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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, C22.2302- (1+4), C22.3132+ (1+4), C42⋊8C4⋊54C2, C23⋊Q8.16C2, (C22×C4).147C23, (C2×C42).614C22, C22.362(C22×D4), C23.Q8.20C2, C23.11D4.28C2, (C22×Q8).451C22, C23.81C23⋊64C2, C23.78C23⋊29C2, C2.88(C22.19C24), C24.C22.43C2, C23.63C23⋊113C2, C2.C42.262C22, C2.30(C22.31C24), C2.28(C22.35C24), C2.50(C22.36C24), 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

Subgroups: 404 in 225 conjugacy classes, 96 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2, C4 [×19], C22 [×3], C22 [×4], C22 [×7], C2×C4 [×10], C2×C4 [×37], Q8 [×8], C23, C23 [×7], C42 [×4], C42 [×4], C22⋊C4 [×12], C4⋊C4 [×20], C22×C4 [×6], C22×C4 [×8], C2×Q8 [×6], C24, C2.C42 [×2], C2.C42 [×10], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×3], C2×C22⋊C4 [×4], C2×C4⋊C4 [×4], C2×C4⋊C4 [×8], C4×Q8 [×4], C42⋊2C2 [×4], C22×Q8, C42⋊8C4, C23.63C23 [×2], C24.C22 [×2], C23⋊Q8, C23.78C23, C23.Q8 [×2], C23.11D4, C23.81C23, C23.81C23 [×2], C2×C4×Q8, C2×C42⋊2C2, C42.192D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ (1+4), 2- (1+4) [×3], C22.19C24, C23.38C23, C22.31C24, C22.35C24 [×2], C22.36C24 [×2], C42.192D4

Generators and relations
 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 >

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

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

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

Matrix representation ►G ⊆ 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] >;
 

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+4)2- (1+4)
kernelC42.192D4C42⋊8C4C23.63C23C24.C22C23⋊Q8C23.78C23C23.Q8C23.11D4C23.81C23C2×C4×Q8C2×C42⋊2C2C42C2×C4C22C22
# reps112211213114813

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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