Copied to
clipboard

## G = C2×C23⋊2D4order 128 = 27

### Direct product of C2 and C23⋊2D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C2×C23⋊2D4
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — D4×C23 — C2×C23⋊2D4
 Lower central C1 — C23 — C2×C23⋊2D4
 Upper central C1 — C24 — C2×C23⋊2D4
 Jennings C1 — C23 — C2×C23⋊2D4

Generators and relations for C2×C232D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf=bc=cb, bd=db, ebe-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1908 in 906 conjugacy classes, 180 normal (8 characteristic)
C1, C2, C2 [×14], C2 [×12], C4 [×14], C22, C22 [×34], C22 [×108], C2×C4 [×12], C2×C4 [×46], D4 [×96], C23, C23 [×26], C23 [×156], C22⋊C4 [×24], C22×C4 [×20], C22×C4 [×18], C2×D4 [×168], C24, C24 [×18], C24 [×36], C2.C42 [×4], C2×C22⋊C4 [×12], C2×C22⋊C4 [×12], C23×C4, C23×C4 [×3], C22×D4 [×12], C22×D4 [×36], C25 [×3], C2×C2.C42, C232D4 [×8], C22×C22⋊C4 [×3], D4×C23 [×3], C2×C232D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×24], C23 [×15], C2×D4 [×36], C4○D4 [×2], C24, C22≀C2 [×12], C4⋊D4 [×12], C41D4 [×4], C22×D4 [×6], C2×C4○D4, C232D4 [×8], C2×C22≀C2 [×3], C2×C4⋊D4 [×3], C2×C41D4, C2×C232D4

Smallest permutation representation of C2×C232D4
On 64 points
Generators in S64
(1 32)(2 29)(3 30)(4 31)(5 60)(6 57)(7 58)(8 59)(9 51)(10 52)(11 49)(12 50)(13 18)(14 19)(15 20)(16 17)(21 39)(22 40)(23 37)(24 38)(25 61)(26 62)(27 63)(28 64)(33 56)(34 53)(35 54)(36 55)(41 47)(42 48)(43 45)(44 46)
(1 57)(2 28)(3 59)(4 26)(5 16)(6 32)(7 14)(8 30)(9 22)(10 47)(11 24)(12 45)(13 63)(15 61)(17 60)(18 27)(19 58)(20 25)(21 56)(23 54)(29 64)(31 62)(33 39)(34 44)(35 37)(36 42)(38 49)(40 51)(41 52)(43 50)(46 53)(48 55)
(1 11)(2 12)(3 9)(4 10)(5 37)(6 38)(7 39)(8 40)(13 36)(14 33)(15 34)(16 35)(17 54)(18 55)(19 56)(20 53)(21 58)(22 59)(23 60)(24 57)(25 46)(26 47)(27 48)(28 45)(29 50)(30 51)(31 52)(32 49)(41 62)(42 63)(43 64)(44 61)
(1 55)(2 56)(3 53)(4 54)(5 41)(6 42)(7 43)(8 44)(9 20)(10 17)(11 18)(12 19)(13 49)(14 50)(15 51)(16 52)(21 28)(22 25)(23 26)(24 27)(29 33)(30 34)(31 35)(32 36)(37 62)(38 63)(39 64)(40 61)(45 58)(46 59)(47 60)(48 57)
(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 4)(2 3)(5 42)(6 41)(7 44)(8 43)(9 12)(10 11)(13 16)(14 15)(17 18)(19 20)(21 25)(22 28)(23 27)(24 26)(29 30)(31 32)(33 34)(35 36)(37 63)(38 62)(39 61)(40 64)(45 59)(46 58)(47 57)(48 60)(49 52)(50 51)(53 56)(54 55)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2AA 4A ··· 4P order 1 2 ··· 2 2 ··· 2 4 ··· 4 size 1 1 ··· 1 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 D4 D4 C4○D4 kernel C2×C23⋊2D4 C2×C2.C42 C23⋊2D4 C22×C22⋊C4 D4×C23 C22×C4 C24 C23 # reps 1 1 8 3 3 12 12 4

Matrix representation of C2×C232D4 in GL8(ℤ)

 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 1 0 0 0 0 0 0 0 0 1
,
 1 2 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -2 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 1 0 0 0 0 0 0 0 0 -1
,
 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 -1 0 0 0 0 0 0 0 0 -1
,
 -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 1 0 0 0 0 0 0 0 0 1
,
 -1 0 0 0 0 0 0 0 1 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 0 1 0 0 0 0 0 0 -1 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 -1 -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 0 0 0 0 0 0 1 0

G:=sub<GL(8,Integers())| [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,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,2,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1],[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,-1,0,0,0,0,0,0,0,0,-1],[-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,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C2×C232D4 in GAP, Magma, Sage, TeX

C_2\times C_2^3\rtimes_2D_4
% in TeX

G:=Group("C2xC2^3:2D4");
// GroupNames label

G:=SmallGroup(128,1116);
// by ID

G=gap.SmallGroup(128,1116);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723]);
// Polycyclic

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

׿
×
𝔽