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G = C8.C22:C4  order 128 = 27

2nd semidirect product of C8.C22 and C4 acting via C4/C2=C2

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

Aliases: C4oD4.5D4, C4.64(C4xD4), C4:C4.314D4, C8.C22:2C4, C42:6C4:2C2, (C2xD4).275D4, M4(2):4(C2xC4), (C2xQ8).216D4, C22.36(C4xD4), (C22xC4).23D4, D4.4(C22:C4), C23.558(C2xD4), Q8.4(C22:C4), C4.130(C4:D4), C22.C42:3C2, C2.4(D4.9D4), C22.97C22wrC2, C2.4(D4.10D4), C23.38D4:21C2, C22.43(C4:D4), (C22xC4).680C23, (C2xC42).281C22, (C2xM4(2)).7C22, (C22xQ8).14C22, C42:C2.17C22, C23.67C23:3C2, C4.67(C22.D4), C2.21(C23.23D4), C23.33C23.3C2, (C2xQ8):7(C2xC4), (C2xC4wrC2).9C2, C4oD4.8(C2xC4), C4.16(C2xC22:C4), (C2xC4).1001(C2xD4), (C2xC4:C4).58C22, (C2xC4).10(C22xC4), (C2xC8.C22).1C2, (C2xC4).319(C4oD4), (C2xC4oD4).17C22, SmallGroup(128,614)

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

Derived series
C1 — C2xC4 — C8.C22:C4
Chief series
C1 — C2 — C4 — C2xC4 — C22xC4 — C2xC4oD4 — C23.33C23 — C8.C22:C4
Lower central
C1 — C2 — C2xC4 — C8.C22:C4
Upper central
C1 — C22 — C22xC4 — C8.C22:C4
Jennings
C1 — C2 — C2 — C22xC4 — C8.C22:C4

Generators and relations for C8.C22:C4
 G = < a,b,c,d | a8=b2=c2=d4=1, bab=a3, cac=a5, dad-1=a-1c, cbc=a4b, bd=db, dcd-1=a4c >

Subgroups: 332 in 167 conjugacy classes, 56 normal (38 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C42, C22:C4, C4:C4, C4:C4, C2xC8, M4(2), M4(2), SD16, Q16, C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C2xQ8, C4oD4, C4oD4, C2.C42, Q8:C4, C4wrC2, C2xC42, C2xC4:C4, C2xC4:C4, C42:C2, C42:C2, C4xD4, C4xQ8, C2xM4(2), C2xSD16, C2xQ16, C8.C22, C8.C22, C22xQ8, C2xC4oD4, C42:6C4, C22.C42, C23.67C23, C23.38D4, C2xC4wrC2, C23.33C23, C2xC8.C22, C8.C22:C4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, C22xC4, C2xD4, C4oD4, C2xC22:C4, C4xD4, C22wrC2, C4:D4, C22.D4, C23.23D4, D4.9D4, D4.10D4, C8.C22:C4

Smallest permutation representation of C8.C22: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 20)(2 23)(3 18)(4 21)(5 24)(6 19)(7 22)(8 17)(9 28)(10 31)(11 26)(12 29)(13 32)(14 27)(15 30)(16 25)

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

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

32 conjugacy classes

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

32 irreducible representations

dim11111111122222244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C4D4D4D4D4D4C4oD4D4.9D4D4.10D4
kernelC8.C22:C4C42:6C4C22.C42C23.67C23C23.38D4C2xC4wrC2C23.33C23C2xC8.C22C8.C22C4:C4C22xC4C2xD4C2xQ8C4oD4C2xC4C2C2
# reps11111111822112422

Matrix representation of C8.C22:C4 ►in GL6(F17)

1620000
1610000
0002013
000080
000900
001040
,
1620000
010000
0000130
000409
004000
0004013
,
100000
010000
001000
000100
0000160
0001016
,
1380000
040000
0001504
000090
0001500
0040130

G:=sub<GL(6,GF(17))| [16,16,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,1,0,0,2,0,9,0,0,0,0,8,0,4,0,0,13,0,0,0],[16,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0,4,0,0,13,0,0,0,0,0,0,9,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,16,0,0,0,0,0,0,16],[13,0,0,0,0,0,8,4,0,0,0,0,0,0,0,0,0,4,0,0,15,0,15,0,0,0,0,9,0,13,0,0,4,0,0,0] >;
 

C8.C22:C4 in GAP, Magma, Sage, TeX 

C_8.C_2^2\rtimes C_4
 
% in TeX
 
G:=Group("C8.C2^2:C4");
 
// GroupNames label
 
G:=SmallGroup(128,614);
 
// by ID
 
G=gap.SmallGroup(128,614);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,1018,248,2804,718,172,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^2=d^4=1,b*a*b=a^3,c*a*c=a^5,d*a*d^-1=a^-1*c,c*b*c=a^4*b,b*d=d*b,d*c*d^-1=a^4*c>;
 
// generators/relations
 

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