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G = M4(2)○2M4(2)  order 128 = 27

Central product of M4(2) and M4(2)

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

Aliases: M4(2)2M4(2), C23.18C42, C42.588C23, C4⋊C4M4(2), (C4×C8)⋊51C22, C22⋊C4M4(2), C4.59(C23×C4), (C2×C4).27C42, C4.32(C2×C42), C24.71(C2×C4), C8.47(C22×C4), C8⋊C469C22, M4(2)(C2×M4(2)), (C2×M4(2))⋊20C4, (C4×M4(2))⋊27C2, M4(2)⋊23(C2×C4), (C2×C8).611C23, (C2×C4).623C24, C42.198(C2×C4), C82M4(2)⋊26C2, C42⋊C2.25C4, C2.1(Q8○M4(2)), C22.18(C2×C42), C2.15(C22×C42), C22.34(C23×C4), M4(2)(C42⋊C2), (C22×C8).421C22, (C2×C42).749C22, (C23×C4).504C22, C23.135(C22×C4), (C22×C4).1488C23, (C22×M4(2)).24C2, C42⋊C2.346C22, (C2×M4(2)).380C22, (C2×C8)⋊25(C2×C4), (C2×C4⋊C4).64C4, C8⋊C4(C8⋊C4), C4⋊C4.243(C2×C4), C22⋊C4.86(C2×C4), (C2×C22⋊C4).32C4, (C2×C4).569(C22×C4), (C22×C4).134(C2×C4), C42⋊C2(C2×M4(2)), (C2×C42⋊C2).49C2, SmallGroup(128,1605)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — M4(2)○2M4(2)
C1C2C22C2×C4C22×C4C23×C4C2×C42⋊C2 — M4(2)○2M4(2)
C1C2 — M4(2)○2M4(2)
C1C2×C4 — M4(2)○2M4(2)
C1C2C2C2×C4 — M4(2)○2M4(2)

Generators and relations for M4(2)○2M4(2)
 G = < a,b,c,d | a8=b2=d2=1, c4=a4, bab=a5, ac=ca, ad=da, bc=cb, bd=db, dcd=a4c >

Subgroups: 332 in 282 conjugacy classes, 252 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×8], C4 [×8], C22, C22 [×6], C22 [×10], C8 [×16], C2×C4 [×2], C2×C4 [×34], C2×C4 [×4], C23, C23 [×6], C23 [×2], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×8], C2×C8 [×24], M4(2) [×32], C22×C4 [×2], C22×C4 [×16], C24, C4×C8 [×8], C8⋊C4 [×8], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×8], C22×C8 [×4], C2×M4(2) [×24], C23×C4, C4×M4(2) [×4], C82M4(2) [×8], C2×C42⋊C2, C22×M4(2) [×2], M4(2)○2M4(2)
Quotients: C1, C2 [×15], C4 [×24], C22 [×35], C2×C4 [×84], C23 [×15], C42 [×16], C22×C4 [×42], C24, C2×C42 [×12], C23×C4 [×3], C22×C42, Q8○M4(2) [×2], M4(2)○2M4(2)

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

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

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

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

68 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4Z8A···8AF
order12222···244444···48···8
size11112···211112···22···2

68 irreducible representations

dim1111111114
type+++++
imageC1C2C2C2C2C4C4C4C4Q8○M4(2)
kernelM4(2)○2M4(2)C4×M4(2)C82M4(2)C2×C42⋊C2C22×M4(2)C2×C22⋊C4C2×C4⋊C4C42⋊C2C2×M4(2)C2
# reps14812448324

Matrix representation of M4(2)○2M4(2) in GL5(𝔽17)

130000
000130
000154
01000
091600
,
160000
011300
001600
000113
000016
,
10000
00041
000013
011300
001600
,
160000
011300
001600
000164
00001

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

M4(2)○2M4(2) in GAP, Magma, Sage, TeX

M_4(2)\circ_2M_4(2)
% in TeX

G:=Group("M4(2)o2M4(2)");
// GroupNames label

G:=SmallGroup(128,1605);
// by ID

G=gap.SmallGroup(128,1605);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,232,723,2019,172]);
// Polycyclic

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

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