p-group, metabelian, nilpotent (class 4), monomial
Aliases: C23.20D8, C16⋊3C4⋊9C2, C16⋊4C4⋊12C2, (C2×C8).179D4, (C2×C4).115D8, C8.74(C4○D4), C22⋊C16.7C2, C2.Q32⋊13C2, C2.16(C4○D16), (C2×C16).12C22, (C2×C8).537C23, C8.18D4.9C2, (C22×C4).357D4, C22.123(C2×D8), C2.D8.22C22, C2.19(Q32⋊C2), C4.19(C8.C22), (C2×Q16).12C22, (C22×C8).175C22, C23.25D4.5C2, C4.44(C22.D4), C2.17(C22.D8), (C2×C4).805(C2×D4), SmallGroup(128,969)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.20D8
G = < a,b,c,d,e | a2=b2=c2=1, d8=e2=c, dad-1=ab=ba, ac=ca, eae-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd7 >
Subgroups: 148 in 66 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, Q8, C23, C16, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C2×Q8, Q8⋊C4, C4.Q8, C2.D8, C2×C16, C42⋊C2, C22⋊Q8, C22×C8, C2×Q16, C22⋊C16, C2.Q32, C16⋊3C4, C16⋊4C4, C23.25D4, C8.18D4, C23.20D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, C22.D4, C2×D8, C8.C22, C22.D8, C4○D16, Q32⋊C2, C23.20D8
Character table of C23.20D8
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | 8E | 8F | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 16 | 16 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | √2 | -√2 | -√2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ12 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | -√2 | √2 | √2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ13 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | √2 | -√2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ14 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | -√2 | √2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ16 | 2 | -2 | -2 | 2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 2 | 0 | 0 | -2 | -2i | 2i | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 2 | 0 | 0 | -2 | 2i | -2i | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | -√-2 | √-2 | ζ167-ζ16 | ζ165-ζ163 | -ζ167+ζ16 | ζ1615+ζ169 | ζ1613+ζ1611 | -ζ165+ζ163 | ζ165+ζ163 | ζ167+ζ16 | complex lifted from C4○D16 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | √-2 | -√-2 | -ζ167+ζ16 | -ζ165+ζ163 | ζ167-ζ16 | ζ1615+ζ169 | ζ1613+ζ1611 | ζ165-ζ163 | ζ165+ζ163 | ζ167+ζ16 | complex lifted from C4○D16 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | √-2 | -√-2 | ζ167-ζ16 | ζ165-ζ163 | -ζ167+ζ16 | ζ167+ζ16 | ζ165+ζ163 | -ζ165+ζ163 | ζ1613+ζ1611 | ζ1615+ζ169 | complex lifted from C4○D16 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | -√-2 | √-2 | -ζ167+ζ16 | -ζ165+ζ163 | ζ167-ζ16 | ζ167+ζ16 | ζ165+ζ163 | ζ165-ζ163 | ζ1613+ζ1611 | ζ1615+ζ169 | complex lifted from C4○D16 |
| ρ23 | 2 | 2 | -2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | √-2 | -√-2 | ζ165-ζ163 | -ζ167+ζ16 | -ζ165+ζ163 | ζ165+ζ163 | ζ1615+ζ169 | ζ167-ζ16 | ζ167+ζ16 | ζ1613+ζ1611 | complex lifted from C4○D16 |
| ρ24 | 2 | 2 | -2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | √-2 | -√-2 | -ζ165+ζ163 | ζ167-ζ16 | ζ165-ζ163 | ζ1613+ζ1611 | ζ167+ζ16 | -ζ167+ζ16 | ζ1615+ζ169 | ζ165+ζ163 | complex lifted from C4○D16 |
| ρ25 | 2 | 2 | -2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | -√-2 | √-2 | ζ165-ζ163 | -ζ167+ζ16 | -ζ165+ζ163 | ζ1613+ζ1611 | ζ167+ζ16 | ζ167-ζ16 | ζ1615+ζ169 | ζ165+ζ163 | complex lifted from C4○D16 |
| ρ26 | 2 | 2 | -2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | -√-2 | √-2 | -ζ165+ζ163 | ζ167-ζ16 | ζ165-ζ163 | ζ165+ζ163 | ζ1615+ζ169 | -ζ167+ζ16 | ζ167+ζ16 | ζ1613+ζ1611 | complex lifted from C4○D16 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ28 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | -2√2 | 2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q32⋊C2, Schur index 2 |
| ρ29 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 2√2 | -2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q32⋊C2, Schur index 2 |
►On 64 points
Generators in S64
(2 56)(4 58)(6 60)(8 62)(10 64)(12 50)(14 52)(16 54)(17 39)(18 26)(19 41)(20 28)(21 43)(22 30)(23 45)(24 32)(25 47)(27 33)(29 35)(31 37)(34 42)(36 44)(38 46)(40 48)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 63)(10 64)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 47)(18 48)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)
(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 45 9 37)(2 22 10 30)(3 43 11 35)(4 20 12 28)(5 41 13 33)(6 18 14 26)(7 39 15 47)(8 32 16 24)(17 61 25 53)(19 59 27 51)(21 57 29 49)(23 55 31 63)(34 50 42 58)(36 64 44 56)(38 62 46 54)(40 60 48 52)
G:=sub<Sym(64)| (2,56)(4,58)(6,60)(8,62)(10,64)(12,50)(14,52)(16,54)(17,39)(18,26)(19,41)(20,28)(21,43)(22,30)(23,45)(24,32)(25,47)(27,33)(29,35)(31,37)(34,42)(36,44)(38,46)(40,48), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,47)(18,48)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46), (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), (1,45,9,37)(2,22,10,30)(3,43,11,35)(4,20,12,28)(5,41,13,33)(6,18,14,26)(7,39,15,47)(8,32,16,24)(17,61,25,53)(19,59,27,51)(21,57,29,49)(23,55,31,63)(34,50,42,58)(36,64,44,56)(38,62,46,54)(40,60,48,52)>;
G:=Group( (2,56)(4,58)(6,60)(8,62)(10,64)(12,50)(14,52)(16,54)(17,39)(18,26)(19,41)(20,28)(21,43)(22,30)(23,45)(24,32)(25,47)(27,33)(29,35)(31,37)(34,42)(36,44)(38,46)(40,48), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,47)(18,48)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46), (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), (1,45,9,37)(2,22,10,30)(3,43,11,35)(4,20,12,28)(5,41,13,33)(6,18,14,26)(7,39,15,47)(8,32,16,24)(17,61,25,53)(19,59,27,51)(21,57,29,49)(23,55,31,63)(34,50,42,58)(36,64,44,56)(38,62,46,54)(40,60,48,52) );
G=PermutationGroup([[(2,56),(4,58),(6,60),(8,62),(10,64),(12,50),(14,52),(16,54),(17,39),(18,26),(19,41),(20,28),(21,43),(22,30),(23,45),(24,32),(25,47),(27,33),(29,35),(31,37),(34,42),(36,44),(38,46),(40,48)], [(1,55),(2,56),(3,57),(4,58),(5,59),(6,60),(7,61),(8,62),(9,63),(10,64),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,47),(18,48),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,39),(26,40),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46)], [(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)], [(1,45,9,37),(2,22,10,30),(3,43,11,35),(4,20,12,28),(5,41,13,33),(6,18,14,26),(7,39,15,47),(8,32,16,24),(17,61,25,53),(19,59,27,51),(21,57,29,49),(23,55,31,63),(34,50,42,58),(36,64,44,56),(38,62,46,54),(40,60,48,52)]])
Matrix representation of C23.20D8 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 13 | 8 | 0 | 0 |
| 13 | 4 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 5 |
| 1 | 15 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 5 |
| 0 | 0 | 10 | 0 |
G:=sub<GL(4,GF(17))| [1,1,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[13,13,0,0,8,4,0,0,0,0,10,0,0,0,0,5],[1,0,0,0,15,16,0,0,0,0,0,10,0,0,5,0] >;
C23.20D8 in GAP, Magma, Sage, TeX
C_2^3._{20}D_8 % in TeX
G:=Group("C2^3.20D8"); // GroupNames label
G:=SmallGroup(128,969);
// by ID
G=gap.SmallGroup(128,969);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,456,422,58,1684,438,242,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=e^2=c,d*a*d^-1=a*b=b*a,a*c=c*a,e*a*e^-1=a*b*c,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*d^7>;
// generators/relations
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