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## G = C2×Q8○M4(2)  order 128 = 27

### Direct product of C2 and Q8○M4(2)

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2×Q8○M4(2)
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×C4○D4 — C2×Q8○M4(2)
 Lower central C1 — C2 — C2×Q8○M4(2)
 Upper central C1 — C2×C4 — C2×Q8○M4(2)
 Jennings C1 — C2 — C2 — C4 — C2×Q8○M4(2)

Generators and relations for C2×Q8○M4(2)
G = < a,b,c,d,e | a2=b4=e2=1, c2=d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 812 in 730 conjugacy classes, 684 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×14], C4 [×2], C4 [×14], C22, C22 [×14], C22 [×26], C8 [×16], C2×C4 [×2], C2×C4 [×70], D4 [×48], Q8 [×16], C23, C23 [×18], C23 [×6], C2×C8 [×56], M4(2) [×64], C22×C4, C22×C4 [×39], C2×D4 [×36], C2×Q8 [×12], C4○D4 [×64], C24 [×3], C22×C8 [×12], C2×M4(2) [×64], C8○D4 [×64], C23×C4 [×3], C22×D4 [×3], C22×Q8, C2×C4○D4 [×24], C22×M4(2) [×6], C2×C8○D4 [×8], Q8○M4(2) [×16], C22×C4○D4, C2×Q8○M4(2)
Quotients: C1, C2 [×31], C4 [×16], C22 [×155], C2×C4 [×120], C23 [×155], C22×C4 [×140], C24 [×31], C23×C4 [×30], C25, Q8○M4(2) [×2], C24×C4, C2×Q8○M4(2)

Smallest permutation representation of C2×Q8○M4(2)
On 32 points
Generators in S32
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 25)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)
(1 21 5 17)(2 22 6 18)(3 23 7 19)(4 24 8 20)(9 27 13 31)(10 28 14 32)(11 29 15 25)(12 30 16 26)
(1 32 5 28)(2 25 6 29)(3 26 7 30)(4 27 8 31)(9 20 13 24)(10 21 14 17)(11 22 15 18)(12 23 16 19)
(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)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2Q 4A 4B 4C 4D 4E ··· 4R 8A ··· 8AF order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 2 ··· 2

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 type + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 Q8○M4(2) kernel C2×Q8○M4(2) C22×M4(2) C2×C8○D4 Q8○M4(2) C22×C4○D4 C22×D4 C22×Q8 C2×C4○D4 C2 # reps 1 6 8 16 1 6 2 24 4

Matrix representation of C2×Q8○M4(2) in GL5(𝔽17)

 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16
,
 1 0 0 0 0 0 16 4 12 4 0 8 1 0 5 0 0 0 4 0 0 0 0 8 13
,
 1 0 0 0 0 0 13 16 0 0 0 0 4 0 0 0 0 0 4 13 0 0 0 0 13
,
 1 0 0 0 0 0 12 10 13 9 0 0 0 15 1 0 4 11 0 11 0 8 1 0 5
,
 16 0 0 0 0 0 1 0 0 14 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,8,0,0,0,4,1,0,0,0,12,0,4,8,0,4,5,0,13],[1,0,0,0,0,0,13,0,0,0,0,16,4,0,0,0,0,0,4,0,0,0,0,13,13],[1,0,0,0,0,0,12,0,4,8,0,10,0,11,1,0,13,15,0,0,0,9,1,11,5],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,14,0,0,16] >;

C2×Q8○M4(2) in GAP, Magma, Sage, TeX

C_2\times Q_8\circ M_4(2)
% in TeX

G:=Group("C2xQ8oM4(2)");
// GroupNames label

G:=SmallGroup(128,2304);
// by ID

G=gap.SmallGroup(128,2304);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,-2,224,723,2019,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=e^2=1,c^2=d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=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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