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G = M4(2).3Q8order 128 = 27

1st non-split extension by M4(2) of Q8 acting via Q8/C4=C2

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

Aliases: M4(2).3Q8, C2.D86C4, C4.Q811C4, C4.27(C4×Q8), (C2×C8).326D4, C2.15(C8○D8), C426C4.7C2, C22.160(C4×D4), C2.15(C8.26D4), C4.C42.1C2, C4.116(C22⋊Q8), C4.26(C42⋊C2), C23.205(C4○D4), C82M4(2).17C2, (C22×C8).393C22, (C2×C42).295C22, C23.25D4.2C2, C22.7(C42.C2), (C22×C4).1378C23, C22.5(C422C2), C42⋊C2.271C22, C4.139(C22.D4), (C2×M4(2)).320C22, C22.7C42.30C2, C42.6C22.10C2, C2.13(C23.63C23), C4⋊C4.80(C2×C4), (C2×C8).40(C2×C4), (C2×C4).273(C2×Q8), (C2×C4).1534(C2×D4), (C2×C4).760(C4○D4), (C2×C4).396(C22×C4), SmallGroup(128,654)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — M4(2).3Q8
C1C2C4C2×C4C22×C4C42⋊C2C82M4(2) — M4(2).3Q8
C1C2C2×C4 — M4(2).3Q8
C1C2×C4C22×C8 — M4(2).3Q8
C1C2C2C22×C4 — M4(2).3Q8

Generators and relations for M4(2).3Q8
 G = < a,b,c,d | a8=b2=1, c4=a4, d2=a4c2, bab=a5, ac=ca, dad-1=a-1b, bc=cb, dbd-1=a4b, dcd-1=a6c3 >

Subgroups: 156 in 93 conjugacy classes, 48 normal (46 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×6], C22 [×3], C22 [×2], C8 [×7], C2×C4 [×6], C2×C4 [×8], C23, C42 [×4], C22⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×6], M4(2) [×2], M4(2) [×3], C22×C4, C22×C4, C4×C8, C8⋊C4, C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C42⋊C2 [×2], C22×C8 [×2], C2×M4(2) [×2], C22.7C42, C426C4 [×2], C4.C42, C82M4(2), C42.6C22, C23.25D4, M4(2).3Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C23.63C23, C8○D8, C8.26D4, M4(2).3Q8

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G···4N4O4P8A8B8C8D8E···8N8O8P
order1222224444444···44488888···888
size1111221111224···48822224···488

38 irreducible representations

dim111111111222224
type++++++++-
imageC1C2C2C2C2C2C2C4C4D4Q8C4○D4C4○D4C8○D8C8.26D4
kernelM4(2).3Q8C22.7C42C426C4C4.C42C82M4(2)C42.6C22C23.25D4C4.Q8C2.D8C2×C8M4(2)C2×C4C23C2C2
# reps112111144226282

Matrix representation of M4(2).3Q8 in GL4(𝔽17) generated by

4000
0400
0009
0080
,
16000
01600
00160
0001
,
0400
13000
0020
0002
,
0100
1000
00013
0010
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,0,8,0,0,9,0],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1],[0,13,0,0,4,0,0,0,0,0,2,0,0,0,0,2],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,13,0] >;

M4(2).3Q8 in GAP, Magma, Sage, TeX

M_4(2)._3Q_8
% in TeX

G:=Group("M4(2).3Q8");
// GroupNames label

G:=SmallGroup(128,654);
// by ID

G=gap.SmallGroup(128,654);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,58,2804,718,172,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=a^4*c^2,b*a*b=a^5,a*c=c*a,d*a*d^-1=a^-1*b,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=a^6*c^3>;
// generators/relations

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