p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×D4).302D4, (C2×Q8).237D4, C2.15(Q8○D8), C4⋊C4.395C23, (C2×C4).295C24, (C2×C8).312C23, C23.246(C2×D4), (C2×Q8).70C23, C4.Q8.12C22, C2.D8.83C22, C2.24(D4○SD16), C23.47D4⋊3C2, C23.25D4⋊8C2, C22⋊C8.17C22, M4(2)⋊C4⋊27C2, C22⋊Q8.23C22, C23.20D4⋊14C2, C23.38D4⋊11C2, C23.48D4⋊13C2, (C22×C8).185C22, C22.555(C22×D4), (C22×C4).1011C23, Q8⋊C4.150C22, C4.83(C22.D4), (C2×M4(2)).77C22, (C22×Q8).293C22, C42⋊C2.124C22, C23.33C23.9C2, C23.38C23.12C2, C22.19(C22.D4), C4.105(C2×C4○D4), (C2×C4).490(C2×D4), (C2×Q8⋊C4)⋊30C2, (C2×C4).297(C4○D4), (C2×C4⋊C4).611C22, (C22×C8)⋊C2.3C2, (C2×C4○D4).140C22, C2.60(C2×C22.D4), SmallGroup(128,1829)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 340 in 193 conjugacy classes, 92 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×17], D4 [×6], Q8 [×8], C23, C23 [×2], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C22⋊C8 [×4], Q8⋊C4 [×8], C4.Q8 [×2], C4.Q8 [×2], C2.D8 [×2], C2.D8 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C42⋊C2 [×2], C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C22×Q8, C2×C4○D4, (C22×C8)⋊C2, C2×Q8⋊C4, C23.38D4, C23.25D4, M4(2)⋊C4, C23.47D4 [×2], C23.48D4 [×2], C23.20D4 [×4], C23.33C23, C23.38C23, (C2×D4).302D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C2×C22.D4, D4○SD16, Q8○D8, (C2×D4).302D4
Generators and relations
G = < a,b,c,d,e | a2=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=ebe-1=ab-1, dcd-1=ece-1=ab2c, ede-1=ab2d3 >
►On 64 points
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 49)(33 46)(34 47)(35 48)(36 41)(37 42)(38 43)(39 44)(40 45)
(1 61 5 57)(2 8 6 4)(3 63 7 59)(9 26 13 30)(10 16 14 12)(11 28 15 32)(17 19 21 23)(18 49 22 53)(20 51 24 55)(25 31 29 27)(33 35 37 39)(34 45 38 41)(36 47 40 43)(42 44 46 48)(50 52 54 56)(58 64 62 60)
(1 53)(2 17)(3 55)(4 19)(5 49)(6 21)(7 51)(8 23)(9 47)(10 39)(11 41)(12 33)(13 43)(14 35)(15 45)(16 37)(18 57)(20 59)(22 61)(24 63)(25 44)(26 36)(27 46)(28 38)(29 48)(30 40)(31 42)(32 34)(50 60)(52 62)(54 64)(56 58)
(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 46 5 42)(2 40 6 36)(3 44 7 48)(4 38 8 34)(9 23 13 19)(10 55 14 51)(11 21 15 17)(12 53 16 49)(18 25 22 29)(20 31 24 27)(26 54 30 50)(28 52 32 56)(33 63 37 59)(35 61 39 57)(41 60 45 64)(43 58 47 62)
G:=sub<Sym(64)| (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,49)(33,46)(34,47)(35,48)(36,41)(37,42)(38,43)(39,44)(40,45), (1,61,5,57)(2,8,6,4)(3,63,7,59)(9,26,13,30)(10,16,14,12)(11,28,15,32)(17,19,21,23)(18,49,22,53)(20,51,24,55)(25,31,29,27)(33,35,37,39)(34,45,38,41)(36,47,40,43)(42,44,46,48)(50,52,54,56)(58,64,62,60), (1,53)(2,17)(3,55)(4,19)(5,49)(6,21)(7,51)(8,23)(9,47)(10,39)(11,41)(12,33)(13,43)(14,35)(15,45)(16,37)(18,57)(20,59)(22,61)(24,63)(25,44)(26,36)(27,46)(28,38)(29,48)(30,40)(31,42)(32,34)(50,60)(52,62)(54,64)(56,58), (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), (1,46,5,42)(2,40,6,36)(3,44,7,48)(4,38,8,34)(9,23,13,19)(10,55,14,51)(11,21,15,17)(12,53,16,49)(18,25,22,29)(20,31,24,27)(26,54,30,50)(28,52,32,56)(33,63,37,59)(35,61,39,57)(41,60,45,64)(43,58,47,62)>;
G:=Group( (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,49)(33,46)(34,47)(35,48)(36,41)(37,42)(38,43)(39,44)(40,45), (1,61,5,57)(2,8,6,4)(3,63,7,59)(9,26,13,30)(10,16,14,12)(11,28,15,32)(17,19,21,23)(18,49,22,53)(20,51,24,55)(25,31,29,27)(33,35,37,39)(34,45,38,41)(36,47,40,43)(42,44,46,48)(50,52,54,56)(58,64,62,60), (1,53)(2,17)(3,55)(4,19)(5,49)(6,21)(7,51)(8,23)(9,47)(10,39)(11,41)(12,33)(13,43)(14,35)(15,45)(16,37)(18,57)(20,59)(22,61)(24,63)(25,44)(26,36)(27,46)(28,38)(29,48)(30,40)(31,42)(32,34)(50,60)(52,62)(54,64)(56,58), (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), (1,46,5,42)(2,40,6,36)(3,44,7,48)(4,38,8,34)(9,23,13,19)(10,55,14,51)(11,21,15,17)(12,53,16,49)(18,25,22,29)(20,31,24,27)(26,54,30,50)(28,52,32,56)(33,63,37,59)(35,61,39,57)(41,60,45,64)(43,58,47,62) );
G=PermutationGroup([(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,57),(8,58),(9,32),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,56),(24,49),(33,46),(34,47),(35,48),(36,41),(37,42),(38,43),(39,44),(40,45)], [(1,61,5,57),(2,8,6,4),(3,63,7,59),(9,26,13,30),(10,16,14,12),(11,28,15,32),(17,19,21,23),(18,49,22,53),(20,51,24,55),(25,31,29,27),(33,35,37,39),(34,45,38,41),(36,47,40,43),(42,44,46,48),(50,52,54,56),(58,64,62,60)], [(1,53),(2,17),(3,55),(4,19),(5,49),(6,21),(7,51),(8,23),(9,47),(10,39),(11,41),(12,33),(13,43),(14,35),(15,45),(16,37),(18,57),(20,59),(22,61),(24,63),(25,44),(26,36),(27,46),(28,38),(29,48),(30,40),(31,42),(32,34),(50,60),(52,62),(54,64),(56,58)], [(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)], [(1,46,5,42),(2,40,6,36),(3,44,7,48),(4,38,8,34),(9,23,13,19),(10,55,14,51),(11,21,15,17),(12,53,16,49),(18,25,22,29),(20,31,24,27),(26,54,30,50),(28,52,32,56),(33,63,37,59),(35,61,39,57),(41,60,45,64),(43,58,47,62)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 16 | 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 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 12 |
| 0 | 0 | 0 | 0 | 5 | 5 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 0 | 5 | 5 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0],[0,2,0,0,0,0,9,0,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,5,5,0,0,0,0,12,5],[0,8,0,0,0,0,15,0,0,0,0,0,0,0,0,0,12,5,0,0,0,0,5,5,0,0,12,5,0,0,0,0,5,5,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 8A | 8B | 8C | 8D | 8E | 8F |
| order | 1 | 2 | 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 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
32 irreducible representations
| dim | 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 | D4 | D4 | C4○D4 | D4○SD16 | Q8○D8 |
| kernel | (C2×D4).302D4 | (C22×C8)⋊C2 | C2×Q8⋊C4 | C23.38D4 | C23.25D4 | M4(2)⋊C4 | C23.47D4 | C23.48D4 | C23.20D4 | C23.33C23 | C23.38C23 | C2×D4 | C2×Q8 | C2×C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 3 | 1 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
(C_2\times D_4)._{302}D_4 % in TeX
G:=Group("(C2xD4).302D4"); // GroupNames label
G:=SmallGroup(128,1829);
// by ID
G=gap.SmallGroup(128,1829);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,100,1018,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^4=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d^-1=e*b*e^-1=a*b^-1,d*c*d^-1=e*c*e^-1=a*b^2*c,e*d*e^-1=a*b^2*d^3>;
// generators/relations