p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.168D4, C24.29C23, C23.445C24, C22.1782- (1+4), (C2×Q8).167D4, C4.59(C4⋊D4), C2.38(Q8⋊5D4), C23.Q8⋊28C2, (C22×C4).537C23, (C2×C42).550C22, C22.296(C22×D4), C24.C22⋊79C2, (C22×D4).166C22, (C22×Q8).433C22, C23.67C23⋊59C2, C24.3C22.45C2, C2.C42.183C22, C2.32(C22.26C24), C2.37(C22.50C24), C2.22(C23.38C23), (C4×C4⋊C4)⋊89C2, (C2×C4×Q8)⋊23C2, (C2×C4⋊Q8)⋊13C2, (C2×C4).354(C2×D4), C2.37(C2×C4⋊D4), (C2×C4).148(C4○D4), (C2×C4⋊C4).301C22, (C2×C4.4D4).24C2, C22.322(C2×C4○D4), (C2×C22⋊C4).179C22, SmallGroup(128,1277)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 532 in 288 conjugacy classes, 112 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×4], C4 [×18], C22 [×3], C22 [×4], C22 [×14], C2×C4 [×18], C2×C4 [×30], D4 [×4], Q8 [×12], C23, C23 [×14], C42 [×4], C42 [×8], C22⋊C4 [×24], C4⋊C4 [×18], C22×C4 [×3], C22×C4 [×10], C2×D4 [×6], C2×Q8 [×4], C2×Q8 [×10], C24 [×2], C2.C42 [×6], C2×C42 [×3], C2×C42 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4, C2×C4⋊C4 [×8], C4×Q8 [×4], C4.4D4 [×8], C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], C4×C4⋊C4, C24.C22 [×4], C24.3C22, C23.67C23, C23.Q8 [×4], C2×C4×Q8, C2×C4.4D4 [×2], C2×C4⋊Q8, C42.168D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4 [×3], 2- (1+4) [×2], C2×C4⋊D4, C22.26C24, C23.38C23, Q8⋊5D4 [×2], C22.50C24 [×2], C42.168D4
Generators and relations
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=c-1 >
(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 12 21 43)(2 9 22 44)(3 10 23 41)(4 11 24 42)(5 56 40 26)(6 53 37 27)(7 54 38 28)(8 55 39 25)(13 64 48 36)(14 61 45 33)(15 62 46 34)(16 63 47 35)(17 30 52 60)(18 31 49 57)(19 32 50 58)(20 29 51 59)
(1 47 39 59)(2 46 40 58)(3 45 37 57)(4 48 38 60)(5 32 22 15)(6 31 23 14)(7 30 24 13)(8 29 21 16)(9 34 26 19)(10 33 27 18)(11 36 28 17)(12 35 25 20)(41 61 53 49)(42 64 54 52)(43 63 55 51)(44 62 56 50)
(1 2 21 22)(3 4 23 24)(5 39 40 8)(6 7 37 38)(9 12 44 43)(10 42 41 11)(13 57 48 31)(14 30 45 60)(15 59 46 29)(16 32 47 58)(17 33 52 61)(18 64 49 36)(19 35 50 63)(20 62 51 34)(25 56 55 26)(27 54 53 28)
G:=sub<Sym(64)| (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), (1,12,21,43)(2,9,22,44)(3,10,23,41)(4,11,24,42)(5,56,40,26)(6,53,37,27)(7,54,38,28)(8,55,39,25)(13,64,48,36)(14,61,45,33)(15,62,46,34)(16,63,47,35)(17,30,52,60)(18,31,49,57)(19,32,50,58)(20,29,51,59), (1,47,39,59)(2,46,40,58)(3,45,37,57)(4,48,38,60)(5,32,22,15)(6,31,23,14)(7,30,24,13)(8,29,21,16)(9,34,26,19)(10,33,27,18)(11,36,28,17)(12,35,25,20)(41,61,53,49)(42,64,54,52)(43,63,55,51)(44,62,56,50), (1,2,21,22)(3,4,23,24)(5,39,40,8)(6,7,37,38)(9,12,44,43)(10,42,41,11)(13,57,48,31)(14,30,45,60)(15,59,46,29)(16,32,47,58)(17,33,52,61)(18,64,49,36)(19,35,50,63)(20,62,51,34)(25,56,55,26)(27,54,53,28)>;
G:=Group( (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), (1,12,21,43)(2,9,22,44)(3,10,23,41)(4,11,24,42)(5,56,40,26)(6,53,37,27)(7,54,38,28)(8,55,39,25)(13,64,48,36)(14,61,45,33)(15,62,46,34)(16,63,47,35)(17,30,52,60)(18,31,49,57)(19,32,50,58)(20,29,51,59), (1,47,39,59)(2,46,40,58)(3,45,37,57)(4,48,38,60)(5,32,22,15)(6,31,23,14)(7,30,24,13)(8,29,21,16)(9,34,26,19)(10,33,27,18)(11,36,28,17)(12,35,25,20)(41,61,53,49)(42,64,54,52)(43,63,55,51)(44,62,56,50), (1,2,21,22)(3,4,23,24)(5,39,40,8)(6,7,37,38)(9,12,44,43)(10,42,41,11)(13,57,48,31)(14,30,45,60)(15,59,46,29)(16,32,47,58)(17,33,52,61)(18,64,49,36)(19,35,50,63)(20,62,51,34)(25,56,55,26)(27,54,53,28) );
G=PermutationGroup([(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)], [(1,12,21,43),(2,9,22,44),(3,10,23,41),(4,11,24,42),(5,56,40,26),(6,53,37,27),(7,54,38,28),(8,55,39,25),(13,64,48,36),(14,61,45,33),(15,62,46,34),(16,63,47,35),(17,30,52,60),(18,31,49,57),(19,32,50,58),(20,29,51,59)], [(1,47,39,59),(2,46,40,58),(3,45,37,57),(4,48,38,60),(5,32,22,15),(6,31,23,14),(7,30,24,13),(8,29,21,16),(9,34,26,19),(10,33,27,18),(11,36,28,17),(12,35,25,20),(41,61,53,49),(42,64,54,52),(43,63,55,51),(44,62,56,50)], [(1,2,21,22),(3,4,23,24),(5,39,40,8),(6,7,37,38),(9,12,44,43),(10,42,41,11),(13,57,48,31),(14,30,45,60),(15,59,46,29),(16,32,47,58),(17,33,52,61),(18,64,49,36),(19,35,50,63),(20,62,51,34),(25,56,55,26),(27,54,53,28)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 2 | 4 | 0 | 0 | 0 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 4 | 4 | 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 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 2 | 4 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(5))| [2,3,0,0,0,0,4,3,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[1,4,0,0,0,0,2,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,1,0,0,0,0,3,1,0,0,0,0,0,0,1,1,0,0,0,0,3,4,0,0,0,0,0,0,3,0,0,0,0,0,0,2],[2,0,0,0,0,0,4,3,0,0,0,0,0,0,4,4,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4Z | 4AA | 4AB |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | 2- (1+4) |
| kernel | C42.168D4 | C4×C4⋊C4 | C24.C22 | C24.3C22 | C23.67C23 | C23.Q8 | C2×C4×Q8 | C2×C4.4D4 | C2×C4⋊Q8 | C42 | C2×Q8 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 1 | 2 | 1 | 4 | 4 | 12 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{168}D_4 % in TeX
G:=Group("C4^2.168D4"); // GroupNames label
G:=SmallGroup(128,1277);
// by ID
G=gap.SmallGroup(128,1277);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,456,758,723,184,675,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations