p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.233D4, C42.349C23, D4⋊4(C4○D4), D8⋊C4⋊8C2, C4⋊D8⋊22C2, Q8⋊4(C4○D4), C4⋊SD16⋊6C2, Q8⋊Q8⋊6C2, D4⋊D4⋊19C2, D4⋊Q8⋊23C2, C4⋊C8.51C22, C4⋊C4.68C23, (C2×C8).42C23, SD16⋊C4⋊9C2, C8⋊C4.8C22, (C2×C4).313C24, C42.6C4⋊6C2, (C2×D8).60C22, (C4×D4).78C22, (C2×D4).92C23, (C22×C4).453D4, C23.253(C2×D4), C4⋊Q8.269C22, C4.125(C8⋊C22), C4.Q8.17C22, C2.D8.88C22, C22⋊C8.26C22, (C4×Q8).303C22, (C2×Q8).378C23, D4⋊C4.32C22, C4⋊1D4.142C22, C4⋊D4.168C22, C23.19D4⋊18C2, C22.26C24⋊7C2, (C2×C42).840C22, Q8⋊C4.33C22, (C2×SD16).14C22, C22.573(C22×D4), C2.32(D8⋊C22), (C22×C4).1029C23, C42⋊C2.323C22, C2.114(C22.19C24), (C4×C4○D4)⋊12C2, C4.198(C2×C4○D4), C2.36(C2×C8⋊C22), (C2×C4).1221(C2×D4), (C2×C4○D4).314C22, SmallGroup(128,1847)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.233D4
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=dad=a-1b2, cbc-1=a2b, bd=db, dcd=a2c3 >
Subgroups: 428 in 215 conjugacy classes, 90 normal (44 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×D8, C2×SD16, C2×C4○D4, C2×C4○D4, C42.6C4, SD16⋊C4, D8⋊C4, D4⋊D4, C4⋊D8, C4⋊SD16, D4⋊Q8, Q8⋊Q8, C23.19D4, C4×C4○D4, C22.26C24, C42.233D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8⋊C22, C22×D4, C2×C4○D4, C22.19C24, C2×C8⋊C22, D8⋊C22, C42.233D4
►On 64 points
(1 20 5 24)(2 47 6 43)(3 22 7 18)(4 41 8 45)(9 28 13 32)(10 56 14 52)(11 30 15 26)(12 50 16 54)(17 63 21 59)(19 57 23 61)(25 38 29 34)(27 40 31 36)(33 51 37 55)(35 53 39 49)(42 62 46 58)(44 64 48 60)
(1 26 58 49)(2 31 59 54)(3 28 60 51)(4 25 61 56)(5 30 62 53)(6 27 63 50)(7 32 64 55)(8 29 57 52)(9 48 33 18)(10 45 34 23)(11 42 35 20)(12 47 36 17)(13 44 37 22)(14 41 38 19)(15 46 39 24)(16 43 40 21)
(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 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 40)(9 64)(10 63)(11 62)(12 61)(13 60)(14 59)(15 58)(16 57)(17 25)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 56)(48 55)
G:=sub<Sym(64)| (1,20,5,24)(2,47,6,43)(3,22,7,18)(4,41,8,45)(9,28,13,32)(10,56,14,52)(11,30,15,26)(12,50,16,54)(17,63,21,59)(19,57,23,61)(25,38,29,34)(27,40,31,36)(33,51,37,55)(35,53,39,49)(42,62,46,58)(44,64,48,60), (1,26,58,49)(2,31,59,54)(3,28,60,51)(4,25,61,56)(5,30,62,53)(6,27,63,50)(7,32,64,55)(8,29,57,52)(9,48,33,18)(10,45,34,23)(11,42,35,20)(12,47,36,17)(13,44,37,22)(14,41,38,19)(15,46,39,24)(16,43,40,21), (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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,40)(9,64)(10,63)(11,62)(12,61)(13,60)(14,59)(15,58)(16,57)(17,25)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,56)(48,55)>;
G:=Group( (1,20,5,24)(2,47,6,43)(3,22,7,18)(4,41,8,45)(9,28,13,32)(10,56,14,52)(11,30,15,26)(12,50,16,54)(17,63,21,59)(19,57,23,61)(25,38,29,34)(27,40,31,36)(33,51,37,55)(35,53,39,49)(42,62,46,58)(44,64,48,60), (1,26,58,49)(2,31,59,54)(3,28,60,51)(4,25,61,56)(5,30,62,53)(6,27,63,50)(7,32,64,55)(8,29,57,52)(9,48,33,18)(10,45,34,23)(11,42,35,20)(12,47,36,17)(13,44,37,22)(14,41,38,19)(15,46,39,24)(16,43,40,21), (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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,40)(9,64)(10,63)(11,62)(12,61)(13,60)(14,59)(15,58)(16,57)(17,25)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,56)(48,55) );
G=PermutationGroup([[(1,20,5,24),(2,47,6,43),(3,22,7,18),(4,41,8,45),(9,28,13,32),(10,56,14,52),(11,30,15,26),(12,50,16,54),(17,63,21,59),(19,57,23,61),(25,38,29,34),(27,40,31,36),(33,51,37,55),(35,53,39,49),(42,62,46,58),(44,64,48,60)], [(1,26,58,49),(2,31,59,54),(3,28,60,51),(4,25,61,56),(5,30,62,53),(6,27,63,50),(7,32,64,55),(8,29,57,52),(9,48,33,18),(10,45,34,23),(11,42,35,20),(12,47,36,17),(13,44,37,22),(14,41,38,19),(15,46,39,24),(16,43,40,21)], [(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)], [(1,39),(2,38),(3,37),(4,36),(5,35),(6,34),(7,33),(8,40),(9,64),(10,63),(11,62),(12,61),(13,60),(14,59),(15,58),(16,57),(17,25),(18,32),(19,31),(20,30),(21,29),(22,28),(23,27),(24,26),(41,54),(42,53),(43,52),(44,51),(45,50),(46,49),(47,56),(48,55)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4J | 4K | ··· | 4Q | 4R | 4S | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D4 | C8⋊C22 | D8⋊C22 |
| kernel | C42.233D4 | C42.6C4 | SD16⋊C4 | D8⋊C4 | D4⋊D4 | C4⋊D8 | C4⋊SD16 | D4⋊Q8 | Q8⋊Q8 | C23.19D4 | C4×C4○D4 | C22.26C24 | C42 | C22×C4 | D4 | Q8 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 |
Matrix representation of C42.233D4 ►in GL6(𝔽17)
| 0 | 16 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 15 | 0 |
| 0 | 0 | 1 | 0 | 0 | 15 |
| 0 | 0 | 1 | 0 | 0 | 16 |
| 0 | 0 | 0 | 1 | 16 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 | 8 | 8 |
| 0 | 0 | 16 | 1 | 9 | 8 |
| 0 | 0 | 0 | 1 | 16 | 7 |
| 0 | 0 | 10 | 0 | 1 | 7 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 9 | 0 |
| 0 | 0 | 13 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 13 | 0 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,1,0,0,0,1,0,0,1,0,0,15,0,0,16,0,0,0,15,16,0],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,10,16,0,10,0,0,10,1,1,0,0,0,8,9,16,1,0,0,8,8,7,7],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0,0,0,0,0,9,0,0,13,0,0,0,8,4,0] >;
C42.233D4 in GAP, Magma, Sage, TeX
C_4^2._{233}D_4 % in TeX
G:=Group("C4^2.233D4"); // GroupNames label
G:=SmallGroup(128,1847);
// by ID
G=gap.SmallGroup(128,1847);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,521,304,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=a^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^2,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=a^2*c^3>;
// generators/relations