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G = M4(2).44D4order 128 = 27

8th non-split extension by M4(2) of D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: M4(2).44D4, C8:C22:1C4, C4.63(C4xD4), (C2xC8).27D4, C4oD4.45D4, (C2xD4).71D4, C8.C22:1C4, (C2xQ8).69D4, Q8oM4(2):5C2, C22.35(C4xD4), C4.125C22wrC2, C4.C42:4C2, D4.3(C22:C4), C23.7(C4oD4), M4(2).4(C2xC4), Q8.3(C22:C4), C4.129(C4:D4), M4(2):4C4:2C2, D8:C22.1C2, C23.24D4:2C2, (C22xC8).39C22, C42:C22:11C2, C22.42(C4:D4), (C22xC4).679C23, C42:C2.16C22, C2.20(C23.23D4), (C2xM4(2)).178C22, C22.21(C22.D4), C4oD4.7(C2xC4), (C2xD4).74(C2xC4), C4.15(C2xC22:C4), (C2xC4).9(C22xC4), (C2xQ8).65(C2xC4), (C2xC4).1000(C2xD4), (C22xC8):C2:23C2, (C2xC4).318(C4oD4), (C2xC4oD4).16C22, SmallGroup(128,613)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — M4(2).44D4
C1C2C22C2xC4C22xC4C2xC4oD4Q8oM4(2) — M4(2).44D4
C1C2C2xC4 — M4(2).44D4
C1C4C22xC4 — M4(2).44D4
C1C2C2C22xC4 — M4(2).44D4

Generators and relations for M4(2).44D4
 G = < a,b,c,d | a8=b2=1, c4=a4, d2=b, bab=a5, cac-1=dad-1=a3b, bc=cb, bd=db, dcd-1=a4bc3 >

Subgroups: 300 in 155 conjugacy classes, 54 normal (36 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C42, C22:C4, C4:C4, C2xC8, C2xC8, M4(2), M4(2), D8, SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C4oD4, C22:C8, D4:C4, Q8:C4, C4wrC2, C42:C2, C22xC8, C2xM4(2), C2xM4(2), C8oD4, C4oD8, C8:C22, C8:C22, C8.C22, C8.C22, C2xC4oD4, C4.C42, M4(2):4C4, (C22xC8):C2, C23.24D4, C42:C22, Q8oM4(2), D8:C22, M4(2).44D4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, C22xC4, C2xD4, C4oD4, C2xC22:C4, C4xD4, C22wrC2, C4:D4, C22.D4, C23.23D4, M4(2).44D4

Smallest permutation representation of M4(2).44D4
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)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)
(1 14 23 26 5 10 19 30)(2 9 20 25 6 13 24 29)(3 16 17 28 7 12 21 32)(4 11 22 27 8 15 18 31)
(1 30)(2 25 6 29)(3 32)(4 27 8 31)(5 26)(7 28)(9 24 13 20)(10 23)(11 18 15 22)(12 17)(14 19)(16 21)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J8A···8L8M8N
order1222222244444444448···888
size1122244811222448884···488

32 irreducible representations

dim111111111122222224
type+++++++++++++
imageC1C2C2C2C2C2C2C2C4C4D4D4D4D4D4C4oD4C4oD4M4(2).44D4
kernelM4(2).44D4C4.C42M4(2):4C4(C22xC8):C2C23.24D4C42:C22Q8oM4(2)D8:C22C8:C22C8.C22C2xC8M4(2)C2xD4C2xQ8C4oD4C2xC4C23C1
# reps111111114422112224

Matrix representation of M4(2).44D4 in GL4(F17) generated by

0161310
016010
16100
0201
,
1001
0101
00160
00016
,
14141214
314120
001012
00100
,
143125
33128
001012
00107
G:=sub<GL(4,GF(17))| [0,0,16,0,16,16,1,2,13,0,0,0,10,10,0,1],[1,0,0,0,0,1,0,0,0,0,16,0,1,1,0,16],[14,3,0,0,14,14,0,0,12,12,10,10,14,0,12,0],[14,3,0,0,3,3,0,0,12,12,10,10,5,8,12,7] >;

M4(2).44D4 in GAP, Magma, Sage, TeX

M_4(2)._{44}D_4
% in TeX

G:=Group("M4(2).44D4");
// GroupNames label

G:=SmallGroup(128,613);
// by ID

G=gap.SmallGroup(128,613);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,1018,248,2804,718,172,2028]);
// Polycyclic

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

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