p-group, metabelian, nilpotent (class 4), monomial
Aliases: C23.24D8, C4○D8⋊1C4, D8.7(C2×C4), C8.92(C2×D4), C4○(C2.D16), (C22×C16)⋊5C2, (C2×C8).277D4, (C2×C4).168D8, Q16.7(C2×C4), C2.D16⋊20C2, C4○(C2.Q32), C2.1(C4○D16), C8.29(C22×C4), (C2×C4).77SD16, C4.10(C2×SD16), C22.53(C2×D8), C2.Q32⋊20C2, C8.28(C22⋊C4), (C2×C16).63C22, (C2×C8).494C23, (C22×C4).584D4, C4.56(D4⋊C4), (C2×D8).100C22, C23.25D4⋊1C2, (C2×Q16).99C22, C2.D8.143C22, C22.7(D4⋊C4), (C22×C8).551C22, (C2×C4○D8).3C2, (C2×C4)○(C2.D16), (C2×C8).179(C2×C4), (C2×C4).756(C2×D4), C4.50(C2×C22⋊C4), (C2×C4)○(C2.Q32), C2.28(C2×D4⋊C4), (C2×C4).271(C22⋊C4), SmallGroup(128,870)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.24D8
G = < a,b,c,d,e | a2=b2=c2=1, d8=c, e2=b, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bcd7 >
Subgroups: 260 in 114 conjugacy classes, 52 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C16, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, D8, SD16, Q16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4.Q8, C2.D8, C2×C16, C2×C16, C42⋊C2, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C4○D8, C2×C4○D4, C2.D16, C2.Q32, C23.25D4, C22×C16, C2×C4○D8, C23.24D8
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, C2×D4⋊C4, C4○D16, C23.24D8
►On 64 points
(1 44)(2 45)(3 46)(4 47)(5 48)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 63)(18 64)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 64)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 33)(32 34)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 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 55 56 16)(2 15 57 54)(3 53 58 14)(4 13 59 52)(5 51 60 12)(6 11 61 50)(7 49 62 10)(8 9 63 64)(17 26 35 44)(18 43 36 25)(19 24 37 42)(20 41 38 23)(21 22 39 40)(27 32 45 34)(28 33 46 31)(29 30 47 48)
G:=sub<Sym(64)| (1,44)(2,45)(3,46)(4,47)(5,48)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,63)(18,64)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62), (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,33)(32,34), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (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), (1,55,56,16)(2,15,57,54)(3,53,58,14)(4,13,59,52)(5,51,60,12)(6,11,61,50)(7,49,62,10)(8,9,63,64)(17,26,35,44)(18,43,36,25)(19,24,37,42)(20,41,38,23)(21,22,39,40)(27,32,45,34)(28,33,46,31)(29,30,47,48)>;
G:=Group( (1,44)(2,45)(3,46)(4,47)(5,48)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,63)(18,64)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62), (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,33)(32,34), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (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), (1,55,56,16)(2,15,57,54)(3,53,58,14)(4,13,59,52)(5,51,60,12)(6,11,61,50)(7,49,62,10)(8,9,63,64)(17,26,35,44)(18,43,36,25)(19,24,37,42)(20,41,38,23)(21,22,39,40)(27,32,45,34)(28,33,46,31)(29,30,47,48) );
G=PermutationGroup([[(1,44),(2,45),(3,46),(4,47),(5,48),(6,33),(7,34),(8,35),(9,36),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,63),(18,64),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,61),(7,62),(8,63),(9,64),(10,49),(11,50),(12,51),(13,52),(14,53),(15,54),(16,55),(17,35),(18,36),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,33),(32,34)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64)], [(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)], [(1,55,56,16),(2,15,57,54),(3,53,58,14),(4,13,59,52),(5,51,60,12),(6,11,61,50),(7,49,62,10),(8,9,63,64),(17,26,35,44),(18,43,36,25),(19,24,37,42),(20,41,38,23),(21,22,39,40),(27,32,45,34),(28,33,46,31),(29,30,47,48)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D8 | SD16 | D8 | C4○D16 |
| kernel | C23.24D8 | C2.D16 | C2.Q32 | C23.25D4 | C22×C16 | C2×C4○D8 | C4○D8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 3 | 1 | 2 | 4 | 2 | 16 |
Matrix representation of C23.24D8 ►in GL3(𝔽17) generated by
| 1 | 0 | 0 |
| 0 | 4 | 4 |
| 0 | 9 | 13 |
| 16 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 4 | 0 | 0 |
| 0 | 10 | 6 |
| 0 | 5 | 15 |
| 13 | 0 | 0 |
| 0 | 10 | 4 |
| 0 | 5 | 7 |
G:=sub<GL(3,GF(17))| [1,0,0,0,4,9,0,4,13],[16,0,0,0,1,0,0,0,1],[1,0,0,0,16,0,0,0,16],[4,0,0,0,10,5,0,6,15],[13,0,0,0,10,5,0,4,7] >;
C23.24D8 in GAP, Magma, Sage, TeX
C_2^3._{24}D_8 % in TeX
G:=Group("C2^3.24D8"); // GroupNames label
G:=SmallGroup(128,870);
// by ID
G=gap.SmallGroup(128,870);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,352,1123,570,360,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=c,e^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*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=b*c*d^7>;
// generators/relations