Copied to
clipboard

?

G = C2×(C22×C8)⋊C2  order 128 = 27

Direct product of C2 and (C22×C8)⋊C2

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

Aliases: C2×(C22×C8)⋊C2, (C23×C8)⋊4C2, C24.72(C2×C4), C22⋊C8⋊72C22, (C2×C8).388C23, (C2×C4).628C24, (C22×C8)⋊61C22, (C22×D4).33C4, C4.177(C22×D4), (C22×C4).773D4, (C22×Q8).26C4, C22.37(C8○D4), (C2×M4(2))⋊63C22, (C22×M4(2))⋊14C2, C22.158(C23×C4), (C23×C4).505C22, C23.289(C22×C4), C23.127(C22⋊C4), (C22×C4).1492C23, C2.6(C2×C8○D4), (C2×C22⋊C8)⋊40C2, (C2×C4○D4).21C4, C4.66(C2×C22⋊C4), C4○((C22×C8)⋊C2), (C2×D4).218(C2×C4), (C2×C4).1397(C2×D4), (C2×Q8).197(C2×C4), (C22×C4).318(C2×C4), (C2×C4).239(C22×C4), (C22×C4○D4).13C2, C2.21(C22×C22⋊C4), C22.17(C2×C22⋊C4), (C2×C4).277(C22⋊C4), (C2×C4○D4).267C22, (C2×C4)○((C22×C8)⋊C2), SmallGroup(128,1610)

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

Derived series
C1 — C22 — C2×(C22×C8)⋊C2
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×C4○D4 — C2×(C22×C8)⋊C2
Lower central
C1 — C22 — C2×(C22×C8)⋊C2
Upper central
C1 — C22×C4 — C2×(C22×C8)⋊C2
Jennings
C1 — C2 — C2 — C2×C4 — C2×(C22×C8)⋊C2

Subgroups: 636 in 396 conjugacy classes, 180 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×2], C4 [×6], C4 [×4], C22, C22 [×10], C22 [×32], C8 [×8], C2×C4 [×2], C2×C4 [×30], C2×C4 [×20], D4 [×24], Q8 [×8], C23, C23 [×10], C23 [×16], C2×C8 [×8], C2×C8 [×16], M4(2) [×8], C22×C4 [×2], C22×C4 [×22], C22×C4 [×8], C2×D4 [×12], C2×D4 [×12], C2×Q8 [×4], C2×Q8 [×4], C4○D4 [×32], C24, C24 [×2], C22⋊C8 [×16], C22×C8 [×8], C22×C8 [×4], C2×M4(2) [×4], C2×M4(2) [×4], C23×C4, C23×C4 [×2], C22×D4, C22×D4 [×2], C22×Q8, C2×C4○D4 [×8], C2×C4○D4 [×8], C2×C22⋊C8 [×4], (C22×C8)⋊C2 [×8], C23×C8, C22×M4(2), C22×C4○D4, C2×(C22×C8)⋊C2

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○D4 [×4], C23×C4, C22×D4 [×2], (C22×C8)⋊C2 [×4], C22×C22⋊C4, C2×C8○D4 [×2], C2×(C22×C8)⋊C2

Generators and relations
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=bd4, ede=cd=dc, ce=ec >

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

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

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

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

Matrix representation ►G ⊆ GL5(𝔽17)

160000
01000
00100
000160
000016
,
10000
016000
02100
000016
000160
,
10000
01000
00100
000160
000016
,
10000
02000
00200
00002
00020
,
160000
04400
091300
000013
00040

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

56 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4H4I4J4K4L4M4N4O4P8A···8P8Q···8X
order12···2222222224···4444444448···88···8
size11···1222244441···1222244442···24···4

56 irreducible representations

dim11111111122
type+++++++
imageC1C2C2C2C2C2C4C4C4D4C8○D4
kernelC2×(C22×C8)⋊C2C2×C22⋊C8(C22×C8)⋊C2C23×C8C22×M4(2)C22×C4○D4C22×D4C22×Q8C2×C4○D4C22×C4C22
# reps148111628816

In GAP, Magma, Sage, TeX 

C_2\times (C_2^2\times C_8)\rtimes C_2
 
% in TeX
 
G:=Group("C2x(C2^2xC8):C2");
 
// GroupNames label
 
G:=SmallGroup(128,1610);
 
// by ID
 
G=gap.SmallGroup(128,1610);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,723,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=b*d^4,e*d*e=c*d=d*c,c*e=e*c>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀