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## G = C2×D8⋊2C4order 128 = 27

### Direct product of C2 and D8⋊2C4

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×D82C4
G = < a,b,c,d | a2=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b5c >

Subgroups: 276 in 116 conjugacy classes, 52 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×4], C22 [×3], C22 [×6], C8 [×4], C2×C4 [×6], C2×C4 [×9], D4 [×7], Q8 [×3], C23, C23, C16 [×2], C4⋊C4 [×3], C2×C8 [×6], D8 [×2], D8, SD16 [×4], Q16 [×2], Q16, C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8, C4○D4 [×6], C4.Q8 [×2], C4.Q8, C2×C16, M5(2) [×2], M5(2), C2×C4⋊C4, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8 [×4], C4○D8 [×2], C2×C4○D4, D82C4 [×4], C2×C4.Q8, C2×M5(2), C2×C4○D8, C2×D82C4
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, D82C4 [×2], C2×D4⋊C4, C2×D82C4

Smallest permutation representation of C2×D82C4
On 32 points
Generators in S32
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 24)(16 23)
(1 15)(2 10)(3 13)(4 16)(5 11)(6 14)(7 9)(8 12)(17 26 21 30)(18 29 22 25)(19 32 23 28)(20 27 24 31)

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 D8 SD16 D8 D8⋊2C4 kernel C2×D8⋊2C4 D8⋊2C4 C2×C4.Q8 C2×M5(2) C2×C4○D8 C2×D8 C2×Q16 C4○D8 C2×C8 C22×C4 C2×C4 C2×C4 C23 C2 # reps 1 4 1 1 1 2 2 4 3 1 2 4 2 4

Matrix representation of C2×D82C4 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
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 12 5 0 0 0 0 12 12 0 0 0 0 0 0 5 5 0 0 0 0 12 5
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 5 5 0 0 0 0 12 5 0 0 12 5 0 0 0 0 12 12 0 0
,
 4 0 0 0 0 0 0 13 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 5 5 0 0 0 0 5 12

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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,0,0,0,0,5,12,0,0,0,0,5,5],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,5,12,0,0,0,0,5,5,0,0],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,5,5,0,0,0,0,5,12] >;

C2×D82C4 in GAP, Magma, Sage, TeX

C_2\times D_8\rtimes_2C_4
% in TeX

G:=Group("C2xD8:2C4");
// GroupNames label

G:=SmallGroup(128,876);
// by ID

G=gap.SmallGroup(128,876);
# by ID

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

G:=Group<a,b,c,d|a^2=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^5*c>;
// generators/relations

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