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G = C42.187D4  order 128 = 27

169th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.187D4, C24.367C23, C23.528C24, C22.3052+ 1+4, C22.2232- 1+4, (C22×C4)⋊13Q8, C42⋊8C4⋊52C2, C23.67(C2×Q8), C4.86(C22⋊Q8), C23⋊Q8.15C2, (C22×C4).138C23, (C23×C4).430C22, (C2×C42).605C22, C22.353(C22×D4), C23.7Q8.58C2, C23.4Q8.15C2, C22.133(C22×Q8), (C22×Q8).155C22, C23.83C23⋊60C2, C23.78C23⋊27C2, C2.39(C22.29C24), C23.65C23⋊104C2, C2.C42.253C22, C2.47(C22.36C24), C2.20(C23.41C23), C2.39(C23.38C23), (C2×C4⋊Q8)⋊17C2, (C2×C4).387(C2×D4), (C2×C4).130(C2×Q8), C2.43(C2×C22⋊Q8), (C2×C4).660(C4○D4), (C2×C4⋊C4).357C22, C22.400(C2×C4○D4), (C2×C42⋊C2).47C2, (C2×C22⋊C4).217C22, SmallGroup(128,1360)

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

Derived series
C1 — C23 — C42.187D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C23.7Q8 — C42.187D4
Lower central
C1 — C23 — C42.187D4
Upper central
C1 — C23 — C42.187D4
Jennings
C1 — C23 — C42.187D4

Generators and relations for C42.187D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b2, dad=ab2, cbc-1=b-1, bd=db, dcd=a2c-1 >

Subgroups: 452 in 244 conjugacy classes, 108 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊Q8, C23×C4, C22×Q8, C23.7Q8, C42⋊8C4, C23.65C23, C23⋊Q8, C23.78C23, C23.4Q8, C23.83C23, C2×C42⋊C2, C2×C4⋊Q8, C42.187D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C22⋊Q8, C22.29C24, C23.38C23, C22.36C24, C23.41C23, C42.187D4

Smallest permutation representation of C42.187D4
►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 43 39 11)(2 44 40 12)(3 41 37 9)(4 42 38 10)(5 13 17 45)(6 14 18 46)(7 15 19 47)(8 16 20 48)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 36 57 61)(30 33 58 62)(31 34 59 63)(32 35 60 64)

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E···4N4O···4V
order12···22244444···44···4
size11···14422224···48···8

32 irreducible representations

dim111111111122244
type+++++++++++-+-
imageC1C2C2C2C2C2C2C2C2C2D4Q8C4○D42+ 1+42- 1+4
kernelC42.187D4C23.7Q8C42⋊8C4C23.65C23C23⋊Q8C23.78C23C23.4Q8C23.83C23C2×C42⋊C2C2×C4⋊Q8C42C22×C4C2×C4C22C22
# reps121222221144422

Matrix representation of C42.187D4 ►in GL8(𝔽5)

40000000
04000000
00200000
00030000
00000010
00000001
00004000
00000400
,
40000000
04000000
00100000
00010000
00002000
00000300
00000020
00000003
,
04000000
10000000
00040000
00100000
00000100
00004000
00000001
00000040
,
10000000
04000000
00400000
00040000
00001000
00000100
00000040
00000004

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

C42.187D4 in GAP, Magma, Sage, TeX 

C_4^2._{187}D_4
 
% in TeX
 
G:=Group("C4^2.187D4");
 
// GroupNames label
 
G:=SmallGroup(128,1360);
 
// by ID
 
G=gap.SmallGroup(128,1360);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,184,185,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d=a^2*c^-1>;
 
// generators/relations
 

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