Copied to
clipboard

G = C2×C24.3C22  order 128 = 27

Direct product of C2 and C24.3C22

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

Aliases: C2×C24.3C22, C25.16C22, C23.174C24, C24.184C23, (C22×C4)⋊45D4, (C22×D4)⋊20C4, C24.66(C2×C4), (D4×C23).6C2, (C22×C42)⋊12C2, (C2×C42)⋊85C22, C23.828(C2×D4), C22.111(C4×D4), (C23×C4).36C22, C22.65(C23×C4), C23.74(C22×C4), C22.72(C22×D4), C23.362(C4○D4), C22.44(C4⋊1D4), (C22×C4).449C23, C22.159(C4⋊D4), C22.74(C4.4D4), (C22×D4).464C22, C2.11(C2×C4×D4), (C2×C4)⋊14(C2×D4), C4⋊2(C2×C22⋊C4), (C2×D4)⋊37(C2×C4), (C22×C4⋊C4)⋊6C2, C2.2(C2×C4⋊1D4), C2.5(C2×C4⋊D4), (C2×C4)⋊9(C22⋊C4), C2.3(C2×C4.4D4), (C2×C4⋊C4)⋊100C22, (C22×C22⋊C4)⋊5C2, C22.66(C2×C4○D4), C2.9(C22×C22⋊C4), (C2×C22⋊C4)⋊70C22, (C2×C4).449(C22×C4), (C22×C4).410(C2×C4), C22.132(C2×C22⋊C4), SmallGroup(128,1024)

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

Derived series
C1 — C22 — C2×C24.3C22
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C42 — C2×C24.3C22
Lower central
C1 — C22 — C2×C24.3C22
Upper central
C1 — C24 — C2×C24.3C22
Jennings
C1 — C23 — C2×C24.3C22

Subgroups: 1324 in 680 conjugacy classes, 236 normal (14 characteristic)
C1, C2 [×3], C2 [×12], C2 [×8], C4 [×8], C4 [×12], C22 [×3], C22 [×32], C22 [×72], C2×C4 [×36], C2×C4 [×44], D4 [×32], C23, C23 [×22], C23 [×104], C42 [×8], C22⋊C4 [×32], C4⋊C4 [×8], C22×C4 [×30], C22×C4 [×20], C2×D4 [×16], C2×D4 [×48], C24, C24 [×12], C24 [×24], C2×C42 [×4], C2×C42 [×4], C2×C22⋊C4 [×16], C2×C22⋊C4 [×16], C2×C4⋊C4 [×4], C2×C4⋊C4 [×4], C23×C4, C23×C4 [×4], C22×D4 [×12], C22×D4 [×8], C25 [×2], C24.3C22 [×8], C22×C42, C22×C22⋊C4 [×4], C22×C4⋊C4, D4×C23, C2×C24.3C22

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×16], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×24], C4○D4 [×4], C24, C2×C22⋊C4 [×12], C4×D4 [×8], C4⋊D4 [×8], C4.4D4 [×4], C4⋊1D4 [×4], C23×C4, C22×D4 [×4], C2×C4○D4 [×2], C24.3C22 [×8], C22×C22⋊C4, C2×C4×D4 [×2], C2×C4⋊D4 [×2], C2×C4.4D4, C2×C4⋊1D4, C2×C24.3C22

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=1, f2=e, g2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, fbf-1=bc=cb, gbg-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, gfg-1=df=fd, dg=gd, ef=fe, eg=ge >

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

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

10000000
01000000
00400000
00040000
00001000
00000100
00000010
00000001
,
40000000
31000000
00010000
00100000
00000400
00004000
00000010
00000034
,
10000000
01000000
00100000
00010000
00001000
00000100
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000010
00000001
,
40000000
04000000
00100000
00010000
00001000
00000100
00000040
00000004
,
30000000
12000000
00010000
00100000
00000400
00004000
00000033
00000002
,
14000000
24000000
00010000
00400000
00000400
00001000
00000010
00000001

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[3,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,2],[1,2,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

56 conjugacy classes

class 1 2A···2O2P···2W4A···4X4Y···4AF
order12···22···24···44···4
size11···14···42···24···4

56 irreducible representations

dim111111122
type+++++++
imageC1C2C2C2C2C2C4D4C4○D4
kernelC2×C24.3C22C24.3C22C22×C42C22×C22⋊C4C22×C4⋊C4D4×C23C22×D4C22×C4C23
# reps18141116168

In GAP, Magma, Sage, TeX 

C_2\times C_2^4._3C_2^2
 
% in TeX
 
G:=Group("C2xC2^4.3C2^2");
 
// GroupNames label
 
G:=SmallGroup(128,1024);
 
// by ID
 
G=gap.SmallGroup(128,1024);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,568,758,184]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=1,f^2=e,g^2=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,f*b*f^-1=b*c=c*b,g*b*g^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,g*f*g^-1=d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀