p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23.37D8, C24.158D4, C23.19Q16, C22⋊C8⋊7C4, C4⋊C4.297D4, C4.136(C4×D4), C22.36(C2×D8), C22⋊1(C2.D8), C4.1(C22⋊Q8), C2.2(C22⋊D8), (C22×C4).48Q8, C23.71(C4⋊C4), (C22×C4).282D4, C23.757(C2×D4), C22.29(C2×Q16), C22.4Q16⋊33C2, C22.77C22≀C2, C2.2(C22⋊Q16), C22.67(C8⋊C22), (C22×C8).101C22, (C23×C4).248C22, C2.2(C22.D8), C23.7Q8.12C2, C2.8(C23.8Q8), (C22×C4).1349C23, C2.2(C23.48D4), C22.56(C8.C22), C2.10(M4(2)⋊C4), C22.81(C22.D4), (C2×C8)⋊4(C2×C4), (C2×C2.D8)⋊2C2, C2.9(C2×C2.D8), (C2×C4).51(C4⋊C4), (C2×C4).979(C2×D4), (C2×C4).199(C2×Q8), (C2×C4⋊C4).51C22, (C22×C4⋊C4).15C2, (C2×C22⋊C8).25C2, C22.109(C2×C4⋊C4), (C2×C4).745(C4○D4), (C2×C4).548(C22×C4), (C22×C4).271(C2×C4), SmallGroup(128,584)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.37D8
G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=c, dad-1=eae-1=ab=ba, ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 356 in 180 conjugacy classes, 72 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C24, C2.C42, C22⋊C8, C2.D8, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C23×C4, C23×C4, C22.4Q16, C23.7Q8, C2×C22⋊C8, C2×C2.D8, C22×C4⋊C4, C23.37D8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, C2.D8, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C2×D8, C2×Q16, C8⋊C22, C8.C22, C23.8Q8, C2×C2.D8, M4(2)⋊C4, C22⋊D8, C22⋊Q16, C22.D8, C23.48D4, C23.37D8
►On 64 points
(1 19)(2 61)(3 21)(4 63)(5 23)(6 57)(7 17)(8 59)(9 62)(10 22)(11 64)(12 24)(13 58)(14 18)(15 60)(16 20)(25 46)(26 37)(27 48)(28 39)(29 42)(30 33)(31 44)(32 35)(34 51)(36 53)(38 55)(40 49)(41 56)(43 50)(45 52)(47 54)
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 57)(25 53)(26 54)(27 55)(28 56)(29 49)(30 50)(31 51)(32 52)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 41)(40 42)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 62)(10 63)(11 64)(12 57)(13 58)(14 59)(15 60)(16 61)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)(41 56)(42 49)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)
(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 27 19 38)(2 26 20 37)(3 25 21 36)(4 32 22 35)(5 31 23 34)(6 30 24 33)(7 29 17 40)(8 28 18 39)(9 53 62 46)(10 52 63 45)(11 51 64 44)(12 50 57 43)(13 49 58 42)(14 56 59 41)(15 55 60 48)(16 54 61 47)
G:=sub<Sym(64)| (1,19)(2,61)(3,21)(4,63)(5,23)(6,57)(7,17)(8,59)(9,62)(10,22)(11,64)(12,24)(13,58)(14,18)(15,60)(16,20)(25,46)(26,37)(27,48)(28,39)(29,42)(30,33)(31,44)(32,35)(34,51)(36,53)(38,55)(40,49)(41,56)(43,50)(45,52)(47,54), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,57)(25,53)(26,54)(27,55)(28,56)(29,49)(30,50)(31,51)(32,52)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,41)(40,42), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,62)(10,63)(11,64)(12,57)(13,58)(14,59)(15,60)(16,61)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35)(41,56)(42,49)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55), (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), (1,27,19,38)(2,26,20,37)(3,25,21,36)(4,32,22,35)(5,31,23,34)(6,30,24,33)(7,29,17,40)(8,28,18,39)(9,53,62,46)(10,52,63,45)(11,51,64,44)(12,50,57,43)(13,49,58,42)(14,56,59,41)(15,55,60,48)(16,54,61,47)>;
G:=Group( (1,19)(2,61)(3,21)(4,63)(5,23)(6,57)(7,17)(8,59)(9,62)(10,22)(11,64)(12,24)(13,58)(14,18)(15,60)(16,20)(25,46)(26,37)(27,48)(28,39)(29,42)(30,33)(31,44)(32,35)(34,51)(36,53)(38,55)(40,49)(41,56)(43,50)(45,52)(47,54), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,57)(25,53)(26,54)(27,55)(28,56)(29,49)(30,50)(31,51)(32,52)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,41)(40,42), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,62)(10,63)(11,64)(12,57)(13,58)(14,59)(15,60)(16,61)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35)(41,56)(42,49)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55), (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), (1,27,19,38)(2,26,20,37)(3,25,21,36)(4,32,22,35)(5,31,23,34)(6,30,24,33)(7,29,17,40)(8,28,18,39)(9,53,62,46)(10,52,63,45)(11,51,64,44)(12,50,57,43)(13,49,58,42)(14,56,59,41)(15,55,60,48)(16,54,61,47) );
G=PermutationGroup([[(1,19),(2,61),(3,21),(4,63),(5,23),(6,57),(7,17),(8,59),(9,62),(10,22),(11,64),(12,24),(13,58),(14,18),(15,60),(16,20),(25,46),(26,37),(27,48),(28,39),(29,42),(30,33),(31,44),(32,35),(34,51),(36,53),(38,55),(40,49),(41,56),(43,50),(45,52),(47,54)], [(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14),(17,58),(18,59),(19,60),(20,61),(21,62),(22,63),(23,64),(24,57),(25,53),(26,54),(27,55),(28,56),(29,49),(30,50),(31,51),(32,52),(33,43),(34,44),(35,45),(36,46),(37,47),(38,48),(39,41),(40,42)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,62),(10,63),(11,64),(12,57),(13,58),(14,59),(15,60),(16,61),(25,36),(26,37),(27,38),(28,39),(29,40),(30,33),(31,34),(32,35),(41,56),(42,49),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55)], [(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)], [(1,27,19,38),(2,26,20,37),(3,25,21,36),(4,32,22,35),(5,31,23,34),(6,30,24,33),(7,29,17,40),(8,28,18,39),(9,53,62,46),(10,52,63,45),(11,51,64,44),(12,50,57,43),(13,49,58,42),(14,56,59,41),(15,55,60,48),(16,54,61,47)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | Q8 | D4 | C4○D4 | D8 | Q16 | C8⋊C22 | C8.C22 |
| kernel | C23.37D8 | C22.4Q16 | C23.7Q8 | C2×C22⋊C8 | C2×C2.D8 | C22×C4⋊C4 | C22⋊C8 | C4⋊C4 | C22×C4 | C22×C4 | C24 | C2×C4 | C23 | C23 | C22 | C22 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 4 | 1 | 2 | 1 | 4 | 4 | 4 | 1 | 1 |
Matrix representation of C23.37D8 ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 14 | 3 | 0 | 0 |
| 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 16 | 7 | 0 | 0 |
| 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,16,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,14,14,0,0,0,3,14,0,0,0,0,0,1,0,0,0,0,2,16],[4,0,0,0,0,0,16,7,0,0,0,7,1,0,0,0,0,0,16,0,0,0,0,15,1] >;
C23.37D8 in GAP, Magma, Sage, TeX
C_2^3._{37}D_8 % in TeX
G:=Group("C2^3.37D8"); // GroupNames label
G:=SmallGroup(128,584);
// by ID
G=gap.SmallGroup(128,584);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,2019,1018,248]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=c,d*a*d^-1=e*a*e^-1=a*b=b*a,a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations