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G = C42.192D4  order 128 = 27

174th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.192D4, C24.41C23, C23.537C24, C22.3132+ 1+4, C22.2302- 1+4, C42⋊8C4⋊54C2, C23⋊Q8.16C2, (C2×C42).614C22, (C22×C4).147C23, C22.362(C22×D4), C23.Q8.20C2, C23.11D4.28C2, (C22×Q8).451C22, C23.78C23⋊29C2, C23.81C23⋊64C2, C2.88(C22.19C24), C24.C22.43C2, C23.63C23⋊113C2, C2.C42.262C22, C2.28(C22.35C24), C2.50(C22.36C24), C2.30(C22.31C24), C2.43(C23.38C23), (C2×C4×Q8)⋊30C2, (C2×C4).396(C2×D4), (C2×C4).171(C4○D4), (C2×C4⋊C4).363C22, C22.409(C2×C4○D4), (C2×C42⋊2C2).10C2, (C2×C22⋊C4).225C22, SmallGroup(128,1369)

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

Derived series
C1 — C23 — C42.192D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C22⋊C4 — C24.C22 — C42.192D4
Lower central
C1 — C23 — C42.192D4
Upper central
C1 — C23 — C42.192D4
Jennings
C1 — C23 — C42.192D4

Generators and relations for C42.192D4
 G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=a2c-1 >

Subgroups: 404 in 225 conjugacy classes, 96 normal (34 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C24, C2.C42, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C42⋊2C2, C22×Q8, C42⋊8C4, C23.63C23, C24.C22, C23⋊Q8, C23.78C23, C23.Q8, C23.11D4, C23.81C23, C23.81C23, C2×C4×Q8, C2×C42⋊2C2, C42.192D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.19C24, C23.38C23, C22.31C24, C22.35C24, C22.36C24, C42.192D4

Smallest permutation representation of C42.192D4
►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 15 28 47)(2 16 25 48)(3 13 26 45)(4 14 27 46)(5 50 38 18)(6 51 39 19)(7 52 40 20)(8 49 37 17)(9 59 41 30)(10 60 42 31)(11 57 43 32)(12 58 44 29)(21 33 53 63)(22 34 54 64)(23 35 55 61)(24 36 56 62)

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

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

32 conjugacy classes

class 1 2A···2G2H4A4B4C4D4E···4P4Q···4W
order12···2244444···44···4
size11···1822224···48···8

32 irreducible representations

dim111111111112244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC42.192D4C42⋊8C4C23.63C23C24.C22C23⋊Q8C23.78C23C23.Q8C23.11D4C23.81C23C2×C4×Q8C2×C42⋊2C2C42C2×C4C22C22
# reps112211213114813

Matrix representation of C42.192D4 ►in GL8(𝔽5)

01000000
10000000
00100000
00010000
00000040
00000004
00001000
00000100
,
30000000
03000000
00100000
00010000
00000400
00004000
00000004
00000040
,
04000000
10000000
00010000
00400000
00000300
00002000
00000003
00000020
,
01000000
10000000
00010000
00100000
00000300
00003000
00000002
00000020

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

C42.192D4 in GAP, Magma, Sage, TeX 

C_4^2._{192}D_4
 
% in TeX
 
G:=Group("C4^2.192D4");
 
// GroupNames label
 
G:=SmallGroup(128,1369);
 
// by ID
 
G=gap.SmallGroup(128,1369);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,185,192]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=a^2*c^-1>;
 
// generators/relations
 

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