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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), C23.74(C22×C4), C22.65(C23×C4), (C23×C4).36C22, 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, C2.5(C2×C4⋊D4), C2.2(C2×C4⋊1D4), (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, (C22×C4).410(C2×C4), (C2×C4).449(C22×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

Generators and relations for C2×C24.3C22
 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 >

Subgroups: 1324 in 680 conjugacy classes, 236 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C24, C2×C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C23×C4, C23×C4, C22×D4, C22×D4, C25, C24.3C22, C22×C42, C22×C22⋊C4, C22×C4⋊C4, D4×C23, C2×C24.3C22
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C24, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C23×C4, C22×D4, C2×C4○D4, C24.3C22, C22×C22⋊C4, C2×C4×D4, C2×C4⋊D4, C2×C4.4D4, C2×C4⋊1D4, C2×C24.3C22

Smallest permutation representation of C2×C24.3C22
►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)]])
 

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

Matrix representation of C2×C24.3C22 ►in 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] >;
 

C2×C24.3C22 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
 

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