Copied to
clipboard

G = C2×C23.48D4order 128 = 27

Direct product of C2 and C23.48D4

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

Aliases: C2×C23.48D4, C24.182D4, C23.21Q16, C4⋊C4.51C23, C2.D854C22, C2.8(C22×Q16), (C2×C4).288C24, (C2×C8).142C23, C23.664(C2×D4), (C22×C4).439D4, (C2×Q8).65C23, C22.15(C2×Q16), Q8⋊C469C22, C22⋊C8.171C22, (C22×C8).147C22, (C23×C4).558C22, C22.548(C22×D4), C22⋊Q8.158C22, C22.122(C8⋊C22), (C22×C4).1005C23, C4.60(C22.D4), (C22×Q8).291C22, C22.111(C22.D4), (C2×C2.D8)⋊26C2, C4.98(C2×C4○D4), (C2×C4).848(C2×D4), C2.27(C2×C8⋊C22), (C2×Q8⋊C4)⋊24C2, (C22×C4⋊C4).46C2, (C2×C22⋊C8).31C2, (C2×C22⋊Q8).56C2, (C2×C4).846(C4○D4), (C2×C4⋊C4).925C22, C2.53(C2×C22.D4), SmallGroup(128,1822)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C23.48D4
C1C2C4C2×C4C22×C4C2×C4⋊C4C22×C4⋊C4 — C2×C23.48D4
C1C2C2×C4 — C2×C23.48D4
C1C23C23×C4 — C2×C23.48D4
C1C2C2C2×C4 — C2×C23.48D4

Generators and relations for C2×C23.48D4
 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=fbf-1=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >

Subgroups: 412 in 230 conjugacy classes, 108 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×10], C22 [×12], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×34], Q8 [×6], C23, C23 [×6], C23 [×4], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×11], C2×C8 [×4], C2×C8 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×16], C2×Q8 [×2], C2×Q8 [×5], C24, C22⋊C8 [×4], Q8⋊C4 [×8], C2.D8 [×8], C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4 [×6], C2×C4⋊C4 [×4], C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8 [×2], C23×C4, C23×C4, C22×Q8, C2×C22⋊C8, C2×Q8⋊C4 [×2], C2×C2.D8 [×2], C23.48D4 [×8], C22×C4⋊C4, C2×C22⋊Q8, C2×C23.48D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], Q16 [×4], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C2×Q16 [×6], C8⋊C22 [×2], C22×D4, C2×C4○D4 [×2], C23.48D4 [×4], C2×C22.D4, C22×Q16, C2×C8⋊C22, C2×C23.48D4

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

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4N4O4P4Q4R8A···8H
order12···2222244444···444448···8
size11···1222222224···488884···4

38 irreducible representations

dim111111122224
type+++++++++-+
imageC1C2C2C2C2C2C2D4D4C4○D4Q16C8⋊C22
kernelC2×C23.48D4C2×C22⋊C8C2×Q8⋊C4C2×C2.D8C23.48D4C22×C4⋊C4C2×C22⋊Q8C22×C4C24C2×C4C23C22
# reps112281131882

Matrix representation of C2×C23.48D4 in GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
100000
0160000
001000
0001600
000010
000001
,
1600000
0160000
0016000
0001600
000010
000001
,
100000
010000
001000
000100
0000160
0000016
,
040000
400000
0001300
004000
0000314
000033
,
010000
100000
0001600
0016000
00001610
0000101

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,0,0,0,0,3,3,0,0,0,0,14,3],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,16,10,0,0,0,0,10,1] >;

C2×C23.48D4 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{48}D_4
% in TeX

G:=Group("C2xC2^3.48D4");
// GroupNames label

G:=SmallGroup(128,1822);
// by ID

G=gap.SmallGroup(128,1822);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,436,4037,1027,124]);
// 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=f*b*f^-1=b*c=c*b,b*d=d*b,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*d*e^3>;
// generators/relations

׿
×
𝔽