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## 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 — C22 — C2×C4 — C22×C4 — C23×C4 — C22×C22⋊C4 — C2×C23⋊C8
 Lower central C1 — C2 — C22 — C2×C23⋊C8
 Upper central C1 — C23 — C23×C4 — C2×C23⋊C8
 Jennings C1 — C2 — C22 — C22×C4 — C2×C23⋊C8

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

Subgroups: 564 in 228 conjugacy classes, 68 normal (18 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C23, C23, C22⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C24, C24, C24, C22⋊C8, C22⋊C8, C2×C22⋊C4, C2×C22⋊C4, C22×C8, C23×C4, C25, C23⋊C8, C2×C22⋊C8, C22×C22⋊C4, C2×C23⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C22⋊C8, C23⋊C4, C4.D4, C2×C22⋊C4, C22×C8, C2×M4(2), C23⋊C8, C2×C22⋊C8, C2×C23⋊C4, C2×C4.D4, 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 17)(8 18)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)
(1 11)(3 21)(4 25)(5 15)(7 17)(8 29)(9 28)(10 18)(13 32)(14 22)(19 30)(23 26)
(1 19)(2 12)(3 21)(4 14)(5 23)(6 16)(7 17)(8 10)(9 28)(11 30)(13 32)(15 26)(18 29)(20 31)(22 25)(24 27)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
(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)(7,17)(8,18)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27), (1,11)(3,21)(4,25)(5,15)(7,17)(8,29)(9,28)(10,18)(13,32)(14,22)(19,30)(23,26), (1,19)(2,12)(3,21)(4,14)(5,23)(6,16)(7,17)(8,10)(9,28)(11,30)(13,32)(15,26)(18,29)(20,31)(22,25)(24,27), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24), (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)(7,17)(8,18)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27), (1,11)(3,21)(4,25)(5,15)(7,17)(8,29)(9,28)(10,18)(13,32)(14,22)(19,30)(23,26), (1,19)(2,12)(3,21)(4,14)(5,23)(6,16)(7,17)(8,10)(9,28)(11,30)(13,32)(15,26)(18,29)(20,31)(22,25)(24,27), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24), (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),(7,17),(8,18),(9,28),(10,29),(11,30),(12,31),(13,32),(14,25),(15,26),(16,27)], [(1,11),(3,21),(4,25),(5,15),(7,17),(8,29),(9,28),(10,18),(13,32),(14,22),(19,30),(23,26)], [(1,19),(2,12),(3,21),(4,14),(5,23),(6,16),(7,17),(8,10),(9,28),(11,30),(13,32),(15,26),(18,29),(20,31),(22,25),(24,27)], [(1,30),(2,31),(3,32),(4,25),(5,26),(6,27),(7,28),(8,29),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24)], [(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 ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A ··· 4H 4I 4J 4K 4L 8A ··· 8P order 1 2 ··· 2 2 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 4 4 4 4 2 ··· 2 4 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C4 C4 C4 C8 D4 M4(2) C23⋊C4 C4.D4 kernel C2×C23⋊C8 C23⋊C8 C2×C22⋊C8 C22×C22⋊C4 C2×C22⋊C4 C23×C4 C25 C24 C22×C4 C23 C22 C22 # reps 1 4 2 1 4 2 2 16 4 4 2 2

Matrix representation of C2×C23⋊C8 in GL8(𝔽17)

 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 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
,
 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 9 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1
,
 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 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 16 0 0 0 0 0 0 0 0 16
,
 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 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 8 15 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0

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

C2×C23⋊C8 in GAP, Magma, Sage, TeX

C_2\times C_2^3\rtimes C_8
% in TeX

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

G:=SmallGroup(128,188);
// by ID

G=gap.SmallGroup(128,188);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1123,851,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^8=1,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

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