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## G = C23.12SD16order 128 = 27

### 2nd non-split extension by C23 of SD16 acting via SD16/C4=C22

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — C23.12SD16
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C8⋊7D4 — C23.12SD16
 Lower central C1 — C2 — C2×C4 — C2×C8 — C23.12SD16
 Upper central C1 — C22 — C22×C4 — C22×C8 — C23.12SD16
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C2×C4 — C22×C8 — C23.12SD16

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

Smallest permutation representation of C23.12SD16
On 64 points
Generators in S64
(1 55)(2 25)(3 57)(4 27)(5 59)(6 29)(7 61)(8 31)(9 63)(10 17)(11 49)(12 19)(13 51)(14 21)(15 53)(16 23)(18 41)(20 43)(22 45)(24 47)(26 33)(28 35)(30 37)(32 39)(34 58)(36 60)(38 62)(40 64)(42 50)(44 52)(46 54)(48 56)
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 56)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 64)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)
(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 27)(3 37)(4 64)(5 13)(6 23)(7 33)(8 60)(10 19)(11 45)(12 56)(14 31)(15 41)(16 52)(17 34)(18 30)(20 51)(21 46)(22 26)(24 63)(25 42)(28 59)(29 38)(32 55)(35 43)(36 54)(40 50)(44 62)(48 58)(49 61)(53 57)

G:=sub<Sym(64)| (1,55)(2,25)(3,57)(4,27)(5,59)(6,29)(7,61)(8,31)(9,63)(10,17)(11,49)(12,19)(13,51)(14,21)(15,53)(16,23)(18,41)(20,43)(22,45)(24,47)(26,33)(28,35)(30,37)(32,39)(34,58)(36,60)(38,62)(40,64)(42,50)(44,52)(46,54)(48,56), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,56)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55), (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), (2,27)(3,37)(4,64)(5,13)(6,23)(7,33)(8,60)(10,19)(11,45)(12,56)(14,31)(15,41)(16,52)(17,34)(18,30)(20,51)(21,46)(22,26)(24,63)(25,42)(28,59)(29,38)(32,55)(35,43)(36,54)(40,50)(44,62)(48,58)(49,61)(53,57)>;

G:=Group( (1,55)(2,25)(3,57)(4,27)(5,59)(6,29)(7,61)(8,31)(9,63)(10,17)(11,49)(12,19)(13,51)(14,21)(15,53)(16,23)(18,41)(20,43)(22,45)(24,47)(26,33)(28,35)(30,37)(32,39)(34,58)(36,60)(38,62)(40,64)(42,50)(44,52)(46,54)(48,56), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,56)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55), (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), (2,27)(3,37)(4,64)(5,13)(6,23)(7,33)(8,60)(10,19)(11,45)(12,56)(14,31)(15,41)(16,52)(17,34)(18,30)(20,51)(21,46)(22,26)(24,63)(25,42)(28,59)(29,38)(32,55)(35,43)(36,54)(40,50)(44,62)(48,58)(49,61)(53,57) );

G=PermutationGroup([[(1,55),(2,25),(3,57),(4,27),(5,59),(6,29),(7,61),(8,31),(9,63),(10,17),(11,49),(12,19),(13,51),(14,21),(15,53),(16,23),(18,41),(20,43),(22,45),(24,47),(26,33),(28,35),(30,37),(32,39),(34,58),(36,60),(38,62),(40,64),(42,50),(44,52),(46,54),(48,56)], [(1,39),(2,40),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,56),(18,57),(19,58),(20,59),(21,60),(22,61),(23,62),(24,63),(25,64),(26,49),(27,50),(28,51),(29,52),(30,53),(31,54),(32,55)], [(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)], [(2,27),(3,37),(4,64),(5,13),(6,23),(7,33),(8,60),(10,19),(11,45),(12,56),(14,31),(15,41),(16,52),(17,34),(18,30),(20,51),(21,46),(22,26),(24,63),(25,42),(28,59),(29,38),(32,55),(35,43),(36,54),(40,50),(44,62),(48,58),(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

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