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G = C233M4(2)  order 128 = 27

2nd semidirect product of C23 and M4(2) acting via M4(2)/C4=C22

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

Aliases: C233M4(2), C42.296C23, C4.1672+ 1+4, C89D439C2, C4⋊C851C22, (C4×D4).33C4, C24.84(C2×C4), C8⋊C429C22, C22⋊C845C22, (C2×C4).670C24, (C2×C8).431C23, C42.221(C2×C4), (C22×C8)⋊54C22, (C22×D4).42C4, C24.4C434C2, C42.6C450C2, (C4×D4).298C22, C22.4(C2×M4(2)), C2.27(Q8○M4(2)), (C2×M4(2))⋊44C22, (C23×C4).529C22, C23.228(C22×C4), C22.194(C23×C4), (C2×C42).780C22, C2.18(C22×M4(2)), (C22×C4).1281C23, C2.44(C22.11C24), (C2×C4×D4).76C2, (C2×C4⋊C4).76C4, C4⋊C4.228(C2×C4), (C2×C22⋊C8)⋊45C2, (C2×D4).234(C2×C4), C22⋊C4.76(C2×C4), (C2×C22⋊C4).50C4, (C2×C4).275(C22×C4), (C22×C4).138(C2×C4), SmallGroup(128,1705)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C233M4(2)
C1C2C4C2×C4C22×C4C23×C4C2×C4×D4 — C233M4(2)
C1C22 — C233M4(2)
C1C2×C4 — C233M4(2)
C1C2C2C2×C4 — C233M4(2)

Generators and relations for C233M4(2)
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, eae=ac=ca, ad=da, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d5 >

Subgroups: 388 in 228 conjugacy classes, 134 normal (24 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×9], C22, C22 [×6], C22 [×20], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×19], D4 [×8], C23, C23 [×8], C23 [×6], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], C22×C4 [×5], C22×C4 [×8], C22×C4 [×4], C2×D4 [×4], C2×D4 [×4], C24 [×2], C8⋊C4 [×4], C22⋊C8 [×12], C4⋊C8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C22×C8 [×4], C2×M4(2) [×4], C23×C4 [×2], C22×D4, C2×C22⋊C8 [×2], C24.4C4 [×2], C42.6C4 [×2], C89D4 [×8], C2×C4×D4, C233M4(2)
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], M4(2) [×4], C22×C4 [×14], C24, C2×M4(2) [×6], C23×C4, 2+ 1+4 [×2], C22.11C24, C22×M4(2), Q8○M4(2), C233M4(2)

Smallest permutation representation of C233M4(2)
On 32 points
Generators in S32
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)
(2 28)(4 30)(6 32)(8 26)(9 17)(11 19)(13 21)(15 23)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
(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 21)(10 18)(11 23)(12 20)(13 17)(14 22)(15 19)(16 24)(26 30)(28 32)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I2J2K4A4B4C4D4E···4J4K···4P8A···8P
order12222···22244444···44···48···8
size11112···24411112···24···44···4

44 irreducible representations

dim1111111111244
type+++++++
imageC1C2C2C2C2C2C4C4C4C4M4(2)2+ 1+4Q8○M4(2)
kernelC233M4(2)C2×C22⋊C8C24.4C4C42.6C4C89D4C2×C4×D4C2×C22⋊C4C2×C4⋊C4C4×D4C22×D4C23C4C2
# reps1222814282822

Matrix representation of C233M4(2) in GL6(𝔽17)

1600000
0160000
0011600
0001600
001511616
000001
,
1600000
0160000
001000
000100
00150160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
0160000
400000
00160160
001501516
000010
0021600
,
100000
0160000
001000
0021600
000010
001501516

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,15,0,0,0,16,16,1,0,0,0,0,0,16,0,0,0,0,0,16,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,15,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,4,0,0,0,0,16,0,0,0,0,0,0,0,16,15,0,2,0,0,0,0,0,16,0,0,16,15,1,0,0,0,0,16,0,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,2,0,15,0,0,0,16,0,0,0,0,0,0,1,15,0,0,0,0,0,16] >;

C233M4(2) in GAP, Magma, Sage, TeX

C_2^3\rtimes_3M_4(2)
% in TeX

G:=Group("C2^3:3M4(2)");
// GroupNames label

G:=SmallGroup(128,1705);
// by ID

G=gap.SmallGroup(128,1705);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,219,675,124]);
// Polycyclic

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

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