p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.223D4, C42.339C23, Q8⋊1(C4○D4), (C2×C4)⋊17SD16, C4○2(Q8⋊D4), C4○2(Q8⋊Q8), C4○2(C4⋊SD16), Q8⋊D4⋊37C2, Q8⋊Q8⋊49C2, C4⋊SD16⋊48C2, (C4×SD16)⋊29C2, C4⋊C4.56C23, C4.85(C2×SD16), C4⋊C8.334C22, (C2×C8).316C23, (C4×C8).261C22, (C2×C4).301C24, (C2×D4).86C23, (C4×D4).73C22, (C22×C4).803D4, C23.668(C2×D4), C4⋊Q8.263C22, (C2×Q8).372C23, (C4×Q8).300C22, C2.15(C22×SD16), C22.36(C2×SD16), C42.12C4⋊41C2, C4.Q8.151C22, C4○2(C23.46D4), C23.46D4⋊36C2, C4⋊D4.160C22, C4⋊1D4.139C22, C22⋊C8.215C22, (C2×C42).828C22, C22.561(C22×D4), D4⋊C4.183C22, C2.28(D8⋊C22), (C22×C4).1017C23, Q8⋊C4.174C22, (C2×SD16).138C22, (C22×Q8).475C22, C22.26C24.31C2, C2.102(C22.19C24), (C2×C4×Q8)⋊37C2, C4.186(C2×C4○D4), (C2×C4).1581(C2×D4), (C2×C4⋊C4).932C22, SmallGroup(128,1835)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.223D4
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=a-1b2, dad=ab2, bc=cb, dbd=a2b, dcd=c3 >
Subgroups: 404 in 214 conjugacy classes, 98 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C4×Q8, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×SD16, C22×Q8, C2×C4○D4, C42.12C4, C4×SD16, Q8⋊D4, C4⋊SD16, Q8⋊Q8, C23.46D4, C2×C4×Q8, C22.26C24, C42.223D4
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C24, C2×SD16, C22×D4, C2×C4○D4, C22.19C24, C22×SD16, D8⋊C22, C42.223D4
►On 64 points
(1 23 5 19)(2 41 6 45)(3 17 7 21)(4 43 8 47)(9 56 13 52)(10 30 14 26)(11 50 15 54)(12 32 16 28)(18 62 22 58)(20 64 24 60)(25 38 29 34)(27 40 31 36)(33 51 37 55)(35 53 39 49)(42 61 46 57)(44 63 48 59)
(1 28 59 51)(2 29 60 52)(3 30 61 53)(4 31 62 54)(5 32 63 55)(6 25 64 56)(7 26 57 49)(8 27 58 50)(9 41 34 20)(10 42 35 21)(11 43 36 22)(12 44 37 23)(13 45 38 24)(14 46 39 17)(15 47 40 18)(16 48 33 19)
(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 37)(2 40)(3 35)(4 38)(5 33)(6 36)(7 39)(8 34)(9 58)(10 61)(11 64)(12 59)(13 62)(14 57)(15 60)(16 63)(17 30)(18 25)(19 28)(20 31)(21 26)(22 29)(23 32)(24 27)(41 54)(42 49)(43 52)(44 55)(45 50)(46 53)(47 56)(48 51)
G:=sub<Sym(64)| (1,23,5,19)(2,41,6,45)(3,17,7,21)(4,43,8,47)(9,56,13,52)(10,30,14,26)(11,50,15,54)(12,32,16,28)(18,62,22,58)(20,64,24,60)(25,38,29,34)(27,40,31,36)(33,51,37,55)(35,53,39,49)(42,61,46,57)(44,63,48,59), (1,28,59,51)(2,29,60,52)(3,30,61,53)(4,31,62,54)(5,32,63,55)(6,25,64,56)(7,26,57,49)(8,27,58,50)(9,41,34,20)(10,42,35,21)(11,43,36,22)(12,44,37,23)(13,45,38,24)(14,46,39,17)(15,47,40,18)(16,48,33,19), (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), (1,37)(2,40)(3,35)(4,38)(5,33)(6,36)(7,39)(8,34)(9,58)(10,61)(11,64)(12,59)(13,62)(14,57)(15,60)(16,63)(17,30)(18,25)(19,28)(20,31)(21,26)(22,29)(23,32)(24,27)(41,54)(42,49)(43,52)(44,55)(45,50)(46,53)(47,56)(48,51)>;
G:=Group( (1,23,5,19)(2,41,6,45)(3,17,7,21)(4,43,8,47)(9,56,13,52)(10,30,14,26)(11,50,15,54)(12,32,16,28)(18,62,22,58)(20,64,24,60)(25,38,29,34)(27,40,31,36)(33,51,37,55)(35,53,39,49)(42,61,46,57)(44,63,48,59), (1,28,59,51)(2,29,60,52)(3,30,61,53)(4,31,62,54)(5,32,63,55)(6,25,64,56)(7,26,57,49)(8,27,58,50)(9,41,34,20)(10,42,35,21)(11,43,36,22)(12,44,37,23)(13,45,38,24)(14,46,39,17)(15,47,40,18)(16,48,33,19), (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), (1,37)(2,40)(3,35)(4,38)(5,33)(6,36)(7,39)(8,34)(9,58)(10,61)(11,64)(12,59)(13,62)(14,57)(15,60)(16,63)(17,30)(18,25)(19,28)(20,31)(21,26)(22,29)(23,32)(24,27)(41,54)(42,49)(43,52)(44,55)(45,50)(46,53)(47,56)(48,51) );
G=PermutationGroup([[(1,23,5,19),(2,41,6,45),(3,17,7,21),(4,43,8,47),(9,56,13,52),(10,30,14,26),(11,50,15,54),(12,32,16,28),(18,62,22,58),(20,64,24,60),(25,38,29,34),(27,40,31,36),(33,51,37,55),(35,53,39,49),(42,61,46,57),(44,63,48,59)], [(1,28,59,51),(2,29,60,52),(3,30,61,53),(4,31,62,54),(5,32,63,55),(6,25,64,56),(7,26,57,49),(8,27,58,50),(9,41,34,20),(10,42,35,21),(11,43,36,22),(12,44,37,23),(13,45,38,24),(14,46,39,17),(15,47,40,18),(16,48,33,19)], [(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)], [(1,37),(2,40),(3,35),(4,38),(5,33),(6,36),(7,39),(8,34),(9,58),(10,61),(11,64),(12,59),(13,62),(14,57),(15,60),(16,63),(17,30),(18,25),(19,28),(20,31),(21,26),(22,29),(23,32),(24,27),(41,54),(42,49),(43,52),(44,55),(45,50),(46,53),(47,56),(48,51)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4T | 4U | 4V | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | SD16 | C4○D4 | D8⋊C22 |
| kernel | C42.223D4 | C42.12C4 | C4×SD16 | Q8⋊D4 | C4⋊SD16 | Q8⋊Q8 | C23.46D4 | C2×C4×Q8 | C22.26C24 | C42 | C22×C4 | C2×C4 | Q8 | C2 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 8 | 2 |
Matrix representation of C42.223D4 ►in GL4(𝔽17) generated by
| 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 5 | 12 |
| 0 | 0 | 5 | 5 |
| 0 | 13 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [0,16,0,0,16,0,0,0,0,0,0,16,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,0,16,0,0,1,0],[4,0,0,0,0,13,0,0,0,0,5,5,0,0,12,5],[0,4,0,0,13,0,0,0,0,0,1,0,0,0,0,16] >;
C42.223D4 in GAP, Magma, Sage, TeX
C_4^2._{223}D_4 % in TeX
G:=Group("C4^2.223D4"); // GroupNames label
G:=SmallGroup(128,1835);
// by ID
G=gap.SmallGroup(128,1835);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,80,4037,1027,124]);
// 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=a^-1*b^2,d*a*d=a*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^3>;
// generators/relations