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G = C42.196D4  order 128 = 27

178th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.196D4, C24.53C23, C23.558C24, C22.2482- 1+4, C4.8(C4⋊1D4), C42⋊4C4⋊34C2, C23.4Q8⋊36C2, (C2×C42).623C22, (C22×C4).857C23, C22.370(C22×D4), (C22×D4).208C22, (C22×Q8).165C22, C2.C42.561C22, C2.47(C23.38C23), (C2×C4⋊Q8)⋊19C2, (C2×C4).404(C2×D4), C2.15(C2×C4⋊1D4), (C2×C4⋊C4).381C22, (C2×C4.4D4).29C2, (C2×C22⋊C4).238C22, SmallGroup(128,1390)

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

Derived series
C1 — C23 — C42.196D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C22×D4 — C2×C4.4D4 — C42.196D4
Lower central
C1 — C23 — C42.196D4
Upper central
C1 — C23 — C42.196D4
Jennings
C1 — C23 — C42.196D4

Generators and relations for C42.196D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 580 in 306 conjugacy classes, 116 normal (7 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4.4D4, C4⋊Q8, C22×D4, C22×Q8, C42⋊4C4, C23.4Q8, C2×C4.4D4, C2×C4⋊Q8, C42.196D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C4⋊1D4, C22×D4, 2- 1+4, C2×C4⋊1D4, C23.38C23, C42.196D4

Smallest permutation representation of C42.196D4
►On 64 points
Generators in S64
(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)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)

(1 49 34 54)(2 50 35 55)(3 51 36 56)(4 52 33 53)(5 21 61 20)(6 22 62 17)(7 23 63 18)(8 24 64 19)(9 43 14 27)(10 44 15 28)(11 41 16 25)(12 42 13 26)(29 48 37 57)(30 45 38 58)(31 46 39 59)(32 47 40 60)

(1 28 5 59)(2 41 6 47)(3 26 7 57)(4 43 8 45)(9 19 30 53)(10 21 31 49)(11 17 32 55)(12 23 29 51)(13 18 37 56)(14 24 38 52)(15 20 39 54)(16 22 40 50)(25 62 60 35)(27 64 58 33)(34 44 61 46)(36 42 63 48)

(1 46 34 59)(2 45 35 58)(3 48 36 57)(4 47 33 60)(5 44 61 28)(6 43 62 27)(7 42 63 26)(8 41 64 25)(9 17 14 22)(10 20 15 21)(11 19 16 24)(12 18 13 23)(29 56 37 51)(30 55 38 50)(31 54 39 49)(32 53 40 52)
 
G:=sub<Sym(64)| (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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,49,34,54)(2,50,35,55)(3,51,36,56)(4,52,33,53)(5,21,61,20)(6,22,62,17)(7,23,63,18)(8,24,64,19)(9,43,14,27)(10,44,15,28)(11,41,16,25)(12,42,13,26)(29,48,37,57)(30,45,38,58)(31,46,39,59)(32,47,40,60), (1,28,5,59)(2,41,6,47)(3,26,7,57)(4,43,8,45)(9,19,30,53)(10,21,31,49)(11,17,32,55)(12,23,29,51)(13,18,37,56)(14,24,38,52)(15,20,39,54)(16,22,40,50)(25,62,60,35)(27,64,58,33)(34,44,61,46)(36,42,63,48), (1,46,34,59)(2,45,35,58)(3,48,36,57)(4,47,33,60)(5,44,61,28)(6,43,62,27)(7,42,63,26)(8,41,64,25)(9,17,14,22)(10,20,15,21)(11,19,16,24)(12,18,13,23)(29,56,37,51)(30,55,38,50)(31,54,39,49)(32,53,40,52)>;
 
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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,49,34,54)(2,50,35,55)(3,51,36,56)(4,52,33,53)(5,21,61,20)(6,22,62,17)(7,23,63,18)(8,24,64,19)(9,43,14,27)(10,44,15,28)(11,41,16,25)(12,42,13,26)(29,48,37,57)(30,45,38,58)(31,46,39,59)(32,47,40,60), (1,28,5,59)(2,41,6,47)(3,26,7,57)(4,43,8,45)(9,19,30,53)(10,21,31,49)(11,17,32,55)(12,23,29,51)(13,18,37,56)(14,24,38,52)(15,20,39,54)(16,22,40,50)(25,62,60,35)(27,64,58,33)(34,44,61,46)(36,42,63,48), (1,46,34,59)(2,45,35,58)(3,48,36,57)(4,47,33,60)(5,44,61,28)(6,43,62,27)(7,42,63,26)(8,41,64,25)(9,17,14,22)(10,20,15,21)(11,19,16,24)(12,18,13,23)(29,56,37,51)(30,55,38,50)(31,54,39,49)(32,53,40,52) );
 
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),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,49,34,54),(2,50,35,55),(3,51,36,56),(4,52,33,53),(5,21,61,20),(6,22,62,17),(7,23,63,18),(8,24,64,19),(9,43,14,27),(10,44,15,28),(11,41,16,25),(12,42,13,26),(29,48,37,57),(30,45,38,58),(31,46,39,59),(32,47,40,60)], [(1,28,5,59),(2,41,6,47),(3,26,7,57),(4,43,8,45),(9,19,30,53),(10,21,31,49),(11,17,32,55),(12,23,29,51),(13,18,37,56),(14,24,38,52),(15,20,39,54),(16,22,40,50),(25,62,60,35),(27,64,58,33),(34,44,61,46),(36,42,63,48)], [(1,46,34,59),(2,45,35,58),(3,48,36,57),(4,47,33,60),(5,44,61,28),(6,43,62,27),(7,42,63,26),(8,41,64,25),(9,17,14,22),(10,20,15,21),(11,19,16,24),(12,18,13,23),(29,56,37,51),(30,55,38,50),(31,54,39,49),(32,53,40,52)]])
 

32 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E···4P4Q···4V
order12···22244444···44···4
size11···18822224···48···8

32 irreducible representations

dim1111124
type++++++-
imageC1C2C2C2C2D42- 1+4
kernelC42.196D4C42⋊4C4C23.4Q8C2×C4.4D4C2×C4⋊Q8C42C22
# reps11833124

Matrix representation of C42.196D4 ►in GL8(𝔽5)

10000000
01000000
00040000
00100000
00000002
00000030
00000300
00002000
,
40000000
04000000
00100000
00010000
00000400
00001000
00000004
00000010
,
01000000
40000000
00100000
00010000
00000010
00000001
00001000
00000100
,
04000000
40000000
00100000
00040000
00000010
00000004
00004000
00000100

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0],[0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0] >;
 

C42.196D4 in GAP, Magma, Sage, TeX 

C_4^2._{196}D_4
 
% in TeX
 
G:=Group("C4^2.196D4");
 
// GroupNames label
 
G:=SmallGroup(128,1390);
 
// by ID
 
G=gap.SmallGroup(128,1390);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,568,758,723,185,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
 
// generators/relations
 

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