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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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C23.20D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C2×C42⋊C2 — C2×C23.20D4
 Lower central C1 — C2 — C2×C4 — C2×C23.20D4
 Upper central C1 — C23 — C23×C4 — C2×C23.20D4
 Jennings C1 — C2 — C2 — C2×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, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C22⋊C8, Q8⋊C4, C4.Q8, C2.D8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C22⋊Q8, C22⋊Q8, C22×C8, C23×C4, C22×Q8, C2×C22⋊C8, C2×Q8⋊C4, C2×C4.Q8, C2×C2.D8, C23.20D4, C2×C42⋊C2, C2×C22⋊Q8, C2×C23.20D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22.D4, C4○D8, C8.C22, C22×D4, C2×C4○D4, C23.20D4, 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 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(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 63)(50 64)(51 57)(52 58)(53 59)(54 60)(55 61)(56 62)
(1 55)(2 18)(3 49)(4 20)(5 51)(6 22)(7 53)(8 24)(9 45)(10 37)(11 47)(12 39)(13 41)(14 33)(15 43)(16 35)(17 57)(19 59)(21 61)(23 63)(25 42)(26 34)(27 44)(28 36)(29 46)(30 38)(31 48)(32 40)(50 60)(52 62)(54 64)(56 58)
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)(17 55)(18 56)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(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 62 13 58)(11 60 15 64)(17 37 21 33)(18 41 22 45)(19 35 23 39)(20 47 24 43)(25 61 29 57)(27 59 31 63)(34 50 38 54)(36 56 40 52)(42 55 46 51)(44 53 48 49)

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4H 4I ··· 4P 4Q 4R 4S 4T 8A ··· 8H order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 4 4 2 ··· 2 4 ··· 4 8 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 C4○D4 C4○D8 C8.C22 kernel C2×C23.20D4 C2×C22⋊C8 C2×Q8⋊C4 C2×C4.Q8 C2×C2.D8 C23.20D4 C2×C42⋊C2 C2×C22⋊Q8 C22×C4 C24 C2×C4 C22 C22 # reps 1 1 2 1 1 8 1 1 3 1 8 8 2

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

 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 0 0 0 0 0 1 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 0 0 0 0 0 4 4 0 0 0 0 0 0 13 9 0 0 0 0 4 4 0 0 0 0 0 0 8 0 0 0 0 0 0 2
,
 4 8 0 0 0 0 13 13 0 0 0 0 0 0 1 2 0 0 0 0 0 16 0 0 0 0 0 0 0 2 0 0 0 0 8 0

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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