 84)(3 13 27 69 49 85)(4 14 28 70 50 86)(5 15 29 71 51 87)(6 16 30 72 52 88)(7 9 31 65 53 81)(8 10 32 66 54 82)(17 33 77 45 89 60)(18 34 78 46 90 61)(19 35 79 47 91 62)(20 36 80 48 92 63)(21 37 73 41 93 64)(22 38 74 42 94 57)(23 39 75 43 95 58)(24 40 76 44 96 59)
(2 24)(4 18)(6 20)(8 22)(10 38)(12 40)(14 34)(16 36)(26 76)(28 78)(30 80)(32 74)(42 66)(44 68)(46 70)(48 72)(50 90)(52 92)(54 94)(56 96)(57 82)(59 84)(61 86)(63 88)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 37)(10 38)(11 39)(12 40)(13 33)(14 34)(15 35)(16 36)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 73)(32 74)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)(57 82)(58 83)(59 84)(60 85)(61 86)(62 87)(63 88)(64 81)
(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)
G:=sub<Sym(96)| (1,11,25,67,55,83)(2,12,26,68,56,84)(3,13,27,69,49,85)(4,14,28,70,50,86)(5,15,29,71,51,87)(6,16,30,72,52,88)(7,9,31,65,53,81)(8,10,32,66,54,82)(17,33,77,45,89,60)(18,34,78,46,90,61)(19,35,79,47,91,62)(20,36,80,48,92,63)(21,37,73,41,93,64)(22,38,74,42,94,57)(23,39,75,43,95,58)(24,40,76,44,96,59), (2,24)(4,18)(6,20)(8,22)(10,38)(12,40)(14,34)(16,36)(26,76)(28,78)(30,80)(32,74)(42,66)(44,68)(46,70)(48,72)(50,90)(52,92)(54,94)(56,96)(57,82)(59,84)(61,86)(63,88), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,37)(10,38)(11,39)(12,40)(13,33)(14,34)(15,35)(16,36)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,73)(32,74)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,96)(57,82)(58,83)(59,84)(60,85)(61,86)(62,87)(63,88)(64,81), (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)>;
G:=Group( (1,11,25,67,55,83)(2,12,26,68,56,84)(3,13,27,69,49,85)(4,14,28,70,50,86)(5,15,29,71,51,87)(6,16,30,72,52,88)(7,9,31,65,53,81)(8,10,32,66,54,82)(17,33,77,45,89,60)(18,34,78,46,90,61)(19,35,79,47,91,62)(20,36,80,48,92,63)(21,37,73,41,93,64)(22,38,74,42,94,57)(23,39,75,43,95,58)(24,40,76,44,96,59), (2,24)(4,18)(6,20)(8,22)(10,38)(12,40)(14,34)(16,36)(26,76)(28,78)(30,80)(32,74)(42,66)(44,68)(46,70)(48,72)(50,90)(52,92)(54,94)(56,96)(57,82)(59,84)(61,86)(63,88), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,37)(10,38)(11,39)(12,40)(13,33)(14,34)(15,35)(16,36)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,73)(32,74)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,96)(57,82)(58,83)(59,84)(60,85)(61,86)(62,87)(63,88)(64,81), (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) );
G=PermutationGroup([[(1,11,25,67,55,83),(2,12,26,68,56,84),(3,13,27,69,49,85),(4,14,28,70,50,86),(5,15,29,71,51,87),(6,16,30,72,52,88),(7,9,31,65,53,81),(8,10,32,66,54,82),(17,33,77,45,89,60),(18,34,78,46,90,61),(19,35,79,47,91,62),(20,36,80,48,92,63),(21,37,73,41,93,64),(22,38,74,42,94,57),(23,39,75,43,95,58),(24,40,76,44,96,59)], [(2,24),(4,18),(6,20),(8,22),(10,38),(12,40),(14,34),(16,36),(26,76),(28,78),(30,80),(32,74),(42,66),(44,68),(46,70),(48,72),(50,90),(52,92),(54,94),(56,96),(57,82),(59,84),(61,86),(63,88)], [(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,37),(10,38),(11,39),(12,40),(13,33),(14,34),(15,35),(16,36),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,73),(32,74),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72),(49,89),(50,90),(51,91),(52,92),(53,93),(54,94),(55,95),(56,96),(57,82),(58,83),(59,84),(60,85),(61,86),(62,87),(63,88),(64,81)], [(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)]])
120 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6N | 6O | ··· | 6V | 8A | ··· | 8P | 12A | ··· | 12P | 12Q | ··· | 12X | 24A | ··· | 24AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C8 | C12 | C12 | C24 | D4 | M4(2) | C3xD4 | C3xM4(2) |
| kernel | C6xC22:C8 | C3xC22:C8 | C22xC24 | C23xC12 | C2xC22:C8 | C22xC12 | C23xC6 | C22:C8 | C22xC8 | C23xC4 | C22xC6 | C22xC4 | C24 | C23 | C2xC12 | C2xC6 | C2xC4 | C22 |
| # reps | 1 | 4 | 2 | 1 | 2 | 6 | 2 | 8 | 4 | 2 | 16 | 12 | 4 | 32 | 4 | 4 | 8 | 8 |
Matrix representation of C6xC22:C8 ►in GL4(F73) generated by
| 65 | 0 | 0 | 0 |
| 0 | 65 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 53 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 22 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 10 | 1 |
| 0 | 0 | 0 | 63 |
G:=sub<GL(4,GF(73))| [65,0,0,0,0,65,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,1,53,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[22,0,0,0,0,72,0,0,0,0,10,0,0,0,1,63] >;
C6xC22:C8 in GAP, Magma, Sage, TeX
C_6\times C_2^2\rtimes C_8
% in TeX
G:=Group("C6xC2^2:C8"); // GroupNames label
G:=SmallGroup(192,839);
// by ID
G=gap.SmallGroup(192,839);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,124]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations