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## G = C23×SD16order 128 = 27

### Direct product of C23 and SD16

direct product, p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C23×SD16
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — D4×C23 — C23×SD16
 Lower central C1 — C2 — C4 — C23×SD16
 Upper central C1 — C24 — C23×C4 — C23×SD16
 Jennings C1 — C2 — C2 — C4 — C23×SD16

Generators and relations for C23×SD16
G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d3 >

Subgroups: 1500 in 860 conjugacy classes, 476 normal (9 characteristic)
C1, C2, C2 [×14], C2 [×8], C4, C4 [×7], C4 [×8], C22 [×35], C22 [×64], C8 [×8], C2×C4 [×28], C2×C4 [×28], D4 [×8], D4 [×28], Q8 [×8], Q8 [×28], C23 [×15], C23 [×84], C2×C8 [×28], SD16 [×64], C22×C4 [×14], C22×C4 [×14], C2×D4 [×28], C2×D4 [×42], C2×Q8 [×28], C2×Q8 [×42], C24, C24 [×22], C22×C8 [×14], C2×SD16 [×112], C23×C4, C23×C4, C22×D4 [×14], C22×D4 [×7], C22×Q8 [×14], C22×Q8 [×7], C25, C23×C8, C22×SD16 [×28], D4×C23, Q8×C23, C23×SD16
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], SD16 [×8], C2×D4 [×28], C24 [×31], C2×SD16 [×28], C22×D4 [×14], C25, C22×SD16 [×14], D4×C23, C23×SD16

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

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

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

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

56 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2W 4A ··· 4H 4I ··· 4P 8A ··· 8P order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4 2 ··· 2

56 irreducible representations

 dim 1 1 1 1 1 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 D4 D4 SD16 kernel C23×SD16 C23×C8 C22×SD16 D4×C23 Q8×C23 C22×C4 C24 C23 # reps 1 1 28 1 1 7 1 16

Matrix representation of C23×SD16 in GL5(𝔽17)

 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 7 7 0 0 0 5 0
,
 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 16 16

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,7,5,0,0,0,7,0],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,16,0,0,0,0,16] >;

C23×SD16 in GAP, Magma, Sage, TeX

C_2^3\times {\rm SD}_{16}
% in TeX

G:=Group("C2^3xSD16");
// GroupNames label

G:=SmallGroup(128,2307);
// by ID

G=gap.SmallGroup(128,2307);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,-2,448,477,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^3>;
// generators/relations

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