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⋊C4.68C23, C4⋊C8.51C22, (C2×C8).42C23, SD16⋊C4⋊9C2, (C2×C4).313C24, C8⋊C4.8C22, C42.6C4⋊6C2, (C2×D4).92C23, (C4×D4).78C22, (C2×D8).60C22, C23.253(C2×D4), (C22×C4).453D4, C4⋊Q8.269C22, C4.125(C8⋊C22), C2.D8.88C22, C4.Q8.17C22, C22⋊C8.26C22, (C2×Q8).378C23, (C4×Q8).303C22, D4⋊C4.32C22, C4⋊D4.168C22, C23.19D4⋊18C2, C4⋊1D4.142C22, (C2×C42).840C22, C22.26C24⋊7C2, 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
Subgroups: 428 in 215 conjugacy classes, 90 normal (44 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×13], C8 [×4], C2×C4 [×6], C2×C4 [×22], D4 [×2], D4 [×17], Q8 [×2], Q8 [×3], C23, C23 [×3], C42 [×4], C42 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×4], D8 [×4], SD16 [×4], C22×C4 [×3], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×5], C2×Q8, C2×Q8, C4○D4 [×8], C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×D4 [×3], C4×Q8 [×2], C4⋊D4 [×2], C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×D8 [×2], C2×SD16 [×2], C2×C4○D4, C2×C4○D4, C42.6C4, SD16⋊C4 [×2], D8⋊C4 [×2], D4⋊D4 [×2], C4⋊D8, C4⋊SD16, D4⋊Q8, Q8⋊Q8, C23.19D4 [×2], C4×C4○D4, C22.26C24, C42.233D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C8⋊C22 [×2], C22×D4, C2×C4○D4 [×2], C22.19C24, C2×C8⋊C22, D8⋊C22, C42.233D4
Generators and relations
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 >
►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 36 29 40)(27 38 31 34)(33 53 37 49)(35 55 39 51)(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 39 18)(10 45 40 23)(11 42 33 20)(12 47 34 17)(13 44 35 22)(14 41 36 19)(15 46 37 24)(16 43 38 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 37)(2 36)(3 35)(4 34)(5 33)(6 40)(7 39)(8 38)(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)(25,36,29,40)(27,38,31,34)(33,53,37,49)(35,55,39,51)(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)(9,48,39,18)(10,45,40,23)(11,42,33,20)(12,47,34,17)(13,44,35,22)(14,41,36,19)(15,46,37,24)(16,43,38,21), (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,40)(7,39)(8,38)(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)(25,36,29,40)(27,38,31,34)(33,53,37,49)(35,55,39,51)(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)(9,48,39,18)(10,45,40,23)(11,42,33,20)(12,47,34,17)(13,44,35,22)(14,41,36,19)(15,46,37,24)(16,43,38,21), (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,40)(7,39)(8,38)(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),(25,36,29,40),(27,38,31,34),(33,53,37,49),(35,55,39,51),(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),(9,48,39,18),(10,45,40,23),(11,42,33,20),(12,47,34,17),(13,44,35,22),(14,41,36,19),(15,46,37,24),(16,43,38,21)], [(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)], [(1,37),(2,36),(3,35),(4,34),(5,33),(6,40),(7,39),(8,38),(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)])
Matrix representation ►G ⊆ 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] >;
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 |
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