Copied to
clipboard

G = C2×C42.29C22order 128 = 27

Direct product of C2 and C42.29C22

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

Aliases: C2×C42.29C22, C42.237D4, C42.363C23, C4⋊C4.86C23, C8⋊C463C22, (C2×C8).452C23, (C2×C4).331C24, (C22×C4).457D4, C23.874(C2×D4), D4⋊C494C22, C4.22(C4.4D4), (C2×D4).100C23, C42.C234C22, C41D4.145C22, (C2×C42).844C22, (C22×C8).458C22, C22.591(C22×D4), C22.124(C8⋊C22), (C22×C4).1553C23, C22.84(C4.4D4), (C22×D4).367C22, (C2×C8⋊C4)⋊38C2, C4.40(C2×C4○D4), (C2×C4).511(C2×D4), C2.38(C2×C8⋊C22), (C2×D4⋊C4)⋊56C2, (C2×C41D4).23C2, (C2×C42.C2)⋊34C2, C2.42(C2×C4.4D4), (C2×C4).710(C4○D4), (C2×C4⋊C4).622C22, SmallGroup(128,1865)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C42.29C22
C1C2C4C2×C4C22×C4C22×C8C2×C8⋊C4 — C2×C42.29C22
C1C2C2×C4 — C2×C42.29C22
C1C23C2×C42 — C2×C42.29C22
C1C2C2C2×C4 — C2×C42.29C22

Generators and relations for C2×C42.29C22
 G = < a,b,c,d,e | a2=b4=c4=d2=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b-1, ebe-1=bc2, dcd=c-1, ce=ec, ede-1=b2c-1d >

Subgroups: 580 in 244 conjugacy classes, 100 normal (12 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×4], C4 [×8], C22, C22 [×6], C22 [×20], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], D4 [×28], C23, C23 [×16], C42 [×4], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×4], C2×D4 [×26], C24 [×2], C8⋊C4 [×4], D4⋊C4 [×16], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42.C2 [×4], C42.C2 [×2], C41D4 [×4], C41D4 [×2], C22×C8 [×2], C22×D4 [×2], C22×D4 [×2], C2×C8⋊C4, C2×D4⋊C4 [×4], C42.29C22 [×8], C2×C42.C2, C2×C41D4, C2×C42.29C22
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C8⋊C22 [×4], C22×D4, C2×C4○D4 [×2], C42.29C22 [×4], C2×C4.4D4, C2×C8⋊C22 [×2], C2×C42.29C22

Smallest permutation representation of C2×C42.29C22
On 64 points
Generators in S64
(1 53)(2 54)(3 55)(4 56)(5 49)(6 50)(7 51)(8 52)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(33 63)(34 64)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)
(1 60 21 10)(2 57 22 15)(3 62 23 12)(4 59 24 9)(5 64 17 14)(6 61 18 11)(7 58 19 16)(8 63 20 13)(25 45 51 36)(26 42 52 33)(27 47 53 38)(28 44 54 35)(29 41 55 40)(30 46 56 37)(31 43 49 34)(32 48 50 39)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(2 24)(3 7)(4 22)(6 20)(8 18)(9 15)(10 60)(11 13)(12 58)(14 64)(16 62)(19 23)(25 29)(26 50)(28 56)(30 54)(32 52)(33 39)(34 43)(35 37)(36 41)(38 47)(40 45)(42 48)(44 46)(51 55)(57 59)(61 63)
(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)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L8A···8H
order12···222224444444444448···8
size11···188882222444488884···4

32 irreducible representations

dim1111112224
type+++++++++
imageC1C2C2C2C2C2D4D4C4○D4C8⋊C22
kernelC2×C42.29C22C2×C8⋊C4C2×D4⋊C4C42.29C22C2×C42.C2C2×C41D4C42C22×C4C2×C4C22
# reps1148112284

Matrix representation of C2×C42.29C22 in GL8(𝔽17)

160000000
016000000
001600000
000160000
00001000
00000100
00000010
00000001
,
18000000
416000000
00100000
00010000
000000125
0000001212
0000121200
000051200
,
10000000
01000000
001600000
000160000
000001600
00001000
000000016
00000010
,
10000000
416000000
001600000
00010000
00001000
000001600
00000001
00000010
,
415000000
1613000000
00010000
001600000
000000160
000000016
00000100
000016000

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,4,0,0,0,0,0,0,8,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,12,12,0,0,0,0,12,12,0,0,0,0,0,0,5,12,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[1,4,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[4,16,0,0,0,0,0,0,15,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0] >;

C2×C42.29C22 in GAP, Magma, Sage, TeX

C_2\times C_4^2._{29}C_2^2
% in TeX

G:=Group("C2xC4^2.29C2^2");
// GroupNames label

G:=SmallGroup(128,1865);
// by ID

G=gap.SmallGroup(128,1865);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,680,758,723,100,2804,172,4037,124]);
// Polycyclic

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

׿
×
𝔽