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G = D4.3C42  order 128 = 27

3rd non-split extension by D4 of C42 acting via C42/C2×C4=C2

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

Aliases: D4.3C42, Q8.3C42, C4≀C2⋊5C4, C8○D4⋊7C4, C4.164(C4×D4), (C2×C8).194D4, C4.8(C2×C42), D4⋊C4⋊10C4, Q8⋊C4⋊10C4, C22.29(C4×D4), C42⋊6C4⋊12C2, C2.4(C8.26D4), C8.52(C22⋊C4), C42.130(C2×C4), C8○2M4(2)⋊23C2, C4.C42⋊17C2, M4(2).18(C2×C4), C23.200(C4○D4), (C22×C8).382C22, (C2×C42).237C22, C23.24D4.9C2, (C22×C4).1312C23, C22.2(C42⋊C2), C42⋊C2.264C22, (C2×M4(2)).309C22, (C2×C4≀C2).6C2, (C2×C8⋊C4)⋊22C2, C4⋊C4.143(C2×C4), (C2×C8).134(C2×C4), C4○D4.26(C2×C4), (C2×C8○D4).13C2, C2.23(C4×C22⋊C4), (C2×D4).160(C2×C4), (C2×C4).1307(C2×D4), C4.111(C2×C22⋊C4), (C2×Q8).142(C2×C4), (C2×C4).543(C4○D4), (C2×C4).355(C22×C4), (C2×C4○D4).253C22, SmallGroup(128,497)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C4 — D4.3C42
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C2×C8⋊C4 — D4.3C42
Lower central
C1 — C2 — C4 — D4.3C42
Upper central
C1 — C2×C4 — C22×C8 — D4.3C42
Jennings
C1 — C2 — C2 — C22×C4 — D4.3C42

Generators and relations for D4.3C42
 G = < a,b,c,d | a4=b2=c4=1, d4=a2, bab=dad-1=a-1, dcd-1=ac=ca, cbc-1=ab, dbd-1=a2b >

Subgroups: 236 in 146 conjugacy classes, 76 normal (36 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C4×C8, C8⋊C4, D4⋊C4, Q8⋊C4, C4≀C2, C2×C42, C42⋊C2, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C8○D4, C2×C4○D4, C42⋊6C4, C4.C42, C2×C8⋊C4, C8○2M4(2), C23.24D4, C2×C4≀C2, C2×C8○D4, D4.3C42
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C4×C22⋊C4, C8.26D4, D4.3C42

Smallest permutation representation of D4.3C42
►On 32 points
Generators in S32
(1 27 5 31)(2 32 6 28)(3 29 7 25)(4 26 8 30)(9 20 13 24)(10 17 14 21)(11 22 15 18)(12 19 16 23)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G···4P8A···8H8I···8T
order122222224444444···48···88···8
size111122441111224···42···24···4

44 irreducible representations

dim1111111111112224
type+++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4D4C4○D4C4○D4C8.26D4
kernelD4.3C42C42⋊6C4C4.C42C2×C8⋊C4C8○2M4(2)C23.24D4C2×C4≀C2C2×C8○D4D4⋊C4Q8⋊C4C4≀C2C8○D4C2×C8C2×C4C23C2
# reps1111111144884224

Matrix representation of D4.3C42 ►in GL6(𝔽17)

1600000
0160000
004000
000400
0000130
0000013
,
010000
100000
0000016
000040
0001300
0016000
,
100000
0160000
004000
0001300
0000160
000001
,
010000
100000
000010
000001
0013000
0001300

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,13,0,0,0,0,4,0,0,0,0,16,0,0,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

D4.3C42 in GAP, Magma, Sage, TeX 

D_4._3C_4^2
 
% in TeX
 
G:=Group("D4.3C4^2");
 
// GroupNames label
 
G:=SmallGroup(128,497);
 
// by ID
 
G=gap.SmallGroup(128,497);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,100,2019,248,172,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^4=1,d^4=a^2,b*a*b=d*a*d^-1=a^-1,d*c*d^-1=a*c=c*a,c*b*c^-1=a*b,d*b*d^-1=a^2*b>;
 
// generators/relations
 

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