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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, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C24, C4.D4, C4.10D4, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C2×C4.D4, C2×C4.10D4, M4(2).8C22, C22×M4(2), C22×C4○D4, C2×M4(2).8C22
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C24, C2×C22⋊C4, C23×C4, C22×D4, M4(2).8C22, C22×C22⋊C4, C2×M4(2).8C22

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

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

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

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

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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