Copied to
clipboard

## G = C2×C24.3C22order 128 = 27

### Direct product of C2 and C24.3C22

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

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 [×3], C2 [×12], C2 [×8], C4 [×8], C4 [×12], C22 [×3], C22 [×32], C22 [×72], C2×C4 [×36], C2×C4 [×44], D4 [×32], C23, C23 [×22], C23 [×104], C42 [×8], C22⋊C4 [×32], C4⋊C4 [×8], C22×C4 [×30], C22×C4 [×20], C2×D4 [×16], C2×D4 [×48], C24, C24 [×12], C24 [×24], C2×C42 [×4], C2×C42 [×4], C2×C22⋊C4 [×16], C2×C22⋊C4 [×16], C2×C4⋊C4 [×4], C2×C4⋊C4 [×4], C23×C4, C23×C4 [×4], C22×D4 [×12], C22×D4 [×8], C25 [×2], C24.3C22 [×8], C22×C42, C22×C22⋊C4 [×4], C22×C4⋊C4, D4×C23, C2×C24.3C22
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×16], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×24], C4○D4 [×4], C24, C2×C22⋊C4 [×12], C4×D4 [×8], C4⋊D4 [×8], C4.4D4 [×4], C41D4 [×4], C23×C4, C22×D4 [×4], C2×C4○D4 [×2], C24.3C22 [×8], C22×C22⋊C4, C2×C4×D4 [×2], C2×C4⋊D4 [×2], C2×C4.4D4, C2×C41D4, 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 31)(6 32)(7 29)(8 30)(9 27)(10 28)(11 25)(12 26)(13 23)(14 24)(15 21)(16 22)(17 60)(18 57)(19 58)(20 59)(33 63)(34 64)(35 61)(36 62)(37 46)(38 47)(39 48)(40 45)(41 55)(42 56)(43 53)(44 54)
(1 32)(2 39)(3 30)(4 37)(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 33)(2 34)(3 35)(4 36)(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 42)(6 43)(7 44)(8 41)(9 39)(10 40)(11 37)(12 38)(13 34)(14 35)(15 36)(16 33)(17 51)(18 52)(19 49)(20 50)(21 62)(22 63)(23 64)(24 61)(25 46)(26 47)(27 48)(28 45)(29 54)(30 55)(31 56)(32 53)
(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 32)(2 29 58 54)(3 55 59 30)(4 31 60 56)(5 17 42 51)(6 52 43 18)(7 19 44 49)(8 50 41 20)(9 34 39 13)(10 14 40 35)(11 36 37 15)(12 16 38 33)(21 25 62 46)(22 47 63 26)(23 27 64 48)(24 45 61 28)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2W 4A ··· 4X 4Y ··· 4AF order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 C4○D4 kernel C2×C24.3C22 C24.3C22 C22×C42 C22×C22⋊C4 C22×C4⋊C4 D4×C23 C22×D4 C22×C4 C23 # reps 1 8 1 4 1 1 16 16 8

Matrix representation of C2×C24.3C22 in GL8(𝔽5)

 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 0 0 0 0 0 0 0 3 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 0 0 0 0 0 0 0 3 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 0 0 0 0 0 0 0 1 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 3 0 0 0 0 0 0 0 2
,
 1 4 0 0 0 0 0 0 2 4 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 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

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

׿
×
𝔽