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G = C2×C22.34C24  order 128 = 27

Direct product of C2 and C22.34C24

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

Aliases: C2×C22.34C24, C22.41C25, C23.20C24, C24.483C23, C42.543C23, C22.1042+ 1+4, (C4×D4)⋊95C22, (C2×C4).44C24, C4⋊C4.283C23, C4⋊D4⋊65C22, C4⋊1D4⋊44C22, C22⋊C4.8C23, (C2×D4).449C23, C42.C2⋊40C22, C42⋊C2⋊89C22, C2.9(C2×2+ 1+4), (C23×C4).585C22, (C2×C42).918C22, (C22×C4).1181C23, (C22×D4).420C22, C22.D4⋊35C22, (C2×C4×D4)⋊74C2, C4.72(C2×C4○D4), (C2×C4⋊D4)⋊57C2, (C2×C4⋊1D4)⋊23C2, (C2×C42.C2)⋊39C2, C2.18(C22×C4○D4), (C2×C42⋊C2)⋊56C2, (C2×C4).715(C4○D4), (C2×C4⋊C4).700C22, C22.154(C2×C4○D4), (C2×C22.D4)⋊52C2, (C2×C22⋊C4).531C22, SmallGroup(128,2184)

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

Derived series
C1 — C22 — C2×C22.34C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C42⋊C2 — C2×C22.34C24
Lower central
C1 — C22 — C2×C22.34C24
Upper central
C1 — C23 — C2×C22.34C24
Jennings
C1 — C22 — C2×C22.34C24

Generators and relations for C2×C22.34C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=c, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=gdg-1=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 1068 in 630 conjugacy classes, 396 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, C22×C4, C2×D4, C2×D4, C24, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C42.C2, C4⋊1D4, C23×C4, C23×C4, C22×D4, C2×C42⋊C2, C2×C4×D4, C2×C4⋊D4, C2×C22.D4, C2×C42.C2, C2×C4⋊1D4, C22.34C24, C2×C22.34C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, C25, C22.34C24, C22×C4○D4, C2×2+ 1+4, C2×C22.34C24

Smallest permutation representation of C2×C22.34C24
►On 64 points
Generators in S64
(1 49)(2 50)(3 51)(4 52)(5 64)(6 61)(7 62)(8 63)(9 31)(10 32)(11 29)(12 30)(13 19)(14 20)(15 17)(16 18)(21 27)(22 28)(23 25)(24 26)(33 37)(34 38)(35 39)(36 40)(41 55)(42 56)(43 53)(44 54)(45 59)(46 60)(47 57)(48 58)

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

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

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

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

44 conjugacy classes

class 1 2A···2G2H···2Q4A···4L4M···4Z
order12···22···24···44···4
size11···14···42···24···4

44 irreducible representations

dim1111111124
type+++++++++
imageC1C2C2C2C2C2C2C2C4○D42+ 1+4
kernelC2×C22.34C24C2×C42⋊C2C2×C4×D4C2×C4⋊D4C2×C22.D4C2×C42.C2C2×C4⋊1D4C22.34C24C2×C4C22
# reps11264111684

Matrix representation of C2×C22.34C24 ►in GL8(𝔽5)

40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
40000000
01000000
00400000
00010000
00000100
00001000
00000001
00000010
,
20000000
02000000
00200000
00020000
00000001
00000040
00000400
00001000
,
01000000
10000000
00040000
00400000
00000010
00000001
00001000
00000100
,
40000000
04000000
00100000
00010000
00000100
00004000
00000001
00000040

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

C2×C22.34C24 in GAP, Magma, Sage, TeX 

C_2\times C_2^2._{34}C_2^4
 
% in TeX
 
G:=Group("C2xC2^2.34C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,2184);
 
// by ID
 
G=gap.SmallGroup(128,2184);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,387,1123,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=c,g^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e^-1=g*d*g^-1=b*d=d*b,f*e*f=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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