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G = C2xC42.7C22order 128 = 27

Direct product of C2 and C42.7C22

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

Aliases: C2xC42.7C22, C42.258C23, C4:C8:82C22, (C4xC8):68C22, C24.76(C2xC4), C8:C4:53C22, C42.247(C2xC4), (C2xC4).638C24, (C2xC8).473C23, C42:C2.28C4, C22.39(C8oD4), C4.82(C42:C2), C22:C8.226C22, C4o(C42.7C22), (C23xC4).521C22, (C22xC4).910C23, C22.166(C23xC4), C23.292(C22xC4), (C22xC8).507C22, (C2xC42).1023C22, C42:C2.288C22, C22.71(C42:C2), (C2xC4xC8):16C2, (C2xC4:C8):45C2, (C2xC4:C4).67C4, (C2xC8:C4):31C2, C2.10(C2xC8oD4), C4:C4.215(C2xC4), C4.289(C2xC4oD4), (C2xC22:C8).46C2, (C2xC22:C4).44C4, C22:C4.66(C2xC4), (C2xC4).954(C4oD4), (C2xC4).255(C22xC4), (C22xC4).333(C2xC4), C2.38(C2xC42:C2), (C2xC42:C2).56C2, (C2xC4)o(C42.7C22), SmallGroup(128,1651)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2xC42.7C22
C1C2C4C2xC4C22xC4C2xC42C2xC42:C2 — C2xC42.7C22
C1C22 — C2xC42.7C22
C1C22xC4 — C2xC42.7C22
C1C2C2C2xC4 — C2xC42.7C22

Generators and relations for C2xC42.7C22
 G = < a,b,c,d,e | a2=b4=c4=e2=1, d2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1c2, ebe=bc2, cd=dc, ce=ec, ede=b2c2d >

Subgroups: 284 in 206 conjugacy classes, 140 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, C23, C23, C23, C42, C22:C4, C4:C4, C2xC8, C2xC8, C22xC4, C22xC4, C22xC4, C24, C4xC8, C8:C4, C22:C8, C4:C8, C2xC42, C2xC22:C4, C2xC4:C4, C42:C2, C22xC8, C23xC4, C2xC4xC8, C2xC8:C4, C2xC22:C8, C2xC4:C8, C42.7C22, C2xC42:C2, C2xC42.7C22
Quotients: C1, C2, C4, C22, C2xC4, C23, C22xC4, C4oD4, C24, C42:C2, C8oD4, C23xC4, C2xC4oD4, C42.7C22, C2xC42:C2, C2xC8oD4, C2xC42.7C22

Smallest permutation representation of C2xC42.7C22
On 64 points
Generators in S64
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 63)(10 64)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(1 43 19 9)(2 14 20 48)(3 45 21 11)(4 16 22 42)(5 47 23 13)(6 10 24 44)(7 41 17 15)(8 12 18 46)(25 57 53 37)(26 34 54 62)(27 59 55 39)(28 36 56 64)(29 61 49 33)(30 38 50 58)(31 63 51 35)(32 40 52 60)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 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 31)(2 56)(3 25)(4 50)(5 27)(6 52)(7 29)(8 54)(9 39)(10 64)(11 33)(12 58)(13 35)(14 60)(15 37)(16 62)(17 49)(18 26)(19 51)(20 28)(21 53)(22 30)(23 55)(24 32)(34 42)(36 44)(38 46)(40 48)(41 57)(43 59)(45 61)(47 63)

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

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

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

56 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4P4Q···4V8A···8P8Q···8X
order12···2224···44···44···48···88···8
size11···1441···12···24···42···24···4

56 irreducible representations

dim111111111122
type+++++++
imageC1C2C2C2C2C2C2C4C4C4C4oD4C8oD4
kernelC2xC42.7C22C2xC4xC8C2xC8:C4C2xC22:C8C2xC4:C8C42.7C22C2xC42:C2C2xC22:C4C2xC4:C4C42:C2C2xC4C22
# reps1112281448816

Matrix representation of C2xC42.7C22 in GL5(F17)

160000
016000
001600
00010
00001
,
160000
013000
00400
000016
00010
,
160000
01000
00100
00040
00004
,
130000
00100
01000
000150
000015
,
10000
01000
001600
000160
00001

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,13,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,16,0],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[13,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,15,0,0,0,0,0,15],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1] >;

C2xC42.7C22 in GAP, Magma, Sage, TeX

C_2\times C_4^2._7C_2^2
% in TeX

G:=Group("C2xC4^2.7C2^2");
// GroupNames label

G:=SmallGroup(128,1651);
// by ID

G=gap.SmallGroup(128,1651);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,723,100,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^4=e^2=1,d^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1*c^2,e*b*e=b*c^2,c*d=d*c,c*e=e*c,e*d*e=b^2*c^2*d>;
// generators/relations

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