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G = C23×SD16order 128 = 27

Direct product of C23 and SD16

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

Aliases: C23×SD16, C83C24, C4.2C25, Q81C24, D4.1C24, C24.196D4, (C23×C8)⋊13C2, (C2×C8)⋊16C23, (Q8×C23)⋊13C2, (C2×Q8)⋊19C23, C2.37(D4×C23), C4.28(C22×D4), (C2×C4).608C24, (C22×C8)⋊69C22, (D4×C23).21C2, (C22×C4).628D4, C23.894(C2×D4), (C2×D4).489C23, (C22×Q8)⋊67C22, (C23×C4).712C22, C22.165(C22×D4), (C22×C4).1590C23, (C22×D4).602C22, (C2×C4).881(C2×D4), SmallGroup(128,2307)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C23×SD16
C1C2C4C2×C4C22×C4C23×C4D4×C23 — C23×SD16
C1C2C4 — C23×SD16
C1C24C23×C4 — C23×SD16
C1C2C2C4 — 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···2O2P···2W4A···4H4I···4P8A···8P
order12···22···24···44···48···8
size11···14···42···24···42···2

56 irreducible representations

dim11111222
type+++++++
imageC1C2C2C2C2D4D4SD16
kernelC23×SD16C23×C8C22×SD16D4×C23Q8×C23C22×C4C24C23
# reps1128117116

Matrix representation of C23×SD16 in GL5(𝔽17)

10000
016000
001600
00010
00001
,
10000
016000
00100
000160
000016
,
160000
01000
00100
000160
000016
,
10000
01000
001600
00077
00050
,
160000
016000
001600
00010
0001616

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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