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G = (C2×C8).200D4  order 128 = 27

168th non-split extension by C2×C8 of D4 acting via D4/C2=C22

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

Aliases: (C2×C8).200D4, C4.125(C4×D4), D4⋊C4⋊11C4, C2.3(C8⋊2D4), C2.5(C8⋊D4), (C22×C4).131D4, C22.170(C4×D4), C23.787(C2×D4), C2.10(D8⋊C4), C22.4Q16⋊49C2, C4.4(C42⋊2C2), C4.10(C42⋊C2), C22.88(C8⋊C22), (C2×C42).303C22, (C22×C8).398C22, (C22×D4).40C22, C2.17(SD16⋊C4), C22.133(C4⋊D4), (C22×C4).1387C23, C23.65C23⋊4C2, C22.61(C4.4D4), C4.95(C22.D4), C22.77(C8.C22), C24.3C22.9C2, C2.3(C42.28C22), C2.2(C42.29C22), C2.21(C24.C22), C4⋊C4.84(C2×C4), (C2×C8⋊C4)⋊24C2, (C2×C8).147(C2×C4), (C2×D4).100(C2×C4), (C2×C4).1344(C2×D4), (C2×C4⋊C4).73C22, (C2×D4⋊C4).34C2, (C2×C4).582(C4○D4), (C2×C4).405(C22×C4), SmallGroup(128,668)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — (C2×C8).200D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C2×C8⋊C4 — (C2×C8).200D4
Lower central
C1 — C2 — C2×C4 — (C2×C8).200D4
Upper central
C1 — C23 — C2×C42 — (C2×C8).200D4
Jennings
C1 — C2 — C2 — C22×C4 — (C2×C8).200D4

Generators and relations for (C2×C8).200D4
 G = < a,b,c,d | a2=b8=c4=1, d2=a, ab=ba, ac=ca, ad=da, cbc-1=ab3, dbd-1=ab-1, dcd-1=b6c-1 >

Subgroups: 340 in 142 conjugacy classes, 56 normal (44 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×6], C22 [×7], C22 [×10], C8 [×4], C2×C4 [×6], C2×C4 [×16], D4 [×6], C23, C23 [×8], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C24, C2.C42, C8⋊C4 [×2], D4⋊C4 [×4], D4⋊C4 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C2×C4⋊C4, C22×C8 [×2], C22×D4, C22.4Q16 [×2], C23.65C23, C24.3C22, C2×C8⋊C4, C2×D4⋊C4 [×2], (C2×C8).200D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22×C4, C2×D4 [×2], C4○D4 [×4], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C8⋊C22 [×3], C8.C22, C24.C22, SD16⋊C4, D8⋊C4, C8⋊D4, C8⋊2D4, C42.28C22, C42.29C22, (C2×C8).200D4

Smallest permutation representation of (C2×C8).200D4
►On 64 points
Generators in S64
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 49)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 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 22 39 11)(2 27 40 44)(3 20 33 9)(4 25 34 42)(5 18 35 15)(6 31 36 48)(7 24 37 13)(8 29 38 46)(10 55 21 60)(12 53 23 58)(14 51 17 64)(16 49 19 62)(26 61 43 56)(28 59 45 54)(30 57 47 52)(32 63 41 50)

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

32 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G4H4I···4N8A···8H
order12···222444444444···48···8
size11···188222244448···84···4

32 irreducible representations

dim111111122244
type+++++++++-
imageC1C2C2C2C2C2C4D4D4C4○D4C8⋊C22C8.C22
kernel(C2×C8).200D4C22.4Q16C23.65C23C24.3C22C2×C8⋊C4C2×D4⋊C4D4⋊C4C2×C8C22×C4C2×C4C22C22
# reps121112822831

Matrix representation of (C2×C8).200D4 ►in GL8(𝔽17)

10000000
01000000
001600000
000160000
00001000
00000100
00000010
00000001
,
160000000
016000000
001390000
00440000
000088140
000098014
000001499
00003089
,
01000000
160000000
001300000
00440000
000014098
00000388
000098014
000088140
,
160000000
01000000
00480000
000130000
000088140
00008903
000014098
00000388

G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,13,4,0,0,0,0,0,0,9,4,0,0,0,0,0,0,0,0,8,9,0,3,0,0,0,0,8,8,14,0,0,0,0,0,14,0,9,8,0,0,0,0,0,14,9,9],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,13,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,14,0,9,8,0,0,0,0,0,3,8,8,0,0,0,0,9,8,0,14,0,0,0,0,8,8,14,0],[16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,8,13,0,0,0,0,0,0,0,0,8,8,14,0,0,0,0,0,8,9,0,3,0,0,0,0,14,0,9,8,0,0,0,0,0,3,8,8] >;
 

(C2×C8).200D4 in GAP, Magma, Sage, TeX 

(C_2\times C_8)._{200}D_4
 
% in TeX
 
G:=Group("(C2xC8).200D4");
 
// GroupNames label
 
G:=SmallGroup(128,668);
 
// by ID
 
G=gap.SmallGroup(128,668);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,723,58,2019,248,2804,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^3,d*b*d^-1=a*b^-1,d*c*d^-1=b^6*c^-1>;
 
// generators/relations
 

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