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G = C2×C8.18D4order 128 = 27

Direct product of C2 and C8.18D4

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

Aliases: C2×C8.18D4, C234Q16, C24.143D4, C8.108(C2×D4), (C2×C8).350D4, C221(C2×Q16), C4⋊C4.18C23, (C23×C8).15C2, C2.D845C22, C2.6(C22×Q16), (C2×C4).253C24, (C2×C8).487C23, (C22×Q16)⋊10C2, (C2×Q16)⋊42C22, C4.147(C22×D4), C23.860(C2×D4), (C22×C4).607D4, (C2×Q8).45C23, C4.109(C4⋊D4), Q8⋊C458C22, C22.92(C4○D8), (C22×C8).534C22, (C23×C4).702C22, C22.513(C22×D4), C22⋊Q8.151C22, C22.173(C4⋊D4), (C22×C4).1532C23, (C22×Q8).278C22, (C2×C2.D8)⋊19C2, C4.20(C2×C4○D4), C2.15(C2×C4○D8), C2.71(C2×C4⋊D4), (C2×Q8⋊C4)⋊18C2, (C2×C4).1425(C2×D4), (C2×C22⋊Q8).52C2, (C2×C4).699(C4○D4), (C2×C4⋊C4).587C22, SmallGroup(128,1781)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C8.18D4
C1C2C22C2×C4C22×C4C23×C4C23×C8 — C2×C8.18D4
C1C2C2×C4 — C2×C8.18D4
C1C23C23×C4 — C2×C8.18D4
C1C2C2C2×C4 — C2×C8.18D4

Generators and relations for C2×C8.18D4
 G = < a,b,c,d | a2=b8=c4=1, d2=b4, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b4c-1 >

Subgroups: 436 in 256 conjugacy classes, 116 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×10], C22 [×12], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×26], Q8 [×12], C23, C23 [×6], C23 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×8], C2×C8 [×10], Q16 [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×8], C2×Q8 [×4], C2×Q8 [×10], C24, Q8⋊C4 [×8], C2.D8 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C22⋊Q8 [×8], C22⋊Q8 [×4], C22×C8 [×2], C22×C8 [×4], C22×C8 [×4], C2×Q16 [×4], C2×Q16 [×4], C23×C4, C22×Q8 [×2], C2×Q8⋊C4 [×2], C2×C2.D8, C8.18D4 [×8], C2×C22⋊Q8 [×2], C23×C8, C22×Q16, C2×C8.18D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], Q16 [×4], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C2×Q16 [×6], C4○D8 [×2], C22×D4 [×2], C2×C4○D4, C8.18D4 [×4], C2×C4⋊D4, C22×Q16, C2×C4○D8, C2×C8.18D4

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

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P8A···8P
order12···222224···44···48···8
size11···122222···28···82···2

44 irreducible representations

dim1111111222222
type++++++++++-
imageC1C2C2C2C2C2C2D4D4D4C4○D4Q16C4○D8
kernelC2×C8.18D4C2×Q8⋊C4C2×C2.D8C8.18D4C2×C22⋊Q8C23×C8C22×Q16C2×C8C22×C4C24C2×C4C23C22
# reps1218211431488

Matrix representation of C2×C8.18D4 in GL5(𝔽17)

160000
01000
00100
00010
00001
,
10000
031400
03300
00020
00099
,
160000
00400
04000
0001215
000125
,
10000
00400
04000
0001215
000135

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,3,3,0,0,0,14,3,0,0,0,0,0,2,9,0,0,0,0,9],[16,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,0,0,12,12,0,0,0,15,5],[1,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,0,0,12,13,0,0,0,15,5] >;

C2×C8.18D4 in GAP, Magma, Sage, TeX

C_2\times C_8._{18}D_4
% in TeX

G:=Group("C2xC8.18D4");
// GroupNames label

G:=SmallGroup(128,1781);
// by ID

G=gap.SmallGroup(128,1781);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,568,758,2804,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^4*c^-1>;
// generators/relations

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