p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.230D4, C42.346C23, Q8⋊3(C4○D4), C4⋊SD16⋊5C2, Q8⋊D4⋊5C2, C4.Q16⋊22C2, C4⋊C4.65C23, C4⋊C8.49C22, (C2×C8).39C23, SD16⋊C4⋊7C2, C8⋊C4.6C22, C42.6C4⋊4C2, (C2×C4).310C24, (C4×D4).77C22, (C2×D4).91C23, C23.674(C2×D4), (C22×C4).450D4, C4⋊Q8.267C22, C2.D8.87C22, C22.D8⋊15C2, C4.98(C8.C22), C22⋊C8.23C22, (C2×Q8).377C23, (C4×Q8).302C22, D4⋊C4.31C22, C4⋊D4.166C22, C4⋊1D4.141C22, C22.48(C8⋊C22), (C2×C42).837C22, Q8⋊C4.31C22, (C2×SD16).12C22, C22.570(C22×D4), (C22×C4).1026C23, (C22×Q8).478C22, C22.26C24.32C2, C2.111(C22.19C24), (C2×C4×Q8)⋊39C2, C4.195(C2×C4○D4), (C2×C4).498(C2×D4), C2.34(C2×C8⋊C22), C2.33(C2×C8.C22), (C2×C4⋊C4).938C22, SmallGroup(128,1844)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 404 in 212 conjugacy classes, 92 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×12], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×6], C2×C4 [×22], D4 [×12], Q8 [×4], Q8 [×8], C23, C23 [×2], C42 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], SD16 [×8], C22×C4 [×3], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C2×Q8 [×6], C4○D4 [×4], C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C2.D8 [×4], C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×D4 [×2], C4×D4, C4×Q8 [×4], C4×Q8 [×2], C4⋊D4 [×2], C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×SD16 [×4], C22×Q8, C2×C4○D4, C42.6C4, SD16⋊C4 [×4], Q8⋊D4 [×2], C4⋊SD16 [×2], C4.Q16 [×2], C22.D8 [×2], C2×C4×Q8, C22.26C24, C42.230D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C8⋊C22 [×2], C8.C22 [×2], C22×D4, C2×C4○D4 [×2], C22.19C24, C2×C8⋊C22, C2×C8.C22, C42.230D4
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=a-1b2, dad=ab2, cbc-1=dbd=a2b, dcd=c3 >
►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 34 29 38)(27 36 31 40)(33 55 37 51)(35 49 39 53)(42 61 46 57)(44 63 48 59)
(1 28 59 51)(2 25 60 56)(3 30 61 53)(4 27 62 50)(5 32 63 55)(6 29 64 52)(7 26 57 49)(8 31 58 54)(9 45 38 24)(10 42 39 21)(11 47 40 18)(12 44 33 23)(13 41 34 20)(14 46 35 17)(15 43 36 22)(16 48 37 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 33)(2 36)(3 39)(4 34)(5 37)(6 40)(7 35)(8 38)(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)(25,34,29,38)(27,36,31,40)(33,55,37,51)(35,49,39,53)(42,61,46,57)(44,63,48,59), (1,28,59,51)(2,25,60,56)(3,30,61,53)(4,27,62,50)(5,32,63,55)(6,29,64,52)(7,26,57,49)(8,31,58,54)(9,45,38,24)(10,42,39,21)(11,47,40,18)(12,44,33,23)(13,41,34,20)(14,46,35,17)(15,43,36,22)(16,48,37,19), (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), (1,33)(2,36)(3,39)(4,34)(5,37)(6,40)(7,35)(8,38)(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)(25,34,29,38)(27,36,31,40)(33,55,37,51)(35,49,39,53)(42,61,46,57)(44,63,48,59), (1,28,59,51)(2,25,60,56)(3,30,61,53)(4,27,62,50)(5,32,63,55)(6,29,64,52)(7,26,57,49)(8,31,58,54)(9,45,38,24)(10,42,39,21)(11,47,40,18)(12,44,33,23)(13,41,34,20)(14,46,35,17)(15,43,36,22)(16,48,37,19), (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), (1,33)(2,36)(3,39)(4,34)(5,37)(6,40)(7,35)(8,38)(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),(25,34,29,38),(27,36,31,40),(33,55,37,51),(35,49,39,53),(42,61,46,57),(44,63,48,59)], [(1,28,59,51),(2,25,60,56),(3,30,61,53),(4,27,62,50),(5,32,63,55),(6,29,64,52),(7,26,57,49),(8,31,58,54),(9,45,38,24),(10,42,39,21),(11,47,40,18),(12,44,33,23),(13,41,34,20),(14,46,35,17),(15,43,36,22),(16,48,37,19)], [(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)], [(1,33),(2,36),(3,39),(4,34),(5,37),(6,40),(7,35),(8,38),(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)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 16 | 15 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 2 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 0 | 1 | 16 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 13 | 9 | 0 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 13 | 9 | 0 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,15,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,1,1,0,0,0,0,15,16,0,0],[13,4,0,0,0,0,9,4,0,0,0,0,0,0,0,0,7,12,0,0,0,0,10,0,0,0,10,5,0,0,0,0,7,0,0,0],[13,4,0,0,0,0,9,4,0,0,0,0,0,0,0,0,16,16,0,0,0,0,0,1,0,0,16,16,0,0,0,0,0,1,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | ··· | 4R | 4S | 4T | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 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 | 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 | C42.230D4 | C42.6C4 | SD16⋊C4 | Q8⋊D4 | C4⋊SD16 | C4.Q16 | C22.D8 | C2×C4×Q8 | C22.26C24 | C42 | C22×C4 | Q8 | C4 | C22 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{230}D_4 % in TeX
G:=Group("C4^2.230D4"); // GroupNames label
G:=SmallGroup(128,1844);
// by ID
G=gap.SmallGroup(128,1844);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,521,80,4037,1027,124]);
// 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,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^3>;
// generators/relations