p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.367C23, C4⋊C4.239D4, C4⋊C4.90C23, C2.21(Q8○D8), C22⋊C4.79D4, (C2×C8).456C23, (C2×C4).335C24, (C4×C8).350C22, C4.SD16⋊40C2, C23.453(C2×D4), C4⋊Q8.110C22, (C2×Q8).91C23, C8○2M4(2)⋊38C2, C2.34(D4○SD16), C4.46(C4.4D4), (C2×D4).103C23, C8⋊C4.168C22, C23.38D4⋊36C2, (C22×C8).462C22, C4.4D4.32C22, C22.595(C22×D4), C42.C2.16C22, D4⋊C4.132C22, (C22×C4).1033C23, C23.41C23⋊7C2, Q8⋊C4.203C22, C23.24D4.12C2, C23.36D4.14C2, C22.16(C4.4D4), (C22×Q8).301C22, C42.78C22⋊26C2, C42⋊C2.140C22, C42.30C22⋊17C2, C42.28C22⋊30C2, (C2×M4(2)).372C22, C23.38C23.13C2, C4.44(C2×C4○D4), (C2×C4).137(C2×D4), (C2×Q8⋊C4)⋊58C2, C2.46(C2×C4.4D4), (C2×C4).490(C4○D4), (C2×C4⋊C4).625C22, (C2×C4○D4).150C22, SmallGroup(128,1869)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 340 in 186 conjugacy classes, 92 normal (38 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×2], C22 [×5], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], D4 [×4], Q8 [×10], C23, C23, C42 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×11], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C2×Q8 [×7], C4○D4 [×4], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], Q8⋊C4 [×10], C2×C4⋊C4, C42⋊C2 [×2], C22⋊Q8 [×4], C22.D4 [×2], C4.4D4 [×2], C42.C2 [×2], C42.C2, C4⋊Q8 [×4], C4⋊Q8, C22×C8, C2×M4(2), C22×Q8, C2×C4○D4, C8○2M4(2), C2×Q8⋊C4, C23.24D4, C23.36D4, C23.38D4, C4.SD16 [×2], C42.78C22 [×2], C42.28C22 [×2], C42.30C22 [×2], C23.38C23, C23.41C23, C42.367C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], C2×C4.4D4, D4○SD16, Q8○D8, C42.367C23
Generators and relations
G = < a,b,c,d,e | a4=b4=1, c2=b2, d2=a2b-1, e2=a2b2, ab=ba, cac-1=a-1b2, ad=da, eae-1=ab2, cbc-1=b-1, bd=db, be=eb, dcd-1=bc, ece-1=a2b2c, de=ed >
►On 64 points
(1 23 9 43)(2 24 10 44)(3 17 11 45)(4 18 12 46)(5 19 13 47)(6 20 14 48)(7 21 15 41)(8 22 16 42)(25 49 37 60)(26 50 38 61)(27 51 39 62)(28 52 40 63)(29 53 33 64)(30 54 34 57)(31 55 35 58)(32 56 36 59)
(1 15 5 11)(2 16 6 12)(3 9 7 13)(4 10 8 14)(17 43 21 47)(18 44 22 48)(19 45 23 41)(20 46 24 42)(25 35 29 39)(26 36 30 40)(27 37 31 33)(28 38 32 34)(49 58 53 62)(50 59 54 63)(51 60 55 64)(52 61 56 57)
(1 43 5 47)(2 18 6 22)(3 41 7 45)(4 24 8 20)(9 23 13 19)(10 46 14 42)(11 21 15 17)(12 44 16 48)(25 64 29 60)(26 56 30 52)(27 62 31 58)(28 54 32 50)(33 49 37 53)(34 63 38 59)(35 55 39 51)(36 61 40 57)
(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 60 13 53)(2 61 14 54)(3 62 15 55)(4 63 16 56)(5 64 9 49)(6 57 10 50)(7 58 11 51)(8 59 12 52)(17 31 41 39)(18 32 42 40)(19 25 43 33)(20 26 44 34)(21 27 45 35)(22 28 46 36)(23 29 47 37)(24 30 48 38)
G:=sub<Sym(64)| (1,23,9,43)(2,24,10,44)(3,17,11,45)(4,18,12,46)(5,19,13,47)(6,20,14,48)(7,21,15,41)(8,22,16,42)(25,49,37,60)(26,50,38,61)(27,51,39,62)(28,52,40,63)(29,53,33,64)(30,54,34,57)(31,55,35,58)(32,56,36,59), (1,15,5,11)(2,16,6,12)(3,9,7,13)(4,10,8,14)(17,43,21,47)(18,44,22,48)(19,45,23,41)(20,46,24,42)(25,35,29,39)(26,36,30,40)(27,37,31,33)(28,38,32,34)(49,58,53,62)(50,59,54,63)(51,60,55,64)(52,61,56,57), (1,43,5,47)(2,18,6,22)(3,41,7,45)(4,24,8,20)(9,23,13,19)(10,46,14,42)(11,21,15,17)(12,44,16,48)(25,64,29,60)(26,56,30,52)(27,62,31,58)(28,54,32,50)(33,49,37,53)(34,63,38,59)(35,55,39,51)(36,61,40,57), (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), (1,60,13,53)(2,61,14,54)(3,62,15,55)(4,63,16,56)(5,64,9,49)(6,57,10,50)(7,58,11,51)(8,59,12,52)(17,31,41,39)(18,32,42,40)(19,25,43,33)(20,26,44,34)(21,27,45,35)(22,28,46,36)(23,29,47,37)(24,30,48,38)>;
G:=Group( (1,23,9,43)(2,24,10,44)(3,17,11,45)(4,18,12,46)(5,19,13,47)(6,20,14,48)(7,21,15,41)(8,22,16,42)(25,49,37,60)(26,50,38,61)(27,51,39,62)(28,52,40,63)(29,53,33,64)(30,54,34,57)(31,55,35,58)(32,56,36,59), (1,15,5,11)(2,16,6,12)(3,9,7,13)(4,10,8,14)(17,43,21,47)(18,44,22,48)(19,45,23,41)(20,46,24,42)(25,35,29,39)(26,36,30,40)(27,37,31,33)(28,38,32,34)(49,58,53,62)(50,59,54,63)(51,60,55,64)(52,61,56,57), (1,43,5,47)(2,18,6,22)(3,41,7,45)(4,24,8,20)(9,23,13,19)(10,46,14,42)(11,21,15,17)(12,44,16,48)(25,64,29,60)(26,56,30,52)(27,62,31,58)(28,54,32,50)(33,49,37,53)(34,63,38,59)(35,55,39,51)(36,61,40,57), (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), (1,60,13,53)(2,61,14,54)(3,62,15,55)(4,63,16,56)(5,64,9,49)(6,57,10,50)(7,58,11,51)(8,59,12,52)(17,31,41,39)(18,32,42,40)(19,25,43,33)(20,26,44,34)(21,27,45,35)(22,28,46,36)(23,29,47,37)(24,30,48,38) );
G=PermutationGroup([(1,23,9,43),(2,24,10,44),(3,17,11,45),(4,18,12,46),(5,19,13,47),(6,20,14,48),(7,21,15,41),(8,22,16,42),(25,49,37,60),(26,50,38,61),(27,51,39,62),(28,52,40,63),(29,53,33,64),(30,54,34,57),(31,55,35,58),(32,56,36,59)], [(1,15,5,11),(2,16,6,12),(3,9,7,13),(4,10,8,14),(17,43,21,47),(18,44,22,48),(19,45,23,41),(20,46,24,42),(25,35,29,39),(26,36,30,40),(27,37,31,33),(28,38,32,34),(49,58,53,62),(50,59,54,63),(51,60,55,64),(52,61,56,57)], [(1,43,5,47),(2,18,6,22),(3,41,7,45),(4,24,8,20),(9,23,13,19),(10,46,14,42),(11,21,15,17),(12,44,16,48),(25,64,29,60),(26,56,30,52),(27,62,31,58),(28,54,32,50),(33,49,37,53),(34,63,38,59),(35,55,39,51),(36,61,40,57)], [(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)], [(1,60,13,53),(2,61,14,54),(3,62,15,55),(4,63,16,56),(5,64,9,49),(6,57,10,50),(7,58,11,51),(8,59,12,52),(17,31,41,39),(18,32,42,40),(19,25,43,33),(20,26,44,34),(21,27,45,35),(22,28,46,36),(23,29,47,37),(24,30,48,38)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 13 | 8 | 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 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 | 8 | 0 | 0 | 0 | 0 |
| 13 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 14 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 14 | 3 |
| 4 | 9 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,8,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[13,13,0,0,0,0,8,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,14,0,0,0,0,3,3,0,0,0,0,0,0,3,14,0,0,0,0,3,3],[4,0,0,0,0,0,9,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,13,0,0,0,0,0,0,13,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4O | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 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 | C2 | C2 | C2 | D4 | D4 | C4○D4 | D4○SD16 | Q8○D8 |
| kernel | C42.367C23 | C8○2M4(2) | C2×Q8⋊C4 | C23.24D4 | C23.36D4 | C23.38D4 | C4.SD16 | C42.78C22 | C42.28C22 | C42.30C22 | C23.38C23 | C23.41C23 | C22⋊C4 | C4⋊C4 | C2×C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{367}C_2^3 % in TeX
G:=Group("C4^2.367C2^3"); // GroupNames label
G:=SmallGroup(128,1869);
// by ID
G=gap.SmallGroup(128,1869);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,232,758,100,1018,2804,172,4037,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=b^2,d^2=a^2*b^-1,e^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,e*a*e^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b*c,e*c*e^-1=a^2*b^2*c,d*e=e*d>;
// generators/relations