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## G = C2×M5(2)⋊C2order 128 = 27

### Direct product of C2 and M5(2)⋊C2

direct product, p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C2×M5(2)⋊C2
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C22×D8 — C2×M5(2)⋊C2
 Lower central C1 — C2 — C4 — C8 — C2×M5(2)⋊C2
 Upper central C1 — C22 — C22×C4 — C22×C8 — C2×M5(2)⋊C2
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C2×M5(2)⋊C2

Generators and relations for C2×M5(2)⋊C2
G = < a,b,c,d | a2=b16=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b9, dbd=b3c, cd=dc >

Subgroups: 372 in 128 conjugacy classes, 52 normal (32 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C22 [×3], C22 [×18], C8 [×4], C8 [×2], C2×C4 [×6], D4 [×10], C23, C23 [×10], C16 [×2], C2×C8 [×6], C2×C8, M4(2) [×3], D8 [×4], D8 [×6], C22×C4, C2×D4 [×9], C24, C8.C4 [×2], C8.C4, C2×C16, M5(2) [×2], M5(2), C22×C8, C2×M4(2), C2×D8 [×6], C2×D8 [×3], C22×D4, M5(2)⋊C2 [×4], C2×C8.C4, C2×M5(2), C22×D8, C2×M5(2)⋊C2
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, M5(2)⋊C2 [×2], C2×D4⋊C4, C2×M5(2)⋊C2

Smallest permutation representation of C2×M5(2)⋊C2
On 32 points
Generators in S32
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)
(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 31)(2 24)(3 17)(4 26)(5 19)(6 28)(7 21)(8 30)(9 23)(10 32)(11 25)(12 18)(13 27)(14 20)(15 29)(16 22)
(1 5)(2 30)(4 28)(6 26)(7 15)(8 24)(9 13)(10 22)(12 20)(14 18)(16 32)(19 31)(21 29)(23 27)

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C4 D4 D4 D8 SD16 SD16 M5(2)⋊C2 kernel C2×M5(2)⋊C2 M5(2)⋊C2 C2×C8.C4 C2×M5(2) C22×D8 C2×D8 C2×C8 C22×C4 C2×C4 C2×C4 C23 C2 # reps 1 4 1 1 1 8 3 1 4 2 2 4

Matrix representation of C2×M5(2)⋊C2 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
,
 11 14 0 0 0 0 3 13 0 0 0 0 0 0 5 0 2 0 0 0 14 0 16 1 0 0 13 14 12 0 0 0 14 3 1 0
,
 1 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 12 0 16 0 0 0 1 0 0 16
,
 16 0 0 0 0 0 12 1 0 0 0 0 0 0 16 0 0 0 0 0 1 1 0 0 0 0 2 0 14 3 0 0 14 0 3 3

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],[11,3,0,0,0,0,14,13,0,0,0,0,0,0,5,14,13,14,0,0,0,0,14,3,0,0,2,16,12,1,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,12,1,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,12,0,0,0,0,0,1,0,0,0,0,0,0,16,1,2,14,0,0,0,1,0,0,0,0,0,0,14,3,0,0,0,0,3,3] >;

C2×M5(2)⋊C2 in GAP, Magma, Sage, TeX

C_2\times M_5(2)\rtimes C_2
% in TeX

G:=Group("C2xM5(2):C2");
// GroupNames label

G:=SmallGroup(128,878);
// by ID

G=gap.SmallGroup(128,878);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1123,1466,136,1411,172,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^16=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^9,d*b*d=b^3*c,c*d=d*c>;
// generators/relations

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