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G = C2×Q8○D8  order 128 = 27

Direct product of C2 and Q8○D8

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

Aliases: C2×Q8○D8, C8.5C24, C4.10C25, D4.7C24, Q8.7C24, D8.10C23, Q16.9C23, SD16.1C23, M4(2).19C23, 2- (1+4)⋊8C22, D8○(C2×Q8), Q8○(C2×D8), D4○(C2×Q16), Q16○(C2×D4), C4○D4.40D4, D4.63(C2×D4), Q8.65(C2×D4), (C2×D4).359D4, C4○D8⋊11C22, C8○D4⋊15C22, (C2×Q8).278D4, C2.45(D4×C23), (C2×C4).616C24, (C2×C8).575C23, (C2×Q16)⋊61C22, (C22×Q16)⋊23C2, C4○D4.16C23, C4.127(C22×D4), C23.487(C2×D4), (C2×D8).178C22, (C2×D4).492C23, C8.C22⋊14C22, (C2×Q8).323C23, C22.19(C22×D4), (C22×C8).298C22, (C2×2- (1+4))⋊12C2, (C22×C4).1227C23, (C2×SD16).128C22, (C22×Q8).374C22, (C2×M4(2)).292C22, (C2×Q8)○(C2×D8), (C2×C8○D4)⋊12C2, (C2×C4○D8)⋊32C2, (C2×C4).1115(C2×D4), (C2×C8.C22)⋊35C2, (C2×C4○D4).248C22, SmallGroup(128,2315)

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

Derived series
C1 — C4 — C2×Q8○D8
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×2- (1+4) — C2×Q8○D8
Lower central
C1 — C2 — C4 — C2×Q8○D8
Upper central
C1 — C22 — C2×C4○D4 — C2×Q8○D8
Jennings
C1 — C2 — C2 — C4 — C2×Q8○D8

Subgroups: 996 in 704 conjugacy classes, 428 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×2], C4 [×6], C4 [×12], C22, C22 [×6], C22 [×14], C8 [×8], C2×C4, C2×C4 [×15], C2×C4 [×54], D4 [×16], D4 [×26], Q8 [×16], Q8 [×30], C23 [×3], C23 [×2], C2×C8, C2×C8 [×15], M4(2) [×12], D8 [×4], SD16 [×24], Q16 [×36], C22×C4 [×3], C22×C4 [×12], C2×D4 [×5], C2×D4 [×6], C2×Q8, C2×Q8 [×30], C2×Q8 [×34], C4○D4 [×32], C4○D4 [×60], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×D8, C2×SD16 [×6], C2×Q16 [×33], C4○D8 [×24], C8.C22 [×48], C22×Q8 [×6], C22×Q8 [×2], C2×C4○D4, C2×C4○D4 [×6], C2×C4○D4 [×6], 2- (1+4) [×16], 2- (1+4) [×8], C2×C8○D4, C22×Q16 [×3], C2×C4○D8 [×3], C2×C8.C22 [×6], Q8○D8 [×16], C2×2- (1+4) [×2], C2×Q8○D8

Quotients:
C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], C25, Q8○D8 [×2], D4×C23, C2×Q8○D8

Generators and relations
 G = < a,b,c,d,e | a2=b4=e2=1, c2=d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d3 >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

1600000
0160000
0016000
0001600
0000160
0000016
,
1600000
0160000
0000016
000010
0001600
001000
,
1600000
0160000
0013000
0001300
000040
000004
,
16130000
910000
0031400
003300
0000314
000033
,
16130000
010000
003300
0031400
00001414
0000143

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16,0,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[16,9,0,0,0,0,13,1,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,0,0,0,0,3,3,0,0,0,0,14,3],[16,0,0,0,0,0,13,1,0,0,0,0,0,0,3,3,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,14,3] >;
 

44 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M4A···4H4I···4T8A8B8C8D8E···8J
order12222···222224···44···488888···8
size11112···244442···24···422224···4

44 irreducible representations

dim11111112224
type++++++++++-
imageC1C2C2C2C2C2C2D4D4D4Q8○D8
kernelC2×Q8○D8C2×C8○D4C22×Q16C2×C4○D8C2×C8.C22Q8○D8C2×2- (1+4)C2×D4C2×Q8C4○D4C2
# reps113361623144

In GAP, Magma, Sage, TeX 

C_2\times Q_8\circ D_8
 
% in TeX
 
G:=Group("C2xQ8oD8");
 
// GroupNames label
 
G:=SmallGroup(128,2315);
 
// by ID
 
G=gap.SmallGroup(128,2315);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,-2,448,477,456,521,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=e^2=1,c^2=d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d^3>;
 
// generators/relations
 

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