p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.231D4, C42.347C23, Q8⋊Q8⋊5C2, Q16⋊C4⋊6C2, C4⋊2Q16⋊22C2, C4⋊C8.50C22, C4⋊C4.66C23, (C2×C8).40C23, Q8.6(C4○D4), C8⋊C4.7C22, (C2×C4).311C24, C23.675(C2×D4), (C22×C4).451D4, C4⋊Q8.268C22, C22⋊Q16.3C2, (C2×Q8).77C23, (C4×Q8).74C22, C4.Q8.16C22, C4.99(C8.C22), C22⋊C8.24C22, C42.6C4.1C2, (C2×Q16).57C22, (C2×C42).838C22, Q8⋊C4.32C22, C23.47D4.1C2, C22.571(C22×D4), C22⋊Q8.171C22, (C22×C4).1027C23, C22.37(C8.C22), (C22×Q8).479C22, C2.112(C22.19C24), C23.37C23.29C2, (C2×C4×Q8).54C2, C4.196(C2×C4○D4), (C2×C4).499(C2×D4), C2.34(C2×C8.C22), (C2×C4⋊C4).939C22, SmallGroup(128,1845)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.231D4
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=dbd-1=a2b, dcd-1=a2c3 >
Subgroups: 308 in 190 conjugacy classes, 92 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, Q8, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C8⋊C4, C22⋊C8, Q8⋊C4, C4⋊C8, C4.Q8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×Q16, C22×Q8, C42.6C4, Q16⋊C4, C22⋊Q16, C4⋊2Q16, Q8⋊Q8, C23.47D4, C2×C4×Q8, C23.37C23, C42.231D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8.C22, C22×D4, C2×C4○D4, C22.19C24, C2×C8.C22, C42.231D4
►On 64 points
(1 7 5 3)(2 58 6 62)(4 60 8 64)(9 15 13 11)(10 33 14 37)(12 35 16 39)(17 49 21 53)(18 24 22 20)(19 51 23 55)(25 42 29 46)(26 32 30 28)(27 44 31 48)(34 40 38 36)(41 47 45 43)(50 56 54 52)(57 63 61 59)
(1 52 63 22)(2 49 64 19)(3 54 57 24)(4 51 58 21)(5 56 59 18)(6 53 60 23)(7 50 61 20)(8 55 62 17)(9 43 38 28)(10 48 39 25)(11 45 40 30)(12 42 33 27)(13 47 34 32)(14 44 35 29)(15 41 36 26)(16 46 37 31)
(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 31 5 27)(2 30 6 26)(3 29 7 25)(4 28 8 32)(9 51 13 55)(10 50 14 54)(11 49 15 53)(12 56 16 52)(17 38 21 34)(18 37 22 33)(19 36 23 40)(20 35 24 39)(41 64 45 60)(42 63 46 59)(43 62 47 58)(44 61 48 57)
G:=sub<Sym(64)| (1,7,5,3)(2,58,6,62)(4,60,8,64)(9,15,13,11)(10,33,14,37)(12,35,16,39)(17,49,21,53)(18,24,22,20)(19,51,23,55)(25,42,29,46)(26,32,30,28)(27,44,31,48)(34,40,38,36)(41,47,45,43)(50,56,54,52)(57,63,61,59), (1,52,63,22)(2,49,64,19)(3,54,57,24)(4,51,58,21)(5,56,59,18)(6,53,60,23)(7,50,61,20)(8,55,62,17)(9,43,38,28)(10,48,39,25)(11,45,40,30)(12,42,33,27)(13,47,34,32)(14,44,35,29)(15,41,36,26)(16,46,37,31), (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), (1,31,5,27)(2,30,6,26)(3,29,7,25)(4,28,8,32)(9,51,13,55)(10,50,14,54)(11,49,15,53)(12,56,16,52)(17,38,21,34)(18,37,22,33)(19,36,23,40)(20,35,24,39)(41,64,45,60)(42,63,46,59)(43,62,47,58)(44,61,48,57)>;
G:=Group( (1,7,5,3)(2,58,6,62)(4,60,8,64)(9,15,13,11)(10,33,14,37)(12,35,16,39)(17,49,21,53)(18,24,22,20)(19,51,23,55)(25,42,29,46)(26,32,30,28)(27,44,31,48)(34,40,38,36)(41,47,45,43)(50,56,54,52)(57,63,61,59), (1,52,63,22)(2,49,64,19)(3,54,57,24)(4,51,58,21)(5,56,59,18)(6,53,60,23)(7,50,61,20)(8,55,62,17)(9,43,38,28)(10,48,39,25)(11,45,40,30)(12,42,33,27)(13,47,34,32)(14,44,35,29)(15,41,36,26)(16,46,37,31), (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), (1,31,5,27)(2,30,6,26)(3,29,7,25)(4,28,8,32)(9,51,13,55)(10,50,14,54)(11,49,15,53)(12,56,16,52)(17,38,21,34)(18,37,22,33)(19,36,23,40)(20,35,24,39)(41,64,45,60)(42,63,46,59)(43,62,47,58)(44,61,48,57) );
G=PermutationGroup([[(1,7,5,3),(2,58,6,62),(4,60,8,64),(9,15,13,11),(10,33,14,37),(12,35,16,39),(17,49,21,53),(18,24,22,20),(19,51,23,55),(25,42,29,46),(26,32,30,28),(27,44,31,48),(34,40,38,36),(41,47,45,43),(50,56,54,52),(57,63,61,59)], [(1,52,63,22),(2,49,64,19),(3,54,57,24),(4,51,58,21),(5,56,59,18),(6,53,60,23),(7,50,61,20),(8,55,62,17),(9,43,38,28),(10,48,39,25),(11,45,40,30),(12,42,33,27),(13,47,34,32),(14,44,35,29),(15,41,36,26),(16,46,37,31)], [(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)], [(1,31,5,27),(2,30,6,26),(3,29,7,25),(4,28,8,32),(9,51,13,55),(10,50,14,54),(11,49,15,53),(12,56,16,52),(17,38,21,34),(18,37,22,33),(19,36,23,40),(20,35,24,39),(41,64,45,60),(42,63,46,59),(43,62,47,58),(44,61,48,57)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | ··· | 4R | 4S | 4T | 4U | 4V | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C8.C22 | C8.C22 |
| kernel | C42.231D4 | C42.6C4 | Q16⋊C4 | C22⋊Q16 | C4⋊2Q16 | Q8⋊Q8 | C23.47D4 | C2×C4×Q8 | C23.37C23 | C42 | C22×C4 | Q8 | C4 | C22 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.231D4 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 13 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 2 |
| 0 | 0 | 0 | 1 | 15 | 0 |
| 0 | 0 | 0 | 1 | 16 | 0 |
| 0 | 0 | 16 | 0 | 0 | 16 |
| 15 | 16 | 0 | 0 | 0 | 0 |
| 3 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 16 | 9 | 3 |
| 0 | 0 | 1 | 13 | 14 | 9 |
| 0 | 0 | 5 | 3 | 4 | 1 |
| 0 | 0 | 14 | 5 | 16 | 4 |
| 15 | 16 | 0 | 0 | 0 | 0 |
| 3 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 14 | 5 | 8 |
| 0 | 0 | 14 | 2 | 8 | 12 |
| 0 | 0 | 8 | 6 | 15 | 14 |
| 0 | 0 | 6 | 9 | 14 | 2 |
G:=sub<GL(6,GF(17))| [1,13,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,16,0,0,0,1,1,0,0,0,0,15,16,0,0,0,2,0,0,16],[15,3,0,0,0,0,16,2,0,0,0,0,0,0,13,1,5,14,0,0,16,13,3,5,0,0,9,14,4,16,0,0,3,9,1,4],[15,3,0,0,0,0,16,2,0,0,0,0,0,0,15,14,8,6,0,0,14,2,6,9,0,0,5,8,15,14,0,0,8,12,14,2] >;
C42.231D4 in GAP, Magma, Sage, TeX
C_4^2._{231}D_4 % in TeX
G:=Group("C4^2.231D4"); // GroupNames label
G:=SmallGroup(128,1845);
// by ID
G=gap.SmallGroup(128,1845);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,352,521,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^3>;
// generators/relations