p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42⋊8C4⋊8C2, (C2×D4).118D4, C23.928(C2×D4), (C22×C4).158D4, C22.4Q16⋊28C2, (C22×C8).86C22, C4.39(C42⋊2C2), C2.22(D4.2D4), C22.120(C4○D8), (C2×C42).382C22, (C22×D4).93C22, C22.249(C4⋊D4), C22.150(C8⋊C22), C22.7C42⋊14C2, (C22×C4).1462C23, C2.8(C23.11D4), C22.95(C4.4D4), C4.28(C22.D4), C2.10(C23.19D4), C24.3C22.20C2, C2.9(C42.78C22), C2.7(C42.29C22), C22.118(C22.D4), (C2×C4).1061(C2×D4), (C2×D4⋊C4).18C2, (C2×C4).624(C4○D4), (C2×C4⋊C4).147C22, SmallGroup(128,805)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊8C4⋊C2
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=ab2, dad=a-1b2c2, cbc-1=a2b, dbd=bc2, dcd=a-1bc >
Subgroups: 328 in 129 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, D4⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C22.7C42, C22.4Q16, C42⋊8C4, C24.3C22, C2×D4⋊C4, C42⋊8C4⋊C2
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C4○D8, C8⋊C22, C23.11D4, D4.2D4, C23.19D4, C42.78C22, C42.29C22, C42⋊8C4⋊C2
►On 64 points
(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)
(1 58 22 31)(2 59 23 32)(3 60 24 29)(4 57 21 30)(5 51 33 48)(6 52 34 45)(7 49 35 46)(8 50 36 47)(9 56 20 13)(10 53 17 14)(11 54 18 15)(12 55 19 16)(25 44 64 39)(26 41 61 40)(27 42 62 37)(28 43 63 38)
(1 42 11 46)(2 38 12 50)(3 44 9 48)(4 40 10 52)(5 58 64 54)(6 32 61 16)(7 60 62 56)(8 30 63 14)(13 35 29 27)(15 33 31 25)(17 45 21 41)(18 49 22 37)(19 47 23 43)(20 51 24 39)(26 55 34 59)(28 53 36 57)
(2 17)(4 19)(5 41)(6 51)(7 43)(8 49)(10 23)(12 21)(13 29)(14 55)(15 31)(16 53)(25 52)(26 44)(27 50)(28 42)(30 59)(32 57)(33 40)(34 48)(35 38)(36 46)(37 63)(39 61)(45 64)(47 62)(54 58)(56 60)
G:=sub<Sym(64)| (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), (1,58,22,31)(2,59,23,32)(3,60,24,29)(4,57,21,30)(5,51,33,48)(6,52,34,45)(7,49,35,46)(8,50,36,47)(9,56,20,13)(10,53,17,14)(11,54,18,15)(12,55,19,16)(25,44,64,39)(26,41,61,40)(27,42,62,37)(28,43,63,38), (1,42,11,46)(2,38,12,50)(3,44,9,48)(4,40,10,52)(5,58,64,54)(6,32,61,16)(7,60,62,56)(8,30,63,14)(13,35,29,27)(15,33,31,25)(17,45,21,41)(18,49,22,37)(19,47,23,43)(20,51,24,39)(26,55,34,59)(28,53,36,57), (2,17)(4,19)(5,41)(6,51)(7,43)(8,49)(10,23)(12,21)(13,29)(14,55)(15,31)(16,53)(25,52)(26,44)(27,50)(28,42)(30,59)(32,57)(33,40)(34,48)(35,38)(36,46)(37,63)(39,61)(45,64)(47,62)(54,58)(56,60)>;
G:=Group( (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), (1,58,22,31)(2,59,23,32)(3,60,24,29)(4,57,21,30)(5,51,33,48)(6,52,34,45)(7,49,35,46)(8,50,36,47)(9,56,20,13)(10,53,17,14)(11,54,18,15)(12,55,19,16)(25,44,64,39)(26,41,61,40)(27,42,62,37)(28,43,63,38), (1,42,11,46)(2,38,12,50)(3,44,9,48)(4,40,10,52)(5,58,64,54)(6,32,61,16)(7,60,62,56)(8,30,63,14)(13,35,29,27)(15,33,31,25)(17,45,21,41)(18,49,22,37)(19,47,23,43)(20,51,24,39)(26,55,34,59)(28,53,36,57), (2,17)(4,19)(5,41)(6,51)(7,43)(8,49)(10,23)(12,21)(13,29)(14,55)(15,31)(16,53)(25,52)(26,44)(27,50)(28,42)(30,59)(32,57)(33,40)(34,48)(35,38)(36,46)(37,63)(39,61)(45,64)(47,62)(54,58)(56,60) );
G=PermutationGroup([[(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)], [(1,58,22,31),(2,59,23,32),(3,60,24,29),(4,57,21,30),(5,51,33,48),(6,52,34,45),(7,49,35,46),(8,50,36,47),(9,56,20,13),(10,53,17,14),(11,54,18,15),(12,55,19,16),(25,44,64,39),(26,41,61,40),(27,42,62,37),(28,43,63,38)], [(1,42,11,46),(2,38,12,50),(3,44,9,48),(4,40,10,52),(5,58,64,54),(6,32,61,16),(7,60,62,56),(8,30,63,14),(13,35,29,27),(15,33,31,25),(17,45,21,41),(18,49,22,37),(19,47,23,43),(20,51,24,39),(26,55,34,59),(28,53,36,57)], [(2,17),(4,19),(5,41),(6,51),(7,43),(8,49),(10,23),(12,21),(13,29),(14,55),(15,31),(16,53),(25,52),(26,44),(27,50),(28,42),(30,59),(32,57),(33,40),(34,48),(35,38),(36,46),(37,63),(39,61),(45,64),(47,62),(54,58),(56,60)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D8 | C8⋊C22 |
| kernel | C42⋊8C4⋊C2 | C22.7C42 | C22.4Q16 | C42⋊8C4 | C24.3C22 | C2×D4⋊C4 | C22×C4 | C2×D4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 10 | 8 | 2 |
Matrix representation of C42⋊8C4⋊C2 ►in GL6(𝔽17)
| 0 | 4 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 8 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 8 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 3 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 2 | 0 | 0 |
| 0 | 0 | 8 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 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 | 16 |
G:=sub<GL(6,GF(17))| [0,13,0,0,0,0,4,0,0,0,0,0,0,0,6,8,0,0,0,0,6,11,0,0,0,0,0,0,0,8,0,0,0,0,2,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,6,8,0,0,0,0,6,11,0,0,0,0,0,0,0,8,0,0,0,0,2,0],[3,3,0,0,0,0,3,14,0,0,0,0,0,0,6,8,0,0,0,0,2,11,0,0,0,0,0,0,4,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;
C42⋊8C4⋊C2 in GAP, Magma, Sage, TeX
C_4^2\rtimes_8C_4\rtimes C_2
% in TeX
G:=Group("C4^2:8C4:C2"); // GroupNames label
G:=SmallGroup(128,805);
// by ID
G=gap.SmallGroup(128,805);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,560,141,422,387,58,2028,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=a*b^2,d*a*d=a^-1*b^2*c^2,c*b*c^-1=a^2*b,d*b*d=b*c^2,d*c*d=a^-1*b*c>;
// generators/relations