Copied to
clipboard

## G = C2×D4×Q8order 128 = 27

### Direct product of C2, D4 and Q8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×D4×Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — Q8×C23 — C2×D4×Q8
 Lower central C1 — C22 — C2×D4×Q8
 Upper central C1 — C23 — C2×D4×Q8
 Jennings C1 — C22 — C2×D4×Q8

Generators and relations for C2×D4×Q8
G = < a,b,c,d,e | a2=b4=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 1084 in 784 conjugacy classes, 484 normal (11 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×16], C4 [×18], C22, C22 [×14], C22 [×24], C2×C4 [×42], C2×C4 [×66], D4 [×16], Q8 [×16], Q8 [×48], C23, C23 [×12], C23 [×8], C42 [×12], C22⋊C4 [×24], C4⋊C4 [×48], C22×C4, C22×C4 [×36], C22×C4 [×24], C2×D4 [×12], C2×Q8 [×36], C2×Q8 [×88], C24 [×2], C2×C42 [×3], C2×C22⋊C4 [×6], C2×C4⋊C4 [×12], C4×D4 [×24], C4×Q8 [×8], C22⋊Q8 [×48], C4⋊Q8 [×24], C23×C4 [×6], C22×D4, C22×Q8, C22×Q8 [×22], C22×Q8 [×16], C2×C4×D4 [×3], C2×C4×Q8, C2×C22⋊Q8 [×6], C2×C4⋊Q8 [×3], D4×Q8 [×16], Q8×C23 [×2], C2×D4×Q8
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], Q8 [×8], C23 [×155], C2×D4 [×28], C2×Q8 [×28], C24 [×31], C22×D4 [×14], C22×Q8 [×14], 2- 1+4 [×2], C25, D4×Q8 [×4], D4×C23, Q8×C23, C2×2- 1+4, C2×D4×Q8

Smallest permutation representation of C2×D4×Q8
On 64 points
Generators in S64
(1 14)(2 15)(3 16)(4 13)(5 30)(6 31)(7 32)(8 29)(9 35)(10 36)(11 33)(12 34)(17 41)(18 42)(19 43)(20 44)(21 27)(22 28)(23 25)(24 26)(37 47)(38 48)(39 45)(40 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 16)(2 15)(3 14)(4 13)(5 30)(6 29)(7 32)(8 31)(9 35)(10 34)(11 33)(12 36)(17 41)(18 44)(19 43)(20 42)(21 25)(22 28)(23 27)(24 26)(37 47)(38 46)(39 45)(40 48)(49 62)(50 61)(51 64)(52 63)(53 57)(54 60)(55 59)(56 58)
(1 36 25 6)(2 33 26 7)(3 34 27 8)(4 35 28 5)(9 22 30 13)(10 23 31 14)(11 24 32 15)(12 21 29 16)(17 39 49 60)(18 40 50 57)(19 37 51 58)(20 38 52 59)(41 45 62 54)(42 46 63 55)(43 47 64 56)(44 48 61 53)
(1 48 25 53)(2 45 26 54)(3 46 27 55)(4 47 28 56)(5 64 35 43)(6 61 36 44)(7 62 33 41)(8 63 34 42)(9 19 30 51)(10 20 31 52)(11 17 32 49)(12 18 29 50)(13 37 22 58)(14 38 23 59)(15 39 24 60)(16 40 21 57)

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

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

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

50 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4P 4Q ··· 4AH order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 4 type + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 Q8 D4 2- 1+4 kernel C2×D4×Q8 C2×C4×D4 C2×C4×Q8 C2×C22⋊Q8 C2×C4⋊Q8 D4×Q8 Q8×C23 C2×D4 C2×Q8 C22 # reps 1 3 1 6 3 16 2 8 8 2

Matrix representation of C2×D4×Q8 in GL5(𝔽5)

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

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

C2×D4×Q8 in GAP, Magma, Sage, TeX

C_2\times D_4\times Q_8
% in TeX

G:=Group("C2xD4xQ8");
// GroupNames label

G:=SmallGroup(128,2198);
// by ID

G=gap.SmallGroup(128,2198);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,232,1430,570,136]);
// Polycyclic

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

׿
×
𝔽