p-group, metabelian, nilpotent (class 4), monomial
Aliases: C23.13SD16, C4.21C4≀C2, C2.D8.2C4, (C2×C4).103D8, (C2×C8).303D4, (C2×Q16).1C4, C22⋊C16.4C2, C4.8(C23⋊C4), C8.18D4.2C2, (C22×C4).188D4, C2.5(D8.C4), C2.3(C8.17D4), C4.C42.6C2, (C22×C8).100C22, C22.58(D4⋊C4), C2.15(C22.SD16), (C2×C8).20(C2×C4), (C2×C4).220(C22⋊C4), SmallGroup(128,82)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.13SD16
G = < a,b,c,d,e | a2=b2=c2=1, d8=e2=c, eae-1=ab=ba, ac=ca, dad-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=abd3 >
Character table of C23.13SD16
class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | |
size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 16 | 16 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 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 | -i | -i | i | i | i | -i | i | i | -i | -i | -i | i | linear of order 4 |
ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | -i | i | i | -i | i | -i | -i | i | i | i | -i | linear of order 4 |
ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | i | i | -i | -i | i | -i | i | i | -i | -i | -i | i | linear of order 4 |
ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | i | i | -i | -i | -i | i | -i | -i | i | i | i | -i | linear of order 4 |
ρ9 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 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 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | -1+i | 1-i | -1-i | 1+i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
ρ14 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | -1-i | 1+i | -1+i | 1-i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 1-i | -1+i | 1+i | -1-i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
ρ16 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 1+i | -1-i | 1-i | -1+i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
ρ17 | 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 | complex lifted from SD16 |
ρ18 | 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 | complex lifted from SD16 |
ρ19 | 2 | -2 | -2 | 2 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | √-2 | -√-2 | √-2 | -√-2 | -√2 | √2 | 0 | 0 | 0 | 0 | ζ1611+ζ169 | ζ1615+ζ1613 | ζ163+ζ16 | ζ1613+ζ167 | ζ1611+ζ16 | ζ167+ζ165 | ζ169+ζ163 | ζ1615+ζ165 | complex lifted from D8.C4 |
ρ20 | 2 | -2 | -2 | 2 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | -√-2 | √-2 | -√-2 | √-2 | -√2 | √2 | 0 | 0 | 0 | 0 | ζ167+ζ165 | ζ163+ζ16 | ζ1615+ζ1613 | ζ169+ζ163 | ζ1615+ζ165 | ζ1611+ζ169 | ζ1613+ζ167 | ζ1611+ζ16 | complex lifted from D8.C4 |
ρ21 | 2 | -2 | -2 | 2 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | √-2 | -√-2 | √-2 | -√-2 | √2 | -√2 | 0 | 0 | 0 | 0 | ζ169+ζ163 | ζ1615+ζ165 | ζ1611+ζ16 | ζ1615+ζ1613 | ζ1611+ζ169 | ζ1613+ζ167 | ζ163+ζ16 | ζ167+ζ165 | complex lifted from D8.C4 |
ρ22 | 2 | -2 | -2 | 2 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | -√-2 | √-2 | -√-2 | √-2 | √2 | -√2 | 0 | 0 | 0 | 0 | ζ1613+ζ167 | ζ1611+ζ16 | ζ1615+ζ165 | ζ163+ζ16 | ζ167+ζ165 | ζ169+ζ163 | ζ1615+ζ1613 | ζ1611+ζ169 | complex lifted from D8.C4 |
ρ23 | 2 | -2 | -2 | 2 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | -√-2 | √-2 | -√-2 | √-2 | -√2 | √2 | 0 | 0 | 0 | 0 | ζ1615+ζ1613 | ζ1611+ζ169 | ζ167+ζ165 | ζ1611+ζ16 | ζ1613+ζ167 | ζ163+ζ16 | ζ1615+ζ165 | ζ169+ζ163 | complex lifted from D8.C4 |
ρ24 | 2 | -2 | -2 | 2 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | -√-2 | √-2 | -√-2 | √-2 | √2 | -√2 | 0 | 0 | 0 | 0 | ζ1615+ζ165 | ζ169+ζ163 | ζ1613+ζ167 | ζ1611+ζ169 | ζ1615+ζ1613 | ζ1611+ζ16 | ζ167+ζ165 | ζ163+ζ16 | complex lifted from D8.C4 |
ρ25 | 2 | -2 | -2 | 2 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | √-2 | -√-2 | √-2 | -√-2 | -√2 | √2 | 0 | 0 | 0 | 0 | ζ163+ζ16 | ζ167+ζ165 | ζ1611+ζ169 | ζ1615+ζ165 | ζ169+ζ163 | ζ1615+ζ1613 | ζ1611+ζ16 | ζ1613+ζ167 | complex lifted from D8.C4 |
ρ26 | 2 | -2 | -2 | 2 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | √-2 | -√-2 | √-2 | -√-2 | √2 | -√2 | 0 | 0 | 0 | 0 | ζ1611+ζ16 | ζ1613+ζ167 | ζ169+ζ163 | ζ167+ζ165 | ζ163+ζ16 | ζ1615+ζ165 | ζ1611+ζ169 | ζ1615+ζ1613 | complex lifted from D8.C4 |
ρ27 | 4 | -4 | 4 | -4 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
ρ28 | 4 | 4 | -4 | -4 | 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 | 0 | 0 | 0 | 0 | symplectic lifted from C8.17D4, Schur index 2 |
ρ29 | 4 | 4 | -4 | -4 | 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 | 0 | 0 | 0 | 0 | symplectic lifted from C8.17D4, Schur index 2 |
(1 21)(3 23)(5 25)(7 27)(9 29)(11 31)(13 17)(15 19)(33 41)(34 50)(35 43)(36 52)(37 45)(38 54)(39 47)(40 56)(42 58)(44 60)(46 62)(48 64)(49 57)(51 59)(53 61)(55 63)
(1 29)(2 30)(3 31)(4 32)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)
(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 47 9 39)(2 58 10 50)(3 61 11 53)(4 40 12 48)(5 43 13 35)(6 54 14 62)(7 57 15 49)(8 36 16 44)(17 59 25 51)(18 38 26 46)(19 41 27 33)(20 52 28 60)(21 55 29 63)(22 34 30 42)(23 37 31 45)(24 64 32 56)
G:=sub<Sym(64)| (1,21)(3,23)(5,25)(7,27)(9,29)(11,31)(13,17)(15,19)(33,41)(34,50)(35,43)(36,52)(37,45)(38,54)(39,47)(40,56)(42,58)(44,60)(46,62)(48,64)(49,57)(51,59)(53,61)(55,63), (1,29)(2,30)(3,31)(4,32)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (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), (1,47,9,39)(2,58,10,50)(3,61,11,53)(4,40,12,48)(5,43,13,35)(6,54,14,62)(7,57,15,49)(8,36,16,44)(17,59,25,51)(18,38,26,46)(19,41,27,33)(20,52,28,60)(21,55,29,63)(22,34,30,42)(23,37,31,45)(24,64,32,56)>;
G:=Group( (1,21)(3,23)(5,25)(7,27)(9,29)(11,31)(13,17)(15,19)(33,41)(34,50)(35,43)(36,52)(37,45)(38,54)(39,47)(40,56)(42,58)(44,60)(46,62)(48,64)(49,57)(51,59)(53,61)(55,63), (1,29)(2,30)(3,31)(4,32)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (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), (1,47,9,39)(2,58,10,50)(3,61,11,53)(4,40,12,48)(5,43,13,35)(6,54,14,62)(7,57,15,49)(8,36,16,44)(17,59,25,51)(18,38,26,46)(19,41,27,33)(20,52,28,60)(21,55,29,63)(22,34,30,42)(23,37,31,45)(24,64,32,56) );
G=PermutationGroup([(1,21),(3,23),(5,25),(7,27),(9,29),(11,31),(13,17),(15,19),(33,41),(34,50),(35,43),(36,52),(37,45),(38,54),(39,47),(40,56),(42,58),(44,60),(46,62),(48,64),(49,57),(51,59),(53,61),(55,63)], [(1,29),(2,30),(3,31),(4,32),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(16,28),(33,49),(34,50),(35,51),(36,52),(37,53),(38,54),(39,55),(40,56),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64)], [(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)], [(1,47,9,39),(2,58,10,50),(3,61,11,53),(4,40,12,48),(5,43,13,35),(6,54,14,62),(7,57,15,49),(8,36,16,44),(17,59,25,51),(18,38,26,46),(19,41,27,33),(20,52,28,60),(21,55,29,63),(22,34,30,42),(23,37,31,45),(24,64,32,56)])
Matrix representation of C23.13SD16 ►in GL4(𝔽17) generated by
1 | 0 | 0 | 0 |
0 | 16 | 0 | 0 |
0 | 0 | 16 | 0 |
0 | 0 | 12 | 1 |
16 | 0 | 0 | 0 |
0 | 16 | 0 | 0 |
0 | 0 | 16 | 0 |
0 | 0 | 0 | 16 |
16 | 0 | 0 | 0 |
0 | 16 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
6 | 0 | 0 | 0 |
0 | 5 | 0 | 0 |
0 | 0 | 3 | 9 |
0 | 0 | 8 | 14 |
0 | 1 | 0 | 0 |
16 | 0 | 0 | 0 |
0 | 0 | 5 | 15 |
0 | 0 | 12 | 12 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,16,12,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[6,0,0,0,0,5,0,0,0,0,3,8,0,0,9,14],[0,16,0,0,1,0,0,0,0,0,5,12,0,0,15,12] >;
C23.13SD16 in GAP, Magma, Sage, TeX
C_2^3._{13}{\rm SD}_{16}
% in TeX
G:=Group("C2^3.13SD16");
// GroupNames label
G:=SmallGroup(128,82);
// by ID
G=gap.SmallGroup(128,82);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,456,422,387,520,1690,248,2804,1411,172,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=e^2=c,e*a*e^-1=a*b=b*a,a*c=c*a,d*a*d^-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=a*b*d^3>;
// generators/relations
Export
Subgroup lattice of C23.13SD16 in TeX
Character table of C23.13SD16 in TeX