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G = C2×C23.20D4order 128 = 27

Direct product of C2 and C23.20D4

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

Aliases: C2×C23.20D4, C24.114D4, C4.Q862C22, C2.D853C22, C4⋊C4.389C23, (C2×C4).286C24, (C2×C8).310C23, C23.240(C2×D4), (C22×C4).437D4, (C2×Q8).64C23, Q8⋊C481C22, C22.96(C4○D8), C22⋊C8.187C22, (C23×C4).556C22, (C22×C8).347C22, C22.546(C22×D4), C22⋊Q8.157C22, (C22×C4).1546C23, C4.58(C22.D4), (C22×Q8).290C22, C42⋊C2.316C22, C22.111(C8.C22), C22.109(C22.D4), (C2×C2.D8)⋊25C2, (C2×C4.Q8)⋊33C2, C2.21(C2×C4○D8), C4.96(C2×C4○D4), (C2×Q8⋊C4)⋊40C2, (C2×C4).1217(C2×D4), (C2×C22⋊C8).35C2, C2.27(C2×C8.C22), (C2×C22⋊Q8).55C2, (C2×C4).844(C4○D4), (C2×C4⋊C4).610C22, (C2×C42⋊C2).59C2, C2.51(C2×C22.D4), SmallGroup(128,1820)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C23.20D4
C1C2C4C2×C4C22×C4C2×C4⋊C4C2×C42⋊C2 — C2×C23.20D4
C1C2C2×C4 — C2×C23.20D4
C1C23C23×C4 — C2×C23.20D4
C1C2C2C2×C4 — C2×C23.20D4

Generators and relations for C2×C23.20D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=f2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=bc=cb, bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce3 >

Subgroups: 364 in 208 conjugacy classes, 100 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×6], C22 [×10], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×26], Q8 [×6], C23, C23 [×2], C23 [×6], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×6], C4⋊C4 [×7], C2×C8 [×4], C2×C8 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×8], C2×Q8 [×2], C2×Q8 [×5], C24, C22⋊C8 [×4], Q8⋊C4 [×8], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C2×C4⋊C4, C42⋊C2 [×4], C42⋊C2 [×2], C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8 [×2], C23×C4, C22×Q8, C2×C22⋊C8, C2×Q8⋊C4 [×2], C2×C4.Q8, C2×C2.D8, C23.20D4 [×8], C2×C42⋊C2, C2×C22⋊Q8, C2×C23.20D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C4○D8 [×2], C8.C22 [×2], C22×D4, C2×C4○D4 [×2], C23.20D4 [×4], C2×C22.D4, C2×C4○D8, C2×C8.C22, C2×C23.20D4

Smallest permutation representation of C2×C23.20D4
On 64 points
Generators in S64
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 41)(49 61)(50 62)(51 63)(52 64)(53 57)(54 58)(55 59)(56 60)
(1 55)(2 19)(3 49)(4 21)(5 51)(6 23)(7 53)(8 17)(9 45)(10 37)(11 47)(12 39)(13 41)(14 33)(15 43)(16 35)(18 63)(20 57)(22 59)(24 61)(25 42)(26 34)(27 44)(28 36)(29 46)(30 38)(31 48)(32 40)(50 58)(52 60)(54 62)(56 64)
(1 63)(2 64)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)(17 54)(18 55)(19 56)(20 49)(21 50)(22 51)(23 52)(24 53)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 41)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 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 14 5 10)(2 28 6 32)(3 12 7 16)(4 26 8 30)(9 60 13 64)(11 58 15 62)(17 43 21 47)(18 37 22 33)(19 41 23 45)(20 35 24 39)(25 59 29 63)(27 57 31 61)(34 50 38 54)(36 56 40 52)(42 55 46 51)(44 53 48 49)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4P4Q4R4S4T8A···8H
order12···2224···44···444448···8
size11···1442···24···488884···4

38 irreducible representations

dim1111111122224
type++++++++++-
imageC1C2C2C2C2C2C2C2D4D4C4○D4C4○D8C8.C22
kernelC2×C23.20D4C2×C22⋊C8C2×Q8⋊C4C2×C4.Q8C2×C2.D8C23.20D4C2×C42⋊C2C2×C22⋊Q8C22×C4C24C2×C4C22C22
# reps1121181131882

Matrix representation of C2×C23.20D4 in GL6(𝔽17)

1600000
0160000
001000
000100
0000160
0000016
,
1600000
0160000
0016000
001100
000010
0000016
,
100000
010000
0016000
0001600
000010
000001
,
100000
010000
001000
000100
0000160
0000016
,
1300000
440000
0013900
004400
000080
000002
,
480000
13130000
001200
0001600
000002
000080

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[13,4,0,0,0,0,0,4,0,0,0,0,0,0,13,4,0,0,0,0,9,4,0,0,0,0,0,0,8,0,0,0,0,0,0,2],[4,13,0,0,0,0,8,13,0,0,0,0,0,0,1,0,0,0,0,0,2,16,0,0,0,0,0,0,0,8,0,0,0,0,2,0] >;

C2×C23.20D4 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{20}D_4
% in TeX

G:=Group("C2xC2^3.20D4");
// GroupNames label

G:=SmallGroup(128,1820);
// by ID

G=gap.SmallGroup(128,1820);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,100,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=f^2=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,e*b*e^-1=b*c=c*b,b*d=d*b,f*b*f^-1=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^3>;
// generators/relations

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