Copied to
clipboard

## G = C2×C4×C4○D4order 128 = 27

### Direct product of C2×C4 and C4○D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2×C4×C4○D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C22×C42 — C2×C4×C4○D4
 Lower central C1 — C2 — C2×C4×C4○D4
 Upper central C1 — C2×C42 — C2×C4×C4○D4
 Jennings C1 — C22 — C2×C4×C4○D4

Generators and relations for C2×C4×C4○D4
G = < a,b,c,d,e | a2=b4=c4=e2=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >

Subgroups: 1020 in 864 conjugacy classes, 708 normal (9 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C22×C42, C2×C42⋊C2, C2×C4×D4, C2×C4×Q8, C4×C4○D4, C22×C4○D4, C2×C4×C4○D4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C23×C4, C2×C4○D4, C25, C4×C4○D4, C24×C4, C22×C4○D4, C2×C4×C4○D4

Smallest permutation representation of C2×C4×C4○D4
On 64 points
Generators in S64
(1 25)(2 26)(3 27)(4 28)(5 37)(6 38)(7 39)(8 40)(9 15)(10 16)(11 13)(12 14)(17 31)(18 32)(19 29)(20 30)(21 35)(22 36)(23 33)(24 34)(41 47)(42 48)(43 45)(44 46)(49 62)(50 63)(51 64)(52 61)(53 59)(54 60)(55 57)(56 58)
(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 29 11 7)(2 30 12 8)(3 31 9 5)(4 32 10 6)(13 39 25 19)(14 40 26 20)(15 37 27 17)(16 38 28 18)(21 57 52 45)(22 58 49 46)(23 59 50 47)(24 60 51 48)(33 53 63 41)(34 54 64 42)(35 55 61 43)(36 56 62 44)
(1 19 11 39)(2 20 12 40)(3 17 9 37)(4 18 10 38)(5 27 31 15)(6 28 32 16)(7 25 29 13)(8 26 30 14)(21 43 52 55)(22 44 49 56)(23 41 50 53)(24 42 51 54)(33 47 63 59)(34 48 64 60)(35 45 61 57)(36 46 62 58)
(1 35)(2 36)(3 33)(4 34)(5 41)(6 42)(7 43)(8 44)(9 63)(10 64)(11 61)(12 62)(13 52)(14 49)(15 50)(16 51)(17 59)(18 60)(19 57)(20 58)(21 25)(22 26)(23 27)(24 28)(29 55)(30 56)(31 53)(32 54)(37 47)(38 48)(39 45)(40 46)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2S 4A ··· 4X 4Y ··· 4BH order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 C4○D4 kernel C2×C4×C4○D4 C22×C42 C2×C42⋊C2 C2×C4×D4 C2×C4×Q8 C4×C4○D4 C22×C4○D4 C2×C4○D4 C2×C4 # reps 1 3 3 6 2 16 1 32 16

Matrix representation of C2×C4×C4○D4 in GL4(𝔽5) generated by

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

C2×C4×C4○D4 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_4\circ D_4
% in TeX

G:=Group("C2xC4xC4oD4");
// GroupNames label

G:=SmallGroup(128,2156);
// by ID

G=gap.SmallGroup(128,2156);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,352,136]);
// Polycyclic

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

׿
×
𝔽