p-group, metabelian, nilpotent (class 4), monomial
Aliases: C23.12SD16, C4.20C4≀C2, (C2×D8).1C4, C2.D8.1C4, C22⋊C16⋊4C2, (C2×C8).302D4, C8⋊7D4.2C2, (C2×C4).102D8, C4.7(C23⋊C4), (C22×C4).187D4, C4.C42⋊11C2, C2.4(D8.C4), C2.3(M5(2)⋊C2), (C22×C8).99C22, C22.57(D4⋊C4), C2.14(C22.SD16), (C2×C8).19(C2×C4), (C2×C4).219(C22⋊C4), SmallGroup(128,81)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.12SD16
G = < a,b,c,d,e | a2=b2=c2=e2=1, d8=c, eae=ab=ba, ac=ca, dad-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=abcd3 >
Character table of C23.12SD16
class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | |
size | 1 | 1 | 1 | 1 | 4 | 16 | 2 | 2 | 2 | 2 | 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 | 0 | -2 | 2 | 2 | -2 | 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 | 0 | 2 | 2 | 2 | 2 | 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 | 0 | 2 | -2 | -2 | 2 | 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 | 0 | 2 | -2 | -2 | 2 | 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 | 0 | -2 | 2 | 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 | 0 | -2 | 2 | 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 | 0 | -2 | 2 | 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 | 0 | -2 | 2 | 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 | 0 | -2 | -2 | -2 | -2 | 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 | 0 | -2 | -2 | -2 | -2 | 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 | 0 | -2i | 0 | 0 | 2i | 0 | √-2 | -√-2 | √-2 | -√-2 | √2 | -√2 | 0 | 0 | 0 | 0 | ζ1615+ζ165 | ζ1611+ζ16 | ζ1613+ζ167 | ζ1611+ζ169 | ζ167+ζ165 | ζ169+ζ163 | ζ1615+ζ1613 | ζ163+ζ16 | complex lifted from D8.C4 |
ρ20 | 2 | -2 | -2 | 2 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | √-2 | -√-2 | √-2 | -√-2 | √2 | -√2 | 0 | 0 | 0 | 0 | ζ1613+ζ167 | ζ169+ζ163 | ζ1615+ζ165 | ζ163+ζ16 | ζ1615+ζ1613 | ζ1611+ζ16 | ζ167+ζ165 | ζ1611+ζ169 | complex lifted from D8.C4 |
ρ21 | 2 | -2 | -2 | 2 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | -√-2 | √-2 | -√-2 | √-2 | -√2 | √2 | 0 | 0 | 0 | 0 | ζ163+ζ16 | ζ1615+ζ1613 | ζ1611+ζ169 | ζ1615+ζ165 | ζ1611+ζ16 | ζ167+ζ165 | ζ169+ζ163 | ζ1613+ζ167 | complex lifted from D8.C4 |
ρ22 | 2 | -2 | -2 | 2 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | √-2 | -√-2 | √-2 | -√-2 | -√2 | √2 | 0 | 0 | 0 | 0 | ζ1615+ζ1613 | ζ163+ζ16 | ζ167+ζ165 | ζ1611+ζ16 | ζ1615+ζ165 | ζ1611+ζ169 | ζ1613+ζ167 | ζ169+ζ163 | complex lifted from D8.C4 |
ρ23 | 2 | -2 | -2 | 2 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | √-2 | -√-2 | √-2 | -√-2 | -√2 | √2 | 0 | 0 | 0 | 0 | ζ167+ζ165 | ζ1611+ζ169 | ζ1615+ζ1613 | ζ169+ζ163 | ζ1613+ζ167 | ζ163+ζ16 | ζ1615+ζ165 | ζ1611+ζ16 | complex lifted from D8.C4 |
ρ24 | 2 | -2 | -2 | 2 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | -√-2 | √-2 | -√-2 | √-2 | √2 | -√2 | 0 | 0 | 0 | 0 | ζ1611+ζ16 | ζ1615+ζ165 | ζ169+ζ163 | ζ167+ζ165 | ζ1611+ζ169 | ζ1613+ζ167 | ζ163+ζ16 | ζ1615+ζ1613 | complex lifted from D8.C4 |
ρ25 | 2 | -2 | -2 | 2 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | -√-2 | √-2 | -√-2 | √-2 | √2 | -√2 | 0 | 0 | 0 | 0 | ζ169+ζ163 | ζ1613+ζ167 | ζ1611+ζ16 | ζ1615+ζ1613 | ζ163+ζ16 | ζ1615+ζ165 | ζ1611+ζ169 | ζ167+ζ165 | complex lifted from D8.C4 |
ρ26 | 2 | -2 | -2 | 2 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | -√-2 | √-2 | -√-2 | √-2 | -√2 | √2 | 0 | 0 | 0 | 0 | ζ1611+ζ169 | ζ167+ζ165 | ζ163+ζ16 | ζ1613+ζ167 | ζ169+ζ163 | ζ1615+ζ1613 | ζ1611+ζ16 | ζ1615+ζ165 | complex lifted from D8.C4 |
ρ27 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 4 | -4 | 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 | orthogonal lifted from M5(2)⋊C2 |
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 | orthogonal lifted from M5(2)⋊C2 |
(1 63)(2 26)(3 49)(4 28)(5 51)(6 30)(7 53)(8 32)(9 55)(10 18)(11 57)(12 20)(13 59)(14 22)(15 61)(16 24)(17 41)(19 43)(21 45)(23 47)(25 33)(27 35)(29 37)(31 39)(34 64)(36 50)(38 52)(40 54)(42 56)(44 58)(46 60)(48 62)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(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 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)
(2 28)(3 39)(4 56)(5 13)(6 24)(7 35)(8 52)(10 20)(11 47)(12 64)(14 32)(15 43)(16 60)(17 63)(18 36)(19 31)(21 59)(22 48)(23 27)(25 55)(26 44)(29 51)(30 40)(34 50)(37 45)(38 62)(42 58)(46 54)(49 61)(53 57)
G:=sub<Sym(64)| (1,63)(2,26)(3,49)(4,28)(5,51)(6,30)(7,53)(8,32)(9,55)(10,18)(11,57)(12,20)(13,59)(14,22)(15,61)(16,24)(17,41)(19,43)(21,45)(23,47)(25,33)(27,35)(29,37)(31,39)(34,64)(36,50)(38,52)(40,54)(42,56)(44,58)(46,60)(48,62), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(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), (1,9)(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), (2,28)(3,39)(4,56)(5,13)(6,24)(7,35)(8,52)(10,20)(11,47)(12,64)(14,32)(15,43)(16,60)(17,63)(18,36)(19,31)(21,59)(22,48)(23,27)(25,55)(26,44)(29,51)(30,40)(34,50)(37,45)(38,62)(42,58)(46,54)(49,61)(53,57)>;
G:=Group( (1,63)(2,26)(3,49)(4,28)(5,51)(6,30)(7,53)(8,32)(9,55)(10,18)(11,57)(12,20)(13,59)(14,22)(15,61)(16,24)(17,41)(19,43)(21,45)(23,47)(25,33)(27,35)(29,37)(31,39)(34,64)(36,50)(38,52)(40,54)(42,56)(44,58)(46,60)(48,62), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(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), (1,9)(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), (2,28)(3,39)(4,56)(5,13)(6,24)(7,35)(8,52)(10,20)(11,47)(12,64)(14,32)(15,43)(16,60)(17,63)(18,36)(19,31)(21,59)(22,48)(23,27)(25,55)(26,44)(29,51)(30,40)(34,50)(37,45)(38,62)(42,58)(46,54)(49,61)(53,57) );
G=PermutationGroup([(1,63),(2,26),(3,49),(4,28),(5,51),(6,30),(7,53),(8,32),(9,55),(10,18),(11,57),(12,20),(13,59),(14,22),(15,61),(16,24),(17,41),(19,43),(21,45),(23,47),(25,33),(27,35),(29,37),(31,39),(34,64),(36,50),(38,52),(40,54),(42,56),(44,58),(46,60),(48,62)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,33),(10,34),(11,35),(12,36),(13,37),(14,38),(15,39),(16,40),(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)], [(1,9),(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)], [(2,28),(3,39),(4,56),(5,13),(6,24),(7,35),(8,52),(10,20),(11,47),(12,64),(14,32),(15,43),(16,60),(17,63),(18,36),(19,31),(21,59),(22,48),(23,27),(25,55),(26,44),(29,51),(30,40),(34,50),(37,45),(38,62),(42,58),(46,54),(49,61),(53,57)])
Matrix representation of C23.12SD16 ►in GL4(𝔽17) generated by
13 | 8 | 0 | 0 |
13 | 4 | 0 | 0 |
0 | 0 | 16 | 16 |
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 |
5 | 11 | 0 | 0 |
3 | 16 | 0 | 0 |
0 | 0 | 16 | 6 |
0 | 0 | 2 | 1 |
1 | 0 | 0 | 0 |
1 | 16 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 15 | 16 |
G:=sub<GL(4,GF(17))| [13,13,0,0,8,4,0,0,0,0,16,0,0,0,16,1],[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],[5,3,0,0,11,16,0,0,0,0,16,2,0,0,6,1],[1,1,0,0,0,16,0,0,0,0,1,15,0,0,0,16] >;
C23.12SD16 in GAP, Magma, Sage, TeX
C_2^3._{12}{\rm SD}_{16}
% in TeX
G:=Group("C2^3.12SD16");
// GroupNames label
G:=SmallGroup(128,81);
// by ID
G=gap.SmallGroup(128,81);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,387,520,1690,248,2804,1411,172,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=e^2=1,d^8=c,e*a*e=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=a*b*c*d^3>;
// generators/relations
Export
Subgroup lattice of C23.12SD16 in TeX
Character table of C23.12SD16 in TeX