Copied to
clipboard

## G = C2×C42.7C22order 128 = 27

### Direct product of C2 and C42.7C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C42.7C22
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C2×C42⋊C2 — C2×C42.7C22
 Lower central C1 — C22 — C2×C42.7C22
 Upper central C1 — C22×C4 — C2×C42.7C22
 Jennings C1 — C2 — C2 — C2×C4 — C2×C42.7C22

Generators and relations for C2×C42.7C22
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 [×6], C2 [×2], C4 [×4], C4 [×10], C22, C22 [×6], C22 [×10], C8 [×8], C2×C4 [×2], C2×C4 [×14], C2×C4 [×18], C23, C23 [×2], C23 [×6], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×8], C2×C8 [×8], C2×C8 [×8], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C24, C4×C8 [×4], C8⋊C4 [×4], C22⋊C8 [×8], C4⋊C8 [×8], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×8], C22×C8 [×4], C23×C4, C2×C4×C8, C2×C8⋊C4, C2×C22⋊C8 [×2], C2×C4⋊C8 [×2], C42.7C22 [×8], C2×C42⋊C2, C2×C42.7C22
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C8○D4 [×4], C23×C4, C2×C4○D4 [×2], C42.7C22 [×4], C2×C42⋊C2, C2×C8○D4 [×2], C2×C42.7C22

Smallest permutation representation of C2×C42.7C22
On 64 points
Generators in S64
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(1 43 19 15)(2 12 20 48)(3 45 21 9)(4 14 22 42)(5 47 23 11)(6 16 24 44)(7 41 17 13)(8 10 18 46)(25 63 51 37)(26 34 52 60)(27 57 53 39)(28 36 54 62)(29 59 55 33)(30 38 56 64)(31 61 49 35)(32 40 50 58)
(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 25)(2 56)(3 27)(4 50)(5 29)(6 52)(7 31)(8 54)(9 35)(10 58)(11 37)(12 60)(13 39)(14 62)(15 33)(16 64)(17 49)(18 28)(19 51)(20 30)(21 53)(22 32)(23 55)(24 26)(34 48)(36 42)(38 44)(40 46)(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)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,43,19,15)(2,12,20,48)(3,45,21,9)(4,14,22,42)(5,47,23,11)(6,16,24,44)(7,41,17,13)(8,10,18,46)(25,63,51,37)(26,34,52,60)(27,57,53,39)(28,36,54,62)(29,59,55,33)(30,38,56,64)(31,61,49,35)(32,40,50,58), (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), (1,25)(2,56)(3,27)(4,50)(5,29)(6,52)(7,31)(8,54)(9,35)(10,58)(11,37)(12,60)(13,39)(14,62)(15,33)(16,64)(17,49)(18,28)(19,51)(20,30)(21,53)(22,32)(23,55)(24,26)(34,48)(36,42)(38,44)(40,46)(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)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,43,19,15)(2,12,20,48)(3,45,21,9)(4,14,22,42)(5,47,23,11)(6,16,24,44)(7,41,17,13)(8,10,18,46)(25,63,51,37)(26,34,52,60)(27,57,53,39)(28,36,54,62)(29,59,55,33)(30,38,56,64)(31,61,49,35)(32,40,50,58), (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), (1,25)(2,56)(3,27)(4,50)(5,29)(6,52)(7,31)(8,54)(9,35)(10,58)(11,37)(12,60)(13,39)(14,62)(15,33)(16,64)(17,49)(18,28)(19,51)(20,30)(21,53)(22,32)(23,55)(24,26)(34,48)(36,42)(38,44)(40,46)(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),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,31),(18,32),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(33,47),(34,48),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46)], [(1,43,19,15),(2,12,20,48),(3,45,21,9),(4,14,22,42),(5,47,23,11),(6,16,24,44),(7,41,17,13),(8,10,18,46),(25,63,51,37),(26,34,52,60),(27,57,53,39),(28,36,54,62),(29,59,55,33),(30,38,56,64),(31,61,49,35),(32,40,50,58)], [(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)], [(1,25),(2,56),(3,27),(4,50),(5,29),(6,52),(7,31),(8,54),(9,35),(10,58),(11,37),(12,60),(13,39),(14,62),(15,33),(16,64),(17,49),(18,28),(19,51),(20,30),(21,53),(22,32),(23,55),(24,26),(34,48),(36,42),(38,44),(40,46),(41,57),(43,59),(45,61),(47,63)])

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4H 4I ··· 4P 4Q ··· 4V 8A ··· 8P 8Q ··· 8X order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 ··· 1 4 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4○D4 C8○D4 kernel C2×C42.7C22 C2×C4×C8 C2×C8⋊C4 C2×C22⋊C8 C2×C4⋊C8 C42.7C22 C2×C42⋊C2 C2×C22⋊C4 C2×C4⋊C4 C42⋊C2 C2×C4 C22 # reps 1 1 1 2 2 8 1 4 4 8 8 16

Matrix representation of C2×C42.7C22 in GL5(𝔽17)

 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 16 0 0 0 1 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

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] >;

C2×C42.7C22 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

׿
×
𝔽