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G = C2×C23.33C23order 128 = 27

Direct product of C2 and C23.33C23

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

Aliases: C2×C23.33C23, C22.13C25, C42.535C23, C23.107C24, C24.605C23, C22.982+ 1+4, C22.722- 1+4, D49(C22×C4), C2.9(C24×C4), Q88(C22×C4), (C4×D4)⋊86C22, C4.39(C23×C4), (C4×Q8)⋊82C22, C4⋊C4.514C23, (C2×C4).159C24, C22.3(C23×C4), (C2×D4).497C23, (C2×Q8).480C23, C42⋊C281C22, C2.2(C2×2- 1+4), C2.2(C2×2+ 1+4), C22⋊C4.126C23, (C23×C4).187C22, C23.155(C22×C4), (C2×C42).913C22, (C22×C4).1294C23, (C22×D4).613C22, (C22×Q8).513C22, D4(C2×C4⋊C4), Q8(C2×C4⋊C4), C4⋊C42(C2×D4), C4⋊C42(C2×Q8), (C2×C4×D4)⋊67C2, C4⋊C4(C22×D4), (C2×C4×Q8)⋊42C2, (C2×C4○D4)⋊23C4, C4○D417(C2×C4), (C2×D4)⋊55(C2×C4), (C2×C4)⋊7(C22×C4), (C2×Q8)⋊46(C2×C4), (C22×C4⋊C4)⋊36C2, (C22×C4)⋊41(C2×C4), (C2×C4⋊C4)⋊119C22, (C2×C42⋊C2)⋊49C2, (C22×C4○D4).25C2, (C2×C4○D4).316C22, (C2×C22⋊C4).562C22, (C2×Q8)(C2×C4⋊C4), (C2×D4)2(C2×C4⋊C4), C22⋊C44(C2×C4⋊C4), C4⋊C44(C2×C22⋊C4), SmallGroup(128,2159)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C2×C23.33C23
C1C2C22C23C22×C4C23×C4C22×C4○D4 — C2×C23.33C23
C1C2 — C2×C23.33C23
C1C23 — C2×C23.33C23
C1C22 — C2×C23.33C23

Generators and relations for C2×C23.33C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=d, g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, fbf=bc=cb, bd=db, be=eb, bg=gb, cd=dc, geg-1=ce=ec, gfg-1=cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe >

Subgroups: 1020 in 820 conjugacy classes, 692 normal (11 characteristic)
C1, C2 [×3], C2 [×4], C2 [×12], C4 [×16], C4 [×16], C22, C22 [×18], C22 [×36], C2×C4 [×88], C2×C4 [×40], D4 [×48], Q8 [×16], C23, C23 [×18], C23 [×12], C42 [×24], C22⋊C4 [×24], C4⋊C4 [×40], C22×C4, C22×C4 [×71], C22×C4 [×12], C2×D4 [×36], C2×Q8 [×12], C4○D4 [×64], C24 [×3], C2×C42 [×6], C2×C22⋊C4 [×6], C2×C4⋊C4, C2×C4⋊C4 [×33], C42⋊C2 [×24], C4×D4 [×48], C4×Q8 [×16], C23×C4 [×9], C22×D4 [×3], C22×Q8, C2×C4○D4 [×24], C22×C4⋊C4 [×3], C2×C42⋊C2 [×3], C2×C4×D4 [×6], C2×C4×Q8 [×2], C23.33C23 [×16], C22×C4○D4, C2×C23.33C23
Quotients: C1, C2 [×31], C4 [×16], C22 [×155], C2×C4 [×120], C23 [×155], C22×C4 [×140], C24 [×31], C23×C4 [×30], 2+ 1+4 [×2], 2- 1+4 [×2], C25, C23.33C23 [×4], C24×C4, C2×2+ 1+4, C2×2- 1+4, C2×C23.33C23

Smallest permutation representation of C2×C23.33C23
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)
(5 40)(6 37)(7 38)(8 39)(21 57)(22 58)(23 59)(24 60)(25 45)(26 46)(27 47)(28 48)(33 61)(34 62)(35 63)(36 64)
(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 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)(2 60)(3 57)(4 58)(5 54)(6 55)(7 56)(8 53)(9 39)(10 40)(11 37)(12 38)(13 25)(14 26)(15 27)(16 28)(17 21)(18 22)(19 23)(20 24)(29 33)(30 34)(31 35)(32 36)(41 61)(42 62)(43 63)(44 64)(45 49)(46 50)(47 51)(48 52)
(1 55 19 11)(2 12 20 56)(3 53 17 9)(4 10 18 54)(5 22 40 58)(6 59 37 23)(7 24 38 60)(8 57 39 21)(13 29 49 41)(14 42 50 30)(15 31 51 43)(16 44 52 32)(25 61 45 33)(26 34 46 62)(27 63 47 35)(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), (5,40)(6,37)(7,38)(8,39)(21,57)(22,58)(23,59)(24,60)(25,45)(26,46)(27,47)(28,48)(33,61)(34,62)(35,63)(36,64), (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), (1,2,3,4)(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)(2,60)(3,57)(4,58)(5,54)(6,55)(7,56)(8,53)(9,39)(10,40)(11,37)(12,38)(13,25)(14,26)(15,27)(16,28)(17,21)(18,22)(19,23)(20,24)(29,33)(30,34)(31,35)(32,36)(41,61)(42,62)(43,63)(44,64)(45,49)(46,50)(47,51)(48,52), (1,55,19,11)(2,12,20,56)(3,53,17,9)(4,10,18,54)(5,22,40,58)(6,59,37,23)(7,24,38,60)(8,57,39,21)(13,29,49,41)(14,42,50,30)(15,31,51,43)(16,44,52,32)(25,61,45,33)(26,34,46,62)(27,63,47,35)(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), (5,40)(6,37)(7,38)(8,39)(21,57)(22,58)(23,59)(24,60)(25,45)(26,46)(27,47)(28,48)(33,61)(34,62)(35,63)(36,64), (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), (1,2,3,4)(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)(2,60)(3,57)(4,58)(5,54)(6,55)(7,56)(8,53)(9,39)(10,40)(11,37)(12,38)(13,25)(14,26)(15,27)(16,28)(17,21)(18,22)(19,23)(20,24)(29,33)(30,34)(31,35)(32,36)(41,61)(42,62)(43,63)(44,64)(45,49)(46,50)(47,51)(48,52), (1,55,19,11)(2,12,20,56)(3,53,17,9)(4,10,18,54)(5,22,40,58)(6,59,37,23)(7,24,38,60)(8,57,39,21)(13,29,49,41)(14,42,50,30)(15,31,51,43)(16,44,52,32)(25,61,45,33)(26,34,46,62)(27,63,47,35)(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)], [(5,40),(6,37),(7,38),(8,39),(21,57),(22,58),(23,59),(24,60),(25,45),(26,46),(27,47),(28,48),(33,61),(34,62),(35,63),(36,64)], [(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)], [(1,2,3,4),(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),(2,60),(3,57),(4,58),(5,54),(6,55),(7,56),(8,53),(9,39),(10,40),(11,37),(12,38),(13,25),(14,26),(15,27),(16,28),(17,21),(18,22),(19,23),(20,24),(29,33),(30,34),(31,35),(32,36),(41,61),(42,62),(43,63),(44,64),(45,49),(46,50),(47,51),(48,52)], [(1,55,19,11),(2,12,20,56),(3,53,17,9),(4,10,18,54),(5,22,40,58),(6,59,37,23),(7,24,38,60),(8,57,39,21),(13,29,49,41),(14,42,50,30),(15,31,51,43),(16,44,52,32),(25,61,45,33),(26,34,46,62),(27,63,47,35),(28,36,48,64)])

68 conjugacy classes

class 1 2A···2G2H···2S4A···4AV
order12···22···24···4
size11···12···22···2

68 irreducible representations

dim1111111144
type++++++++-
imageC1C2C2C2C2C2C2C42+ 1+42- 1+4
kernelC2×C23.33C23C22×C4⋊C4C2×C42⋊C2C2×C4×D4C2×C4×Q8C23.33C23C22×C4○D4C2×C4○D4C22C22
# reps133621613222

Matrix representation of C2×C23.33C23 in GL6(𝔽5)

400000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000040
003304
,
100000
010000
004000
000400
000040
000004
,
100000
040000
004000
000400
000040
000004
,
400000
030000
002000
000300
000020
001043
,
100000
010000
000010
001141
001000
000004
,
400000
010000
000400
001000
001141
000231

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,3,0,0,0,1,0,3,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,1,0,0,0,3,0,0,0,0,0,0,2,4,0,0,0,0,0,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,1,0,0,0,0,1,4,0,0,0,0,0,1,0,4],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,4,0,1,2,0,0,0,0,4,3,0,0,0,0,1,1] >;

C2×C23.33C23 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{33}C_2^3
% in TeX

G:=Group("C2xC2^3.33C2^3");
// GroupNames label

G:=SmallGroup(128,2159);
// by ID

G=gap.SmallGroup(128,2159);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,387,1123,136]);
// Polycyclic

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

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