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G = C2×C42.29C22  order 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⋊C4⋊63C22, (C2×C4).331C24, (C2×C8).452C23, (C22×C4).457D4, C23.874(C2×D4), D4⋊C4⋊94C22, C4.22(C4.4D4), (C2×D4).100C23, C42.C2⋊34C22, C4⋊1D4.145C22, (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×C4⋊1D4).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

Derived series
C1 — C2×C4 — C2×C42.29C22
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C2×C8⋊C4 — C2×C42.29C22
Lower central
C1 — C2 — C2×C4 — C2×C42.29C22
Upper central
C1 — C23 — C2×C42 — C2×C42.29C22
Jennings
C1 — C2 — C2 — C2×C4 — C2×C42.29C22

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], C4⋊1D4 [×4], C4⋊1D4 [×2], C22×C8 [×2], C22×D4 [×2], C22×D4 [×2], C2×C8⋊C4, C2×D4⋊C4 [×4], C42.29C22 [×8], C2×C42.C2, C2×C4⋊1D4, 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

Generators and relations
 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 >

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

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

Matrix representation ►G ⊆ 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] >;
 

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×C4⋊1D4C42C22×C4C2×C4C22
# reps1148112284

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
 

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