p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23.38D8, C24.161D4, C23.43SD16, C4⋊C4⋊30D4, C4.61(C4×D4), C4⋊D4⋊10C4, C22.38(C2×D8), C2.4(C22⋊D8), C2.4(Q8⋊D4), (C22×C4).288D4, C23.766(C2×D4), C4.127(C4⋊D4), C22⋊1(D4⋊C4), C22.4Q16⋊12C2, C22.91C22≀C2, (C22×C8).33C22, C22.57(C2×SD16), C22.74(C8⋊C22), (C23×C4).256C22, C2.3(C22.D8), (C22×D4).15C22, C23.120(C22⋊C4), (C22×C4).1361C23, C2.3(C23.46D4), C22.64(C8.C22), C2.25(C23.36D4), C2.13(C23.23D4), C22.86(C22.D4), C4⋊C4⋊7(C2×C4), (C2×D4)⋊5(C2×C4), (C22×C4⋊C4)⋊2C2, (C2×D4⋊C4)⋊3C2, (C2×C22⋊C8)⋊14C2, (C2×C4⋊D4).6C2, (C2×C4).1329(C2×D4), C2.20(C2×D4⋊C4), (C2×C4).757(C4○D4), (C2×C4⋊C4).763C22, (C2×C4).379(C22×C4), (C22×C4).278(C2×C4), (C2×C4).131(C22⋊C4), C22.265(C2×C22⋊C4), SmallGroup(128,606)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.38D8
G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, dad-1=eae=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=bd-1 >
Subgroups: 524 in 228 conjugacy classes, 72 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C22⋊C8, D4⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22×C8, C23×C4, C23×C4, C22×D4, C22×D4, C22.4Q16, C2×C22⋊C8, C2×D4⋊C4, C22×C4⋊C4, C2×C4⋊D4, C23.38D8
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, C4○D4, D4⋊C4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C2×D8, C2×SD16, C8⋊C22, C8.C22, C23.23D4, C2×D4⋊C4, C23.36D4, C22⋊D8, Q8⋊D4, C22.D8, C23.46D4, C23.38D8
►On 64 points
(1 61)(2 43)(3 63)(4 45)(5 57)(6 47)(7 59)(8 41)(9 18)(10 34)(11 20)(12 36)(13 22)(14 38)(15 24)(16 40)(17 29)(19 31)(21 25)(23 27)(26 37)(28 39)(30 33)(32 35)(42 49)(44 51)(46 53)(48 55)(50 62)(52 64)(54 58)(56 60)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)(17 40)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(41 60)(42 61)(43 62)(44 63)(45 64)(46 57)(47 58)(48 59)
(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)
(2 11)(3 7)(4 9)(6 15)(8 13)(10 14)(17 40)(18 64)(19 38)(20 62)(21 36)(22 60)(23 34)(24 58)(26 56)(27 31)(28 54)(30 52)(32 50)(33 45)(35 43)(37 41)(39 47)(42 61)(44 59)(46 57)(48 63)(51 55)
G:=sub<Sym(64)| (1,61)(2,43)(3,63)(4,45)(5,57)(6,47)(7,59)(8,41)(9,18)(10,34)(11,20)(12,36)(13,22)(14,38)(15,24)(16,40)(17,29)(19,31)(21,25)(23,27)(26,37)(28,39)(30,33)(32,35)(42,49)(44,51)(46,53)(48,55)(50,62)(52,64)(54,58)(56,60), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,57), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(17,40)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(41,60)(42,61)(43,62)(44,63)(45,64)(46,57)(47,58)(48,59), (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), (2,11)(3,7)(4,9)(6,15)(8,13)(10,14)(17,40)(18,64)(19,38)(20,62)(21,36)(22,60)(23,34)(24,58)(26,56)(27,31)(28,54)(30,52)(32,50)(33,45)(35,43)(37,41)(39,47)(42,61)(44,59)(46,57)(48,63)(51,55)>;
G:=Group( (1,61)(2,43)(3,63)(4,45)(5,57)(6,47)(7,59)(8,41)(9,18)(10,34)(11,20)(12,36)(13,22)(14,38)(15,24)(16,40)(17,29)(19,31)(21,25)(23,27)(26,37)(28,39)(30,33)(32,35)(42,49)(44,51)(46,53)(48,55)(50,62)(52,64)(54,58)(56,60), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,57), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(17,40)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(41,60)(42,61)(43,62)(44,63)(45,64)(46,57)(47,58)(48,59), (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), (2,11)(3,7)(4,9)(6,15)(8,13)(10,14)(17,40)(18,64)(19,38)(20,62)(21,36)(22,60)(23,34)(24,58)(26,56)(27,31)(28,54)(30,52)(32,50)(33,45)(35,43)(37,41)(39,47)(42,61)(44,59)(46,57)(48,63)(51,55) );
G=PermutationGroup([[(1,61),(2,43),(3,63),(4,45),(5,57),(6,47),(7,59),(8,41),(9,18),(10,34),(11,20),(12,36),(13,22),(14,38),(15,24),(16,40),(17,29),(19,31),(21,25),(23,27),(26,37),(28,39),(30,33),(32,35),(42,49),(44,51),(46,53),(48,55),(50,62),(52,64),(54,58),(56,60)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11),(17,46),(18,47),(19,48),(20,41),(21,42),(22,43),(23,44),(24,45),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,58),(34,59),(35,60),(36,61),(37,62),(38,63),(39,64),(40,57)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,30),(10,31),(11,32),(12,25),(13,26),(14,27),(15,28),(16,29),(17,40),(18,33),(19,34),(20,35),(21,36),(22,37),(23,38),(24,39),(41,60),(42,61),(43,62),(44,63),(45,64),(46,57),(47,58),(48,59)], [(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)], [(2,11),(3,7),(4,9),(6,15),(8,13),(10,14),(17,40),(18,64),(19,38),(20,62),(21,36),(22,60),(23,34),(24,58),(26,56),(27,31),(28,54),(30,52),(32,50),(33,45),(35,43),(37,41),(39,47),(42,61),(44,59),(46,57),(48,63),(51,55)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | C4○D4 | D8 | SD16 | C8⋊C22 | C8.C22 |
| kernel | C23.38D8 | C22.4Q16 | C2×C22⋊C8 | C2×D4⋊C4 | C22×C4⋊C4 | C2×C4⋊D4 | C4⋊D4 | C4⋊C4 | C22×C4 | C24 | C2×C4 | C23 | C23 | C22 | C22 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 4 | 3 | 1 | 4 | 4 | 4 | 1 | 1 |
Matrix representation of C23.38D8 ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 15 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 6 | 11 |
| 0 | 0 | 0 | 3 | 0 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 16 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,15,1,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,16,1,0,0,0,0,1,0,0,0,0,0,6,3,0,0,0,11,0],[16,0,0,0,0,0,1,16,0,0,0,0,16,0,0,0,0,0,1,1,0,0,0,0,16] >;
C23.38D8 in GAP, Magma, Sage, TeX
C_2^3._{38}D_8 % in TeX
G:=Group("C2^3.38D8"); // GroupNames label
G:=SmallGroup(128,606);
// by ID
G=gap.SmallGroup(128,606);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,1018,248,2028,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,d*a*d^-1=e*a*e=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=b*d^-1>;
// generators/relations