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G = C42⋊28D4  order 128 = 27

22nd semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C42⋊28D4, C24.363C23, C23.520C24, C22.2172- (1+4), C22.2982+ (1+4), C42⋊9C4⋊30C2, C23.192(C2×D4), (C22×C4).399D4, C4.100(C4⋊D4), C23.Q8⋊37C2, C23.10D4⋊57C2, (C23×C4).423C22, (C2×C42).601C22, (C22×C4).130C23, C22.345(C22×D4), C24.3C22⋊65C2, (C22×D4).192C22, (C22×Q8).151C22, C23.67C23⋊70C2, C2.35(C22.29C24), C2.C42.247C22, C2.24(C22.31C24), C2.44(C22.36C24), C2.34(C23.38C23), (C2×C4).380(C2×D4), C2.44(C2×C4⋊D4), (C2×C22⋊Q8)⋊27C2, (C2×C4.4D4)⋊20C2, (C2×C4⋊D4).39C2, (C2×C42⋊C2)⋊37C2, (C2×C4).656(C4○D4), (C2×C4⋊C4).887C22, C22.392(C2×C4○D4), (C2×C22⋊C4).212C22, SmallGroup(128,1352)

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

Derived series
C1 — C23 — C42⋊28D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C22×D4 — C2×C4⋊D4 — C42⋊28D4
Lower central
C1 — C23 — C42⋊28D4
Upper central
C1 — C23 — C42⋊28D4
Jennings
C1 — C23 — C42⋊28D4

Subgroups: 628 in 306 conjugacy classes, 108 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×14], C22 [×3], C22 [×4], C22 [×24], C2×C4 [×12], C2×C4 [×38], D4 [×12], Q8 [×4], C23, C23 [×2], C23 [×20], C42 [×4], C42 [×2], C22⋊C4 [×24], C4⋊C4 [×16], C22×C4 [×2], C22×C4 [×14], C22×C4 [×4], C2×D4 [×14], C2×Q8 [×6], C24, C24 [×2], C2.C42 [×4], C2×C42 [×2], C2×C22⋊C4 [×14], C2×C4⋊C4 [×2], C2×C4⋊C4 [×8], C42⋊C2 [×4], C4⋊D4 [×4], C22⋊Q8 [×4], C4.4D4 [×4], C23×C4, C22×D4, C22×D4 [×2], C22×Q8, C42⋊9C4, C24.3C22, C23.67C23, C23.10D4 [×4], C23.Q8 [×4], C2×C42⋊C2, C2×C4⋊D4, C2×C22⋊Q8, C2×C4.4D4, C42⋊28D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, 2+ (1+4) [×2], 2- (1+4) [×2], C2×C4⋊D4, C22.29C24, C23.38C23, C22.31C24 [×2], C22.36C24 [×2], C42⋊28D4

Generators and relations
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1, dad=ab2, cbc-1=b-1, bd=db, dcd=c-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 9 47)(2 16 10 48)(3 13 11 45)(4 14 12 46)(5 62 37 35)(6 63 38 36)(7 64 39 33)(8 61 40 34)(17 41 49 23)(18 42 50 24)(19 43 51 21)(20 44 52 22)(25 57 55 31)(26 58 56 32)(27 59 53 29)(28 60 54 30)

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

12000000
44000000
00400000
00040000
00001030
00000103
00000040
00000004
,
40000000
04000000
00100000
00010000
00000100
00004000
00000001
00000040
,
24000000
03000000
00010000
00400000
00004000
00000100
00000040
00000001
,
24000000
33000000
00010000
00100000
00004000
00000400
00004010
00000401

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4N4O···4T
order12···2222244444···44···4
size11···1448822224···48···8

32 irreducible representations

dim111111111122244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2D4D4C4○D42+ (1+4)2- (1+4)
kernelC42⋊28D4C42⋊9C4C24.3C22C23.67C23C23.10D4C23.Q8C2×C42⋊C2C2×C4⋊D4C2×C22⋊Q8C2×C4.4D4C42C22×C4C2×C4C22C22
# reps111144111144422

In GAP, Magma, Sage, TeX 

C_4^2\rtimes_{28}D_4
 
% in TeX
 
G:=Group("C4^2:28D4");
 
// GroupNames label
 
G:=SmallGroup(128,1352);
 
// by ID
 
G=gap.SmallGroup(128,1352);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,758,723,185,80]);
 
// 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,d*a*d=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

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