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G = C2×C89D4order 128 = 27

Direct product of C2 and C89D4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C2×C89D4, C235M4(2), C42.263C23, C818(C2×D4), (C2×C8)⋊40D4, C4(C89D4), C4⋊C885C22, (C23×C8)⋊14C2, (C4×D4).26C4, C4.183(C4×D4), C8⋊C457C22, C22⋊C874C22, C42.208(C2×C4), (C2×C4).646C24, (C2×C8).401C23, C24.100(C2×C4), (C22×C8)⋊70C22, (C22×D4).38C4, C22.113(C4×D4), C4.192(C22×D4), C221(C2×M4(2)), (C4×D4).284C22, C22.41(C8○D4), (C22×M4(2))⋊23C2, (C2×M4(2))⋊74C22, (C23×C4).524C22, (C2×C42).758C22, C23.103(C22×C4), C22.173(C23×C4), C2.10(C22×M4(2)), (C22×C4).1275C23, (C2×C4⋊C8)⋊47C2, C2.44(C2×C4×D4), (C2×C4×D4).70C2, (C2×C4⋊C4).70C4, (C2×C4)(C89D4), (C2×C8⋊C4)⋊33C2, C2.14(C2×C8○D4), C4⋊C4.221(C2×C4), (C2×C22⋊C8)⋊42C2, C4.297(C2×C4○D4), (C2×D4).230(C2×C4), (C2×C4).1571(C2×D4), C22⋊C4.71(C2×C4), (C2×C22⋊C4).47C4, (C2×C4).956(C4○D4), (C22×C4).384(C2×C4), (C2×C4).261(C22×C4), SmallGroup(128,1659)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C89D4
C1C2C4C2×C4C22×C4C22×C8C23×C8 — C2×C89D4
C1C22 — C2×C89D4
C1C22×C4 — C2×C89D4
C1C2C2C2×C4 — C2×C89D4

Generators and relations for C2×C89D4
 G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b5, dcd=c-1 >

Subgroups: 420 in 282 conjugacy classes, 156 normal (36 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×10], C22 [×22], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×12], C2×C4 [×24], D4 [×8], C23, C23 [×8], C23 [×10], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×12], C2×C8 [×14], M4(2) [×8], C22×C4 [×5], C22×C4 [×8], C22×C4 [×8], C2×D4 [×4], C2×D4 [×4], C24 [×2], C8⋊C4 [×4], C22⋊C8 [×8], C4⋊C8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C22×C8 [×4], C22×C8 [×4], C22×C8 [×4], C2×M4(2) [×4], C2×M4(2) [×4], C23×C4 [×2], C22×D4, C2×C8⋊C4, C2×C22⋊C8 [×2], C2×C4⋊C8, C89D4 [×8], C2×C4×D4, C23×C8, C22×M4(2), C2×C89D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], M4(2) [×4], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C2×M4(2) [×6], C8○D4 [×2], C23×C4, C22×D4, C2×C4○D4, C89D4 [×4], C2×C4×D4, C22×M4(2), C2×C8○D4, C2×C89D4

Smallest permutation representation of C2×C89D4
On 64 points
Generators in S64
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)(49 62)(50 63)(51 64)(52 57)(53 58)(54 59)(55 60)(56 61)
(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 34 47 63)(2 39 48 60)(3 36 41 57)(4 33 42 62)(5 38 43 59)(6 35 44 64)(7 40 45 61)(8 37 46 58)(9 49 22 30)(10 54 23 27)(11 51 24 32)(12 56 17 29)(13 53 18 26)(14 50 19 31)(15 55 20 28)(16 52 21 25)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 52)(26 49)(27 54)(28 51)(29 56)(30 53)(31 50)(32 55)(33 58)(34 63)(35 60)(36 57)(37 62)(38 59)(39 64)(40 61)(42 46)(44 48)

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

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

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

56 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A···4H4I4J4K4L4M···4R8A···8P8Q···8X
order12···22222224···444444···48···88···8
size11···12222441···122224···42···24···4

56 irreducible representations

dim1111111111112222
type+++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4D4C4○D4M4(2)C8○D4
kernelC2×C89D4C2×C8⋊C4C2×C22⋊C8C2×C4⋊C8C89D4C2×C4×D4C23×C8C22×M4(2)C2×C22⋊C4C2×C4⋊C4C4×D4C22×D4C2×C8C2×C4C23C22
# reps1121811142824488

Matrix representation of C2×C89D4 in GL6(𝔽17)

100000
010000
0016000
0001600
0000160
0000016
,
400000
040000
001000
000100
0000315
00001114
,
010000
1600000
0001600
001000
000010
0000316
,
100000
0160000
001000
0001600
000010
0000316

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,11,0,0,0,0,15,14],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,3,0,0,0,0,0,16],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,3,0,0,0,0,0,16] >;

C2×C89D4 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_9D_4
% in TeX

G:=Group("C2xC8:9D4");
// GroupNames label

G:=SmallGroup(128,1659);
// by ID

G=gap.SmallGroup(128,1659);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,184,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^5,d*c*d=c^-1>;
// generators/relations

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