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G = C42⋊9(C2×C4)  order 128 = 27

4th semidirect product of C42 and C2×C4 acting via C2×C4/C2=C22

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

Aliases: C4≀C2⋊2C4, C42⋊9(C2×C4), C4○D4.2Q8, D4.5(C4⋊C4), C4⋊C4.303D4, C4.141(C4×D4), Q8.5(C4⋊C4), C42⋊8C4⋊4C2, (C2×D4).272D4, M4(2)⋊3(C2×C4), (C2×Q8).213D4, (C22×C4).21D4, C22.31(C4×D4), C42⋊6C4⋊17C2, C4.5(C22⋊Q8), C23.555(C2×D4), C22.C42⋊2C2, M4(2)⋊C4⋊3C2, C2.3(D4.9D4), C2.3(D4.8D4), C22.83C22≀C2, (C2×C42).266C22, (C22×C4).676C23, C23.36D4.8C2, C4.5(C22.D4), C22.50(C22⋊Q8), C42⋊C2.11C22, C2.16(C23.8Q8), (C2×M4(2)).174C22, C23.33C23.2C2, C4.10(C2×C4⋊C4), (C2×C4≀C2).2C2, C4○D4.5(C2×C4), (C2×C4).10(C2×Q8), (C2×C4).986(C2×D4), (C2×C4).51(C4○D4), (C2×C4⋊C4).57C22, (C2×C4).182(C22×C4), (C2×C4○D4).12C22, SmallGroup(128,592)

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

Derived series
C1 — C2×C4 — C42⋊9(C2×C4)
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C23.33C23 — C42⋊9(C2×C4)
Lower central
C1 — C2 — C2×C4 — C42⋊9(C2×C4)
Upper central
C1 — C22 — C22×C4 — C42⋊9(C2×C4)
Jennings
C1 — C2 — C2 — C22×C4 — C42⋊9(C2×C4)

Generators and relations for C42⋊9(C2×C4)
 G = < a,b,c,d | a4=b4=c2=d4=1, cac=ab=ba, dad-1=a-1b, cbc=b-1, bd=db, cd=dc >

Subgroups: 284 in 143 conjugacy classes, 56 normal (38 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C2.C42, D4⋊C4, Q8⋊C4, C4≀C2, C4.Q8, C2.D8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C2×M4(2), C2×C4○D4, C42⋊6C4, C22.C42, C42⋊8C4, C23.36D4, C2×C4≀C2, M4(2)⋊C4, C23.33C23, C42⋊9(C2×C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C23.8Q8, D4.8D4, D4.9D4, C42⋊9(C2×C4)

Smallest permutation representation of C42⋊9(C2×C4)
►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 5 16 3)(2 6 15 4)(7 14 11 10)(8 13 12 9)(17 32 19 30)(18 29 20 31)(21 26 23 28)(22 27 24 25)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4R4S4T8A8B8C8D
order1222222244444···4448888
size1111224422224···4888888

32 irreducible representations

dim11111111122222244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C4D4D4D4D4Q8C4○D4D4.8D4D4.9D4
kernelC42⋊9(C2×C4)C42⋊6C4C22.C42C42⋊8C4C23.36D4C2×C4≀C2M4(2)⋊C4C23.33C23C4≀C2C4⋊C4C22×C4C2×D4C2×Q8C4○D4C2×C4C2C2
# reps11111111822112422

Matrix representation of C42⋊9(C2×C4) ►in GL6(𝔽17)

0160000
100000
001000
000400
001040
00110016
,
1600000
0160000
004000
0001300
000440
00100013
,
0160000
1600000
0001600
0016000
0012601
0061210
,
1300000
0130000
0012120
00110015
0071151
00156110

G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,0,1,11,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,10,0,0,0,13,4,0,0,0,0,0,4,0,0,0,0,0,0,13],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,16,12,6,0,0,16,0,6,12,0,0,0,0,0,1,0,0,0,0,1,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,12,11,7,15,0,0,1,0,11,6,0,0,2,0,5,11,0,0,0,15,1,0] >;
 

C42⋊9(C2×C4) in GAP, Magma, Sage, TeX 

C_4^2\rtimes_9(C_2\times C_4)
 
% in TeX
 
G:=Group("C4^2:9(C2xC4)");
 
// GroupNames label
 
G:=SmallGroup(128,592);
 
// by ID
 
G=gap.SmallGroup(128,592);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,352,2019,1018,248,2804,718,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^2=d^4=1,c*a*c=a*b=b*a,d*a*d^-1=a^-1*b,c*b*c=b^-1,b*d=d*b,c*d=d*c>;
 
// generators/relations
 

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