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G = D45M4(2)  order 128 = 27

3rd semidirect product of D4 and M4(2) acting via M4(2)/C2×C4=C2

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

Aliases: D45M4(2), C42.50D4, C42.611C23, D4⋊C829C2, C4⋊C845C22, (C4×C8)⋊50C22, (C4×D4).18C4, C22.21C4≀C2, C42.64(C2×C4), (C4×M4(2))⋊15C2, (C22×D4).25C4, (C22×C4).658D4, C4.22(C2×M4(2)), C4.134(C8⋊C22), C42.6C426C2, (C4×D4).267C22, (C2×C42).167C22, C23.171(C22⋊C4), C2.16(C24.4C4), C2.5(C23.37D4), (C2×C4×D4).8C2, C2.10(C2×C4≀C2), (C2×C4⋊C4).42C4, C4⋊C4.183(C2×C4), (C2×D4).195(C2×C4), (C2×C4).1139(C2×D4), (C2×C4).80(C22⋊C4), (C2×C4).316(C22×C4), (C22×C4).189(C2×C4), C22.166(C2×C22⋊C4), 2-Sylow(CO-(4,5)), SmallGroup(128,222)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D45M4(2)
C1C2C22C2×C4C42C2×C42C2×C4×D4 — D45M4(2)
C1C2C2×C4 — D45M4(2)
C1C2×C4C2×C42 — D45M4(2)
C1C22C22C42 — D45M4(2)

Generators and relations for D45M4(2)
 G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a-1b, bd=db, dcd=c5 >

Subgroups: 340 in 158 conjugacy classes, 52 normal (26 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×6], C22, C22 [×2], C22 [×18], C8 [×6], C2×C4 [×6], C2×C4 [×18], D4 [×4], D4 [×6], C23, C23 [×10], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×9], C2×D4 [×2], C2×D4 [×5], C24, C4×C8 [×2], C8⋊C4 [×2], C22⋊C8, C4⋊C8 [×2], C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4 [×4], C4×D4 [×2], C2×M4(2), C23×C4, C22×D4, D4⋊C8 [×4], C4×M4(2), C42.6C4, C2×C4×D4, D45M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], C4≀C2 [×2], C2×C22⋊C4, C2×M4(2) [×2], C8⋊C22 [×2], C24.4C4, C23.37D4, C2×C4≀C2, D45M4(2)

Smallest permutation representation of D45M4(2)
On 32 points
Generators in S32
(1 12 32 19)(2 13 25 20)(3 14 26 21)(4 15 27 22)(5 16 28 23)(6 9 29 24)(7 10 30 17)(8 11 31 18)
(1 28)(2 24)(3 7)(4 11)(5 32)(6 20)(8 15)(9 25)(10 21)(12 16)(13 29)(14 17)(18 27)(19 23)(22 31)(26 30)
(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 32)(2 29)(3 26)(4 31)(5 28)(6 25)(7 30)(8 27)(9 20)(10 17)(11 22)(12 19)(13 24)(14 21)(15 18)(16 23)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4J4K···4P8A···8H8I8J8K8L
order122222222244444···44···48···88888
size111122444411112···24···44···48888

38 irreducible representations

dim1111111122224
type++++++++
imageC1C2C2C2C2C4C4C4D4D4M4(2)C4≀C2C8⋊C22
kernelD45M4(2)D4⋊C8C4×M4(2)C42.6C4C2×C4×D4C2×C4⋊C4C4×D4C22×D4C42C22×C4D4C22C4
# reps1411124222882

Matrix representation of D45M4(2) in GL4(𝔽17) generated by

16000
01600
00016
0010
,
16000
0100
00160
0001
,
0100
4000
0066
00116
,
1000
01600
00160
00016
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,1,0,0,16,0],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1],[0,4,0,0,1,0,0,0,0,0,6,11,0,0,6,6],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16] >;

D45M4(2) in GAP, Magma, Sage, TeX

D_4\rtimes_5M_4(2)
% in TeX

G:=Group("D4:5M4(2)");
// GroupNames label

G:=SmallGroup(128,222);
// by ID

G=gap.SmallGroup(128,222);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,387,184,1123,570,136,172]);
// Polycyclic

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

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