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G = M4(2).30D4order 128 = 27

11st non-split extension by M4(2) of D4 acting via D4/C4=C2

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

Aliases: M4(2).30D4, (C2×D8)⋊12C4, C4.90(C4×D4), (C2×Q16)⋊12C4, (C2×SD16)⋊7C4, (C2×C8).113D4, C22⋊C4.70D4, C22.61(C4×D4), C23.33(C2×D4), C8.21(C22⋊C4), C82M4(2)⋊1C2, C4.207(C4⋊D4), C4.48(C4.4D4), C22.5(C41D4), C42⋊C2217C2, C23.25D417C2, C22.21(C4⋊D4), C23.C238C2, (C22×C8).225C22, (C22×C4).702C23, C42⋊C2.42C22, (C2×M4(2)).325C22, C2.28(C24.3C22), (C2×C8).75(C2×C4), (C2×C4○D8).8C2, (C2×C4).250(C2×D4), C4.44(C2×C22⋊C4), (C2×D4).115(C2×C4), (C2×Q8).100(C2×C4), (C2×C4).764(C4○D4), (C2×C4).200(C22×C4), (C2×C4○D4).41C22, SmallGroup(128,708)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — M4(2).30D4
C1C2C22C2×C4C22×C4C22×C8C82M4(2) — M4(2).30D4
C1C2C2×C4 — M4(2).30D4
C1C4C22×C4 — M4(2).30D4
C1C2C2C22×C4 — M4(2).30D4

Generators and relations for M4(2).30D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=cac-1=a5, dad=a3b, cbc-1=dbd=a4b, dcd=bc-1 >

Subgroups: 300 in 142 conjugacy classes, 54 normal (36 characteristic)
C1, C2, C2 [×5], C4 [×4], C4 [×6], C22 [×3], C22 [×5], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×10], D4 [×8], Q8 [×4], C23, C23 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×6], C2×C8, M4(2) [×2], M4(2), D8 [×2], SD16 [×4], Q16 [×2], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×8], C4×C8, C8⋊C4, C23⋊C4 [×4], C4≀C2 [×4], C4.Q8, C2.D8, C42⋊C2, C42⋊C2 [×2], C22×C8, C2×M4(2), C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×4], C2×C4○D4 [×2], C82M4(2), C23.C23 [×2], C42⋊C22 [×2], C23.25D4, C2×C4○D8, M4(2).30D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C24.3C22, M4(2).30D4

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J···4O8A8B8C8D8E···8J
order12222224444444444···488888···8
size11222881122244448···822224···4

32 irreducible representations

dim11111111122224
type+++++++++
imageC1C2C2C2C2C2C4C4C4D4D4D4C4○D4M4(2).30D4
kernelM4(2).30D4C82M4(2)C23.C23C42⋊C22C23.25D4C2×C4○D8C2×D8C2×SD16C2×Q16C22⋊C4C2×C8M4(2)C2×C4C1
# reps11221124224244

Matrix representation of M4(2).30D4 in GL4(𝔽17) generated by

00125
001212
141400
31400
,
16000
01600
0010
0001
,
00013
0040
1000
0100
,
00314
001414
31400
141400
G:=sub<GL(4,GF(17))| [0,0,14,3,0,0,14,14,12,12,0,0,5,12,0,0],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[0,0,1,0,0,0,0,1,0,4,0,0,13,0,0,0],[0,0,3,14,0,0,14,14,3,14,0,0,14,14,0,0] >;

M4(2).30D4 in GAP, Magma, Sage, TeX

M_4(2)._{30}D_4
% in TeX

G:=Group("M4(2).30D4");
// GroupNames label

G:=SmallGroup(128,708);
// by ID

G=gap.SmallGroup(128,708);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,2019,1018,2804,172,2028]);
// Polycyclic

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

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