Copied to
clipboard

## G = (C2×D4).Q8order 128 = 27

### 9th non-split extension by C2×D4 of Q8 acting via Q8/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — (C2×D4).Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — Q8○M4(2) — (C2×D4).Q8
 Lower central C1 — C2 — C23 — (C2×D4).Q8
 Upper central C1 — C4 — C22×C4 — (C2×D4).Q8
 Jennings C1 — C2 — C2 — C22×C4 — (C2×D4).Q8

Generators and relations for (C2×D4).Q8
G = < a,b,c,d,e | a2=b4=c2=1, d4=b2, e2=d2, ab=ba, ac=ca, dad-1=ab2, ae=ea, cbc=dbd-1=b-1, be=eb, cd=dc, ece-1=ab2c, ede-1=abd3 >

Subgroups: 228 in 130 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×5], C8 [×6], C2×C4 [×2], C2×C4 [×6], C2×C4 [×8], D4 [×6], Q8 [×2], C23, C23 [×2], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×8], M4(2) [×10], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C22⋊C8 [×2], C23⋊C4 [×4], C4⋊C8 [×4], C42⋊C2 [×2], C22×C8, C2×M4(2), C2×M4(2) [×2], C2×M4(2) [×2], C8○D4 [×4], C2×C4○D4, M4(2)⋊4C4 [×2], (C22×C8)⋊C2, C23.C23, C42.6C22 [×2], Q8○M4(2), (C2×D4).Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C23.8Q8, (C2×D4).Q8

Smallest permutation representation of (C2×D4).Q8
On 32 points
Generators in S32
(1 27)(2 32)(3 29)(4 26)(5 31)(6 28)(7 25)(8 30)(9 17)(10 22)(11 19)(12 24)(13 21)(14 18)(15 23)(16 20)
(1 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)(17 23 21 19)(18 20 22 24)(25 27 29 31)(26 32 30 28)
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)
(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)
(1 14 3 16 5 10 7 12)(2 23 4 17 6 19 8 21)(9 28 11 30 13 32 15 26)(18 29 20 31 22 25 24 27)

G:=sub<Sym(32)| (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,17)(10,22)(11,19)(12,24)(13,21)(14,18)(15,23)(16,20), (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16)(17,23,21,19)(18,20,22,24)(25,27,29,31)(26,32,30,28), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28), (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), (1,14,3,16,5,10,7,12)(2,23,4,17,6,19,8,21)(9,28,11,30,13,32,15,26)(18,29,20,31,22,25,24,27)>;

G:=Group( (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,17)(10,22)(11,19)(12,24)(13,21)(14,18)(15,23)(16,20), (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16)(17,23,21,19)(18,20,22,24)(25,27,29,31)(26,32,30,28), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28), (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), (1,14,3,16,5,10,7,12)(2,23,4,17,6,19,8,21)(9,28,11,30,13,32,15,26)(18,29,20,31,22,25,24,27) );

G=PermutationGroup([(1,27),(2,32),(3,29),(4,26),(5,31),(6,28),(7,25),(8,30),(9,17),(10,22),(11,19),(12,24),(13,21),(14,18),(15,23),(16,20)], [(1,3,5,7),(2,8,6,4),(9,15,13,11),(10,12,14,16),(17,23,21,19),(18,20,22,24),(25,27,29,31),(26,32,30,28)], [(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28)], [(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)], [(1,14,3,16,5,10,7,12),(2,23,4,17,6,19,8,21),(9,28,11,30,13,32,15,26),(18,29,20,31,22,25,24,27)])

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 8A ··· 8L 8M 8N order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 8 ··· 8 8 8 size 1 1 2 2 2 4 4 1 1 2 2 2 4 4 8 8 8 8 4 ··· 4 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C4 D4 D4 Q8 Q8 C4○D4 (C2×D4).Q8 kernel (C2×D4).Q8 M4(2)⋊4C4 (C22×C8)⋊C2 C23.C23 C42.6C22 Q8○M4(2) C23⋊C4 C2×C8 C22×C4 C2×D4 C2×Q8 C2×C4 C1 # reps 1 2 1 1 2 1 8 4 2 1 1 4 4

Matrix representation of (C2×D4).Q8 in GL4(𝔽17) generated by

 0 16 0 0 16 0 0 0 16 16 1 2 0 0 0 16
,
 4 0 0 0 0 4 0 0 0 0 13 0 4 4 0 13
,
 0 0 1 0 1 1 16 15 1 0 0 0 0 0 0 16
,
 8 8 9 1 0 0 9 0 0 8 0 0 8 0 9 9
,
 0 15 0 0 15 0 0 0 0 0 15 0 0 0 0 15
G:=sub<GL(4,GF(17))| [0,16,16,0,16,0,16,0,0,0,1,0,0,0,2,16],[4,0,0,4,0,4,0,4,0,0,13,0,0,0,0,13],[0,1,1,0,0,1,0,0,1,16,0,0,0,15,0,16],[8,0,0,8,8,0,8,0,9,9,0,9,1,0,0,9],[0,15,0,0,15,0,0,0,0,0,15,0,0,0,0,15] >;

(C2×D4).Q8 in GAP, Magma, Sage, TeX

(C_2\times D_4).Q_8
% in TeX

G:=Group("(C2xD4).Q8");
// GroupNames label

G:=SmallGroup(128,600);
// by ID

G=gap.SmallGroup(128,600);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,521,1411,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^4=b^2,e^2=d^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,a*e=e*a,c*b*c=d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,e*c*e^-1=a*b^2*c,e*d*e^-1=a*b*d^3>;
// generators/relations

׿
×
𝔽