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G = C2×C23.36D4  order 128 = 27

Direct product of C2 and C23.36D4

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

Aliases: C2×C23.36D4, C24.170D4, D4⋊5(C22×C4), C4.6(C23×C4), Q8⋊5(C22×C4), C4⋊C4.342C23, (C2×C4).176C24, (C2×C8).391C23, (C22×C4).780D4, C4.141(C22×D4), C23.639(C2×D4), D4⋊C4⋊84C22, Q8⋊C4⋊87C22, (C2×D4).360C23, (C2×Q8).333C23, (C2×M4(2))⋊69C22, (C22×M4(2))⋊19C2, (C23×C4).514C22, (C22×C8).424C22, (C22×C4).900C23, C22.126(C22×D4), C22.105(C8⋊C22), C23.131(C22⋊C4), (C22×D4).553C22, C22.94(C8.C22), (C22×Q8).457C22, C4○D4⋊13(C2×C4), (C2×C4○D4)⋊19C4, (C2×D4)⋊48(C2×C4), (C2×Q8)⋊39(C2×C4), C2.2(C2×C8⋊C22), (C22×C4⋊C4)⋊32C2, C4.98(C2×C22⋊C4), (C2×D4⋊C4)⋊49C2, (C2×C4⋊C4)⋊114C22, C2.2(C2×C8.C22), (C2×Q8⋊C4)⋊50C2, (C2×C4).1406(C2×D4), (C22×C4).324(C2×C4), (C2×C4).461(C22×C4), (C22×C4○D4).19C2, C2.38(C22×C22⋊C4), C22.21(C2×C22⋊C4), (C2×C4).158(C22⋊C4), (C2×C4○D4).273C22, SmallGroup(128,1627)

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

Derived series
C1 — C4 — C2×C23.36D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22×C4○D4 — C2×C23.36D4
Lower central
C1 — C2 — C4 — C2×C23.36D4
Upper central
C1 — C23 — C23×C4 — C2×C23.36D4
Jennings
C1 — C2 — C2 — C2×C4 — C2×C23.36D4

Subgroups: 716 in 408 conjugacy classes, 180 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×10], C22 [×28], C8 [×4], C2×C4 [×2], C2×C4 [×26], C2×C4 [×38], D4 [×4], D4 [×22], Q8 [×4], Q8 [×6], C23, C23 [×6], C23 [×14], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], C2×C8 [×4], M4(2) [×8], C22×C4 [×2], C22×C4 [×12], C22×C4 [×23], C2×D4 [×6], C2×D4 [×15], C2×Q8 [×6], C2×Q8 [×3], C4○D4 [×16], C4○D4 [×24], C24, C24, D4⋊C4 [×8], Q8⋊C4 [×8], C2×C4⋊C4 [×6], C2×C4⋊C4 [×3], C22×C8 [×2], C2×M4(2) [×4], C2×M4(2) [×4], C23×C4, C23×C4 [×2], C22×D4, C22×D4, C22×Q8, C2×C4○D4 [×12], C2×C4○D4 [×6], C2×D4⋊C4 [×2], C2×Q8⋊C4 [×2], C23.36D4 [×8], C22×C4⋊C4, C22×M4(2), C22×C4○D4, C2×C23.36D4

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C8⋊C22 [×2], C8.C22 [×2], C23×C4, C22×D4 [×2], C23.36D4 [×4], C22×C22⋊C4, C2×C8⋊C22, C2×C8.C22, C2×C23.36D4

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

10000000
01000000
001600000
000160000
000016000
000001600
000000160
000000016
,
160000000
016000000
00100000
00010000
0000001615
00000011
00001200
0000161600
,
160000000
016000000
001600000
000160000
000016000
000001600
000000160
000000016
,
10000000
01000000
00100000
00010000
000016000
000001600
000000160
000000016
,
815000000
69000000
00010000
00100000
00004444
0000150150
0000441313
000015020
,
815000000
79000000
00010000
001600000
00004444
000015131513
0000441313
0000151324

G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,2,16,0,0,0,0,16,1,0,0,0,0,0,0,15,1,0,0],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[8,6,0,0,0,0,0,0,15,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,15,4,15,0,0,0,0,4,0,4,0,0,0,0,0,4,15,13,2,0,0,0,0,4,0,13,0],[8,7,0,0,0,0,0,0,15,9,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,15,4,15,0,0,0,0,4,13,4,13,0,0,0,0,4,15,13,2,0,0,0,0,4,13,13,4] >;
 

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4H4I···4T8A···8H
order12···2222222224···44···48···8
size11···1222244442···24···44···4

44 irreducible representations

dim111111112244
type++++++++++-
imageC1C2C2C2C2C2C2C4D4D4C8⋊C22C8.C22
kernelC2×C23.36D4C2×D4⋊C4C2×Q8⋊C4C23.36D4C22×C4⋊C4C22×M4(2)C22×C4○D4C2×C4○D4C22×C4C24C22C22
# reps1228111167122

In GAP, Magma, Sage, TeX 

C_2\times C_2^3._{36}D_4
 
% in TeX
 
G:=Group("C2xC2^3.36D4");
 
// GroupNames label
 
G:=SmallGroup(128,1627);
 
// by ID
 
G=gap.SmallGroup(128,1627);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,352,2804,1411,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=b*d=d*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^3>;
 
// generators/relations
 

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