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## G = C4×2+ 1+4order 128 = 27

### Direct product of C4 and 2+ 1+4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C4×2+ 1+4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C2×C4×D4 — C4×2+ 1+4
 Lower central C1 — C2 — C4×2+ 1+4
 Upper central C1 — C2×C4 — C4×2+ 1+4
 Jennings C1 — C22 — C4×2+ 1+4

Generators and relations for C4×2+ 1+4
G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 1124 in 830 conjugacy classes, 686 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C23×C4, C22×D4, C2×C4○D4, 2+ 1+4, C2×C4×D4, C4×C4○D4, C22.11C24, C23.33C23, C2×2+ 1+4, C4×2+ 1+4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C23×C4, 2+ 1+4, C25, C24×C4, C2×2+ 1+4, C2.C25, C4×2+ 1+4

Smallest permutation representation of C4×2+ 1+4
On 32 points
Generators in S32
(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 17 11 16)(2 18 12 13)(3 19 9 14)(4 20 10 15)(5 25 30 24)(6 26 31 21)(7 27 32 22)(8 28 29 23)
(1 16)(2 13)(3 14)(4 15)(5 25)(6 26)(7 27)(8 28)(9 19)(10 20)(11 17)(12 18)(21 31)(22 32)(23 29)(24 30)
(1 27 11 22)(2 28 12 23)(3 25 9 24)(4 26 10 21)(5 19 30 14)(6 20 31 15)(7 17 32 16)(8 18 29 13)
(1 24)(2 21)(3 22)(4 23)(5 17)(6 18)(7 19)(8 20)(9 27)(10 28)(11 25)(12 26)(13 31)(14 32)(15 29)(16 30)

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2U 4A 4B 4C 4D 4E ··· 4AT order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2

68 irreducible representations

 dim 1 1 1 1 1 1 1 4 4 type + + + + + + + image C1 C2 C2 C2 C2 C2 C4 2+ 1+4 C2.C25 kernel C4×2+ 1+4 C2×C4×D4 C4×C4○D4 C22.11C24 C23.33C23 C2×2+ 1+4 2+ 1+4 C4 C2 # reps 1 9 6 9 6 1 32 2 2

Matrix representation of C4×2+ 1+4 in GL5(𝔽5)

 2 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 2 2 0 0 0 0 3 0 0 0 0 1 0 1 0 0 2 4 0
,
 4 0 0 0 0 0 2 2 0 0 0 1 3 0 0 0 0 1 0 1 0 4 2 1 0
,
 4 0 0 0 0 0 4 0 2 0 0 0 0 3 1 0 4 0 1 0 0 3 4 2 0
,
 4 0 0 0 0 0 4 0 2 0 0 0 0 3 1 0 0 0 1 0 0 0 1 2 0

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

C4×2+ 1+4 in GAP, Magma, Sage, TeX

C_4\times 2_+^{1+4}
% in TeX

G:=Group("C4xES+(2,2)");
// GroupNames label

G:=SmallGroup(128,2161);
// by ID

G=gap.SmallGroup(128,2161);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,387,1123,172]);
// Polycyclic

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

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