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G = C2×C22.35C24  order 128 = 27

Direct product of C2 and C22.35C24

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

Aliases: C2×C22.35C24, C22.42C25, C24.484C23, C23.273C24, C42.544C23, C22.762- 1+4, C4⋊Q8⋊75C22, (C2×C4).45C24, (C4×Q8)⋊87C22, C4⋊C4.284C23, C22⋊C4.9C23, (C2×Q8).424C23, C42.C2⋊41C22, C2.7(C2×2- 1+4), (C23×C4).586C22, (C2×C42).919C22, C22⋊Q8.221C22, (C22×C4).1182C23, C42⋊2C2.12C22, (C22×Q8).489C22, C42⋊C2.338C22, (C2×C4×Q8)⋊46C2, (C2×C4⋊Q8)⋊47C2, C4.73(C2×C4○D4), (C2×C42.C2)⋊40C2, C2.19(C22×C4○D4), (C2×C22⋊Q8).60C2, (C2×C4).716(C4○D4), (C2×C4⋊C4).948C22, C22.155(C2×C4○D4), (C2×C42⋊C2).63C2, (C2×C42⋊2C2).19C2, (C2×C22⋊C4).375C22, SmallGroup(128,2185)

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

Derived series
C1 — C22 — C2×C22.35C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C42⋊C2 — C2×C22.35C24
Lower central
C1 — C22 — C2×C22.35C24
Upper central
C1 — C23 — C2×C22.35C24
Jennings
C1 — C22 — C2×C22.35C24

Generators and relations for C2×C22.35C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=g2=b, e2=c, 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: 620 in 502 conjugacy classes, 396 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C42⋊2C2, C4⋊Q8, C23×C4, C22×Q8, C2×C42⋊C2, C2×C4×Q8, C2×C22⋊Q8, C2×C42.C2, C2×C42.C2, C2×C42⋊2C2, C2×C4⋊Q8, C22.35C24, C2×C22.35C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2- 1+4, C25, C22.35C24, C22×C4○D4, C2×2- 1+4, C2×C22.35C24

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

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

(2 20)(4 18)(5 50)(7 52)(10 30)(12 32)(13 15)(14 39)(16 37)(21 23)(22 28)(24 26)(25 27)(34 54)(36 56)(38 40)(41 43)(42 60)(44 58)(45 47)(46 64)(48 62)(57 59)(61 63)

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

44 conjugacy classes

class 1 2A···2G2H2I4A···4L4M···4AH
order12···2224···44···4
size11···1442···24···4

44 irreducible representations

dim1111111124
type++++++++-
imageC1C2C2C2C2C2C2C2C4○D42- 1+4
kernelC2×C22.35C24C2×C42⋊C2C2×C4×Q8C2×C22⋊Q8C2×C42.C2C2×C42⋊2C2C2×C4⋊Q8C22.35C24C2×C4C22
# reps11225411684

Matrix representation of C2×C22.35C24 ►in GL8(𝔽5)

40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
01000000
10000000
00420000
00010000
00004431
00000020
00000200
00003301
,
30000000
03000000
00300000
00030000
00000040
00003312
00001000
00000002
,
10000000
04000000
00100000
00140000
00001000
00000100
00000040
00002204
,
10000000
01000000
00100000
00010000
00000400
00001000
00003312
00002044

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],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,4,0,0,3,0,0,0,0,4,0,2,3,0,0,0,0,3,2,0,0,0,0,0,0,1,0,0,1],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,3,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,2,0,2],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,3,2,0,0,0,0,4,0,3,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,2,4] >;
 

C2×C22.35C24 in GAP, Magma, Sage, TeX 

C_2\times C_2^2._{35}C_2^4
 
% in TeX
 
G:=Group("C2xC2^2.35C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,2185);
 
// by ID
 
G=gap.SmallGroup(128,2185);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,456,1430,387,1123,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=g^2=b,e^2=c,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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