Copied to
clipboard

## G = C2×C23.C8order 128 = 27

### Direct product of C2 and C23.C8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C23.C8
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C22×M4(2) — C2×C23.C8
 Lower central C1 — C2 — C22 — C2×C23.C8
 Upper central C1 — C2×C4 — C22×C8 — C2×C23.C8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C2×C23.C8

Generators and relations for C2×C23.C8
G = < a,b,c,d,e | a2=b2=c2=d2=1, e8=d, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >

Subgroups: 196 in 118 conjugacy classes, 60 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×2], C22 [×3], C22 [×10], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×2], C2×C4 [×10], C23, C23 [×2], C23 [×4], C16 [×4], C2×C8 [×2], C2×C8 [×6], C2×C8 [×2], M4(2) [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×4], C24, C2×C16 [×2], M5(2) [×4], M5(2) [×2], C22×C8 [×2], C2×M4(2) [×4], C2×M4(2) [×4], C23×C4, C23.C8 [×4], C2×M5(2) [×2], C22×M4(2), C2×C23.C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], C22⋊C8 [×4], C2×C22⋊C4, C22×C8, C2×M4(2), C23.C8 [×2], C2×C22⋊C8, C2×C23.C8

Smallest permutation representation of C2×C23.C8
On 32 points
Generators in S32
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 17)(16 18)
(1 27)(2 10)(3 21)(5 31)(6 14)(7 25)(9 19)(11 29)(13 23)(15 17)(20 28)(24 32)
(1 27)(2 20)(3 29)(4 22)(5 31)(6 24)(7 17)(8 26)(9 19)(10 28)(11 21)(12 30)(13 23)(14 32)(15 25)(16 18)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(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)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 8A ··· 8H 8I 8J 8K 8L 16A ··· 16P order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 8 ··· 8 8 8 8 8 16 ··· 16 size 1 1 1 1 2 2 4 4 1 1 1 1 2 2 4 4 2 ··· 2 4 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 4 type + + + + + image C1 C2 C2 C2 C4 C4 C4 C8 C8 D4 M4(2) C23.C8 kernel C2×C23.C8 C23.C8 C2×M5(2) C22×M4(2) C22×C8 C2×M4(2) C23×C4 C22×C4 C24 C2×C8 C2×C4 C2 # reps 1 4 2 1 2 4 2 12 4 4 4 4

Matrix representation of C2×C23.C8 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 0 16 0 0 0 0 13 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 13 0 0 0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,13,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C2×C23.C8 in GAP, Magma, Sage, TeX

C_2\times C_2^3.C_8
% in TeX

G:=Group("C2xC2^3.C8");
// GroupNames label

G:=SmallGroup(128,846);
// by ID

G=gap.SmallGroup(128,846);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,2019,1411,102,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=1,e^8=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations

׿
×
𝔽