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G = M4(2)⋊10D4order 128 = 27

4th semidirect product of M4(2) and D4 acting via D4/C4=C2

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

Aliases: M4(2)⋊10D4, C42.376C23, (C2×C8)⋊17D4, C8.3(C2×D4), C83D45C2, C85D45C2, (C4×C8)⋊27C22, C4⋊C4.240D4, C8.2D45C2, C4⋊Q811C22, (C2×D8)⋊51C22, C22⋊C4.80D4, C4.12(C22×D4), C8⋊C448C22, C8.12D415C2, C4.41(C41D4), (C2×C8).595C23, (C2×C4).352C24, (C2×Q16)⋊21C22, (C22×SD16)⋊5C2, C4.4D48C22, C23.454(C2×D4), C82M4(2)⋊12C2, C2.35(D4○SD16), (C2×SD16)⋊79C22, (C2×D4).118C23, C41D4.63C22, (C2×Q8).106C23, C22.29C2412C2, C22.18(C41D4), (C22×C8).271C22, C22.612(C22×D4), (C22×C4).1042C23, (C22×D4).378C22, (C22×Q8).311C22, C42⋊C2.325C22, C23.38C2312C2, (C2×M4(2)).272C22, (C2×C4○D8)⋊21C2, (C2×C8⋊C22)⋊24C2, (C2×C4).138(C2×D4), C2.31(C2×C41D4), (C2×C8.C22)⋊24C2, (C2×C4○D4).158C22, SmallGroup(128,1886)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — M4(2)⋊10D4
C1C2C22C2×C4C22×C4C42⋊C2C82M4(2) — M4(2)⋊10D4
C1C2C2×C4 — M4(2)⋊10D4
C1C22C42⋊C2 — M4(2)⋊10D4
C1C2C2C2×C4 — M4(2)⋊10D4

Generators and relations for M4(2)⋊10D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=cac-1=a5, dad=a3, cbc-1=dbd=a4b, dcd=c-1 >

Subgroups: 604 in 282 conjugacy classes, 108 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×18], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×10], D4 [×18], Q8 [×10], C23, C23 [×8], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×6], M4(2) [×4], D8 [×6], SD16 [×20], Q16 [×6], C22×C4, C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×D4 [×11], C2×Q8 [×2], C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×8], C24, C4×C8 [×2], C8⋊C4 [×2], C42⋊C2, C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×4], C41D4 [×2], C4⋊Q8 [×2], C22×C8, C2×M4(2), C2×D8, C2×D8 [×2], C2×SD16 [×2], C2×SD16 [×8], C2×SD16 [×4], C2×Q16, C2×Q16 [×2], C4○D8 [×4], C8⋊C22 [×4], C8.C22 [×4], C22×D4, C22×Q8, C2×C4○D4 [×2], C82M4(2), C85D4 [×2], C8.12D4 [×2], C83D4 [×2], C8.2D4 [×2], C22.29C24, C23.38C23, C22×SD16, C2×C4○D8, C2×C8⋊C22, C2×C8.C22, M4(2)⋊10D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C41D4 [×4], C22×D4 [×3], C2×C41D4, D4○SD16 [×2], M4(2)⋊10D4

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E···8J
order122222222244444444444488888···8
size111122888822224444888822224···4

32 irreducible representations

dim11111111111122224
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D4D4○SD16
kernelM4(2)⋊10D4C82M4(2)C85D4C8.12D4C83D4C8.2D4C22.29C24C23.38C23C22×SD16C2×C4○D8C2×C8⋊C22C2×C8.C22C22⋊C4C4⋊C4C2×C8M4(2)C2
# reps11222211111122444

Matrix representation of M4(2)⋊10D4 in GL6(𝔽17)

010000
1600000
0000107
0012007
00012125
00125125
,
1600000
0160000
001000
000100
00160160
00160016
,
010000
1600000
00160150
0000161
000010
000110
,
010000
100000
00160150
001601616
000010
0011610

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,12,0,0,0,0,12,5,0,0,10,0,12,12,0,0,7,7,5,5],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,16,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,15,16,1,1,0,0,0,1,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,16,0,1,0,0,0,0,0,16,0,0,15,16,1,1,0,0,0,16,0,0] >;

M4(2)⋊10D4 in GAP, Magma, Sage, TeX

M_4(2)\rtimes_{10}D_4
% in TeX

G:=Group("M4(2):10D4");
// GroupNames label

G:=SmallGroup(128,1886);
// by ID

G=gap.SmallGroup(128,1886);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,184,521,2804,172]);
// 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,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

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