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G = M4(2)⋊22D4order 128 = 27

1st semidirect product of M4(2) and D4 acting through Inn(M4(2))

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

Aliases: M4(2)⋊22D4, C42.267C23, (C8×D4)⋊38C2, C8.86(C2×D4), C4.76(C4×D4), C89D432C2, C86D433C2, C4⋊C888C22, (C4×C8)⋊57C22, C22≀C2.4C4, C4⋊D4.19C4, C222(C8○D4), C24.81(C2×C4), C22.15(C4×D4), C8⋊C460C22, C22⋊Q8.18C4, C22⋊C877C22, (C2×C8).405C23, (C2×C4).652C24, (C22×C8)⋊52C22, (C4×D4).55C22, M4(2)(C22⋊C8), C4.198(C22×D4), C82M4(2)⋊32C2, C23.36(C22×C4), C22.D4.4C4, C2.16(Q8○M4(2)), (C22×M4(2))⋊26C2, (C2×M4(2))⋊78C22, (C22×C4).919C23, (C23×C4).527C22, C22.179(C23×C4), C22.19C24.11C2, C42.6C2231C2, C42⋊C2.294C22, C2.50(C2×C4×D4), (C2×C8○D4)⋊23C2, C2.18(C2×C8○D4), C4⋊C4.116(C2×C4), (C2×C22⋊C8)⋊44C2, C4.303(C2×C4○D4), C22⋊C8(C2×M4(2)), (C2×D4).173(C2×C4), (C2×C4).1084(C2×D4), C22⋊C4.36(C2×C4), (C2×C4).67(C22×C4), (C2×Q8).156(C2×C4), (C22×C8)⋊C230C2, (C2×C4).830(C4○D4), (C22×C4).342(C2×C4), (C2×C4○D4).287C22, SmallGroup(128,1665)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — M4(2)⋊22D4
C1C2C4C2×C4C22×C4C2×M4(2)C22×M4(2) — M4(2)⋊22D4
C1C22 — M4(2)⋊22D4
C1C2×C4 — M4(2)⋊22D4
C1C2C2C2×C4 — M4(2)⋊22D4

Generators and relations for M4(2)⋊22D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=cac-1=dad=a5, cbc-1=dbd=a4b, dcd=c-1 >

Subgroups: 380 in 251 conjugacy classes, 142 normal (52 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×4], C22 [×17], C8 [×4], C8 [×6], C2×C4 [×4], C2×C4 [×10], C2×C4 [×14], D4 [×12], Q8 [×2], C23 [×3], C23 [×2], C23 [×5], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×4], C2×C8 [×12], M4(2) [×4], M4(2) [×10], C22×C4 [×6], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×6], C2×Q8, C4○D4 [×4], C24, C4×C8 [×2], C8⋊C4 [×2], C22⋊C8 [×4], C22⋊C8 [×4], C4⋊C8 [×4], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C22×C8 [×2], C22×C8 [×4], C2×M4(2) [×2], C2×M4(2) [×4], C2×M4(2) [×4], C8○D4 [×4], C23×C4, C2×C4○D4, C82M4(2), C2×C22⋊C8, (C22×C8)⋊C2, C42.6C22, C8×D4 [×2], C89D4 [×4], C86D4 [×2], C22.19C24, C22×M4(2), C2×C8○D4, M4(2)⋊22D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C8○D4 [×2], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C2×C8○D4, Q8○M4(2), M4(2)⋊22D4

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

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F4G4H4I···4O8A···8P8Q···8X
order12222222222444444444···48···88···8
size11112222444111122224···42···24···4

50 irreducible representations

dim1111111111111112224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C4C4C4C4D4C4○D4C8○D4Q8○M4(2)
kernelM4(2)⋊22D4C82M4(2)C2×C22⋊C8(C22×C8)⋊C2C42.6C22C8×D4C89D4C86D4C22.19C24C22×M4(2)C2×C8○D4C22≀C2C4⋊D4C22⋊Q8C22.D4M4(2)C2×C4C22C2
# reps1111124211144444482

Matrix representation of M4(2)⋊22D4 in GL4(𝔽17) generated by

0100
4000
0010
0001
,
1000
01600
00160
00016
,
0200
9000
001515
00112
,
0200
9000
001515
00102
G:=sub<GL(4,GF(17))| [0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,9,0,0,2,0,0,0,0,0,15,11,0,0,15,2],[0,9,0,0,2,0,0,0,0,0,15,10,0,0,15,2] >;

M4(2)⋊22D4 in GAP, Magma, Sage, TeX

M_4(2)\rtimes_{22}D_4
% in TeX

G:=Group("M4(2):22D4");
// GroupNames label

G:=SmallGroup(128,1665);
// by ID

G=gap.SmallGroup(128,1665);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,521,124]);
// Polycyclic

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

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