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G = C2×C88D4order 128 = 27

Direct product of C2 and C88D4

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

Aliases: C2×C88D4, C235SD16, C24.141D4, C817(C2×D4), (C2×C8)⋊39D4, (C23×C8)⋊12C2, C4⋊C4.16C23, C221(C2×SD16), C4.Q854C22, (C2×C8).485C23, (C2×C4).251C24, (C22×C8)⋊68C22, (C2×D4).56C23, C4.145(C22×D4), C23.858(C2×D4), (C22×C4).605D4, C22⋊Q865C22, (C2×Q8).44C23, C4.107(C4⋊D4), D4⋊C456C22, C2.9(C22×SD16), Q8⋊C457C22, (C2×SD16)⋊75C22, (C22×SD16)⋊25C2, C22.90(C4○D8), C4⋊D4.146C22, (C23×C4).700C22, C22.511(C22×D4), C22.171(C4⋊D4), (C22×C4).1530C23, (C22×D4).344C22, (C22×Q8).277C22, (C2×C4.Q8)⋊27C2, C4.18(C2×C4○D4), C2.13(C2×C4○D8), C2.69(C2×C4⋊D4), (C2×C22⋊Q8)⋊54C2, (C2×D4⋊C4)⋊16C2, (C2×Q8⋊C4)⋊17C2, (C2×C4).1423(C2×D4), (C2×C4⋊D4).53C2, (C2×C4).697(C4○D4), (C2×C4⋊C4).585C22, SmallGroup(128,1779)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C88D4
C1C2C22C2×C4C22×C4C23×C4C23×C8 — C2×C88D4
C1C2C2×C4 — C2×C88D4
C1C23C23×C4 — C2×C88D4
C1C2C2C2×C4 — C2×C88D4

Generators and relations for C2×C88D4
 G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b3, dcd=c-1 >

Subgroups: 564 in 282 conjugacy classes, 116 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×10], C22 [×22], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×22], D4 [×14], Q8 [×6], C23, C23 [×6], C23 [×12], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×8], C2×C8 [×10], SD16 [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×7], C2×D4 [×2], C2×D4 [×13], C2×Q8 [×2], C2×Q8 [×5], C24, C24, D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×4], C4⋊D4 [×2], C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8 [×2], C22×C8 [×4], C22×C8 [×4], C2×SD16 [×4], C2×SD16 [×4], C23×C4, C22×D4, C22×D4, C22×Q8, C2×D4⋊C4, C2×Q8⋊C4, C2×C4.Q8, C88D4 [×8], C2×C4⋊D4, C2×C22⋊Q8, C23×C8, C22×SD16, C2×C88D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], SD16 [×4], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C2×SD16 [×6], C4○D8 [×2], C22×D4 [×2], C2×C4○D4, C88D4 [×4], C2×C4⋊D4, C22×SD16, C2×C4○D8, C2×C88D4

Smallest permutation representation of C2×C88D4
On 64 points
Generators in S64
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 57)(16 58)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(25 44)(26 45)(27 46)(28 47)(29 48)(30 41)(31 42)(32 43)
(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 9 23 45)(2 12 24 48)(3 15 17 43)(4 10 18 46)(5 13 19 41)(6 16 20 44)(7 11 21 47)(8 14 22 42)(25 56 58 40)(26 51 59 35)(27 54 60 38)(28 49 61 33)(29 52 62 36)(30 55 63 39)(31 50 64 34)(32 53 57 37)
(1 39)(2 34)(3 37)(4 40)(5 35)(6 38)(7 33)(8 36)(9 63)(10 58)(11 61)(12 64)(13 59)(14 62)(15 57)(16 60)(17 53)(18 56)(19 51)(20 54)(21 49)(22 52)(23 55)(24 50)(25 46)(26 41)(27 44)(28 47)(29 42)(30 45)(31 48)(32 43)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A···4H4I···4N8A···8P
order12···22222224···44···48···8
size11···12222882···28···82···2

44 irreducible representations

dim111111111222222
type++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4D4C4○D4SD16C4○D8
kernelC2×C88D4C2×D4⋊C4C2×Q8⋊C4C2×C4.Q8C88D4C2×C4⋊D4C2×C22⋊Q8C23×C8C22×SD16C2×C8C22×C4C24C2×C4C23C22
# reps111181111431488

Matrix representation of C2×C88D4 in GL5(𝔽17)

160000
01000
00100
00010
00001
,
10000
012500
0121200
000512
00055
,
160000
00400
04000
000160
00001
,
10000
01000
001600
000160
00001

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],[1,0,0,0,0,0,12,12,0,0,0,5,12,0,0,0,0,0,5,5,0,0,0,12,5],[16,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,0,0,16,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1] >;

C2×C88D4 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_8D_4
% in TeX

G:=Group("C2xC8:8D4");
// GroupNames label

G:=SmallGroup(128,1779);
// by ID

G=gap.SmallGroup(128,1779);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,2804,172]);
// 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^3,d*c*d=c^-1>;
// generators/relations

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