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## G = C2×M4(2).8C22order 128 = 27

### Direct product of C2 and M4(2).8C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×M4(2).8C22
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×C4○D4 — C2×M4(2).8C22
 Lower central C1 — C2 — C22 — C2×M4(2).8C22
 Upper central C1 — C2×C4 — C23×C4 — C2×M4(2).8C22
 Jennings C1 — C2 — C2 — C2×C4 — C2×M4(2).8C22

Generators and relations for C2×M4(2).8C22
G = < a,b,c,d,e | a2=b8=c2=d2=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b5, dbd=bc, cd=dc, ece-1=b4c, ede-1=b4cd >

Subgroups: 636 in 378 conjugacy classes, 172 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×8], C4 [×4], C22 [×3], C22 [×4], C22 [×26], C8 [×8], C2×C4 [×4], C2×C4 [×28], C2×C4 [×20], D4 [×24], Q8 [×8], C23, C23 [×10], C23 [×10], C2×C8 [×12], M4(2) [×8], M4(2) [×12], C22×C4 [×2], C22×C4 [×22], C22×C4 [×8], C2×D4 [×12], C2×D4 [×12], C2×Q8 [×4], C2×Q8 [×4], C4○D4 [×32], C24, C24 [×2], C4.D4 [×8], C4.10D4 [×8], C22×C8 [×2], C2×M4(2) [×12], C2×M4(2) [×6], C23×C4, C23×C4 [×2], C22×D4, C22×D4 [×2], C22×Q8, C2×C4○D4 [×8], C2×C4○D4 [×8], C2×C4.D4 [×2], C2×C4.10D4 [×2], M4(2).8C22 [×8], C22×M4(2) [×2], C22×C4○D4, C2×M4(2).8C22
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], M4(2).8C22 [×2], C22×C22⋊C4, C2×M4(2).8C22

Smallest permutation representation of C2×M4(2).8C22
On 32 points
Generators in S32
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(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 10)(2 15)(3 12)(4 9)(5 14)(6 11)(7 16)(8 13)(17 25)(18 30)(19 27)(20 32)(21 29)(22 26)(23 31)(24 28)
(1 31)(2 20)(3 29)(4 18)(5 27)(6 24)(7 25)(8 22)(9 30)(10 23)(11 28)(12 21)(13 26)(14 19)(15 32)(16 17)
(1 8 3 2 5 4 7 6)(9 12 11 14 13 16 15 10)(17 20 19 22 21 24 23 18)(25 28 27 30 29 32 31 26)

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 4 type + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D4 M4(2).8C22 kernel C2×M4(2).8C22 C2×C4.D4 C2×C4.10D4 M4(2).8C22 C22×M4(2) C22×C4○D4 C23×C4 C22×D4 C2×C4○D4 C22×C4 C2 # reps 1 2 2 8 2 1 4 4 8 8 4

Matrix representation of C2×M4(2).8C22 in GL6(𝔽17)

 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 1 0 0 0 0 0 0 1
,
 0 13 0 0 0 0 13 0 0 0 0 0 0 0 0 0 13 0 0 0 4 1 16 8 0 0 1 0 0 0 0 0 6 11 0 16
,
 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 1 0 0 0 1 13 0 1
,
 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 0 0 0 0 16 0 0 0 0 0 16 4 13 15 0 0 0 1 16 4
,
 0 13 0 0 0 0 13 0 0 0 0 0 0 0 0 0 16 0 0 0 1 13 4 2 0 0 4 0 0 0 0 0 11 11 0 4

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,4,1,6,0,0,0,1,0,11,0,0,13,16,0,0,0,0,0,8,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,16,0,13,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,16,0,0,0,16,0,4,1,0,0,0,0,13,16,0,0,0,0,15,4],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,1,4,11,0,0,0,13,0,11,0,0,16,4,0,0,0,0,0,2,0,4] >;

C2×M4(2).8C22 in GAP, Magma, Sage, TeX

C_2\times M_4(2)._8C_2^2
% in TeX

G:=Group("C2xM4(2).8C2^2");
// GroupNames label

G:=SmallGroup(128,1619);
// by ID

G=gap.SmallGroup(128,1619);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,352,2804,2028,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^2=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^5,d*b*d=b*c,c*d=d*c,e*c*e^-1=b^4*c,e*d*e^-1=b^4*c*d>;
// generators/relations

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