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G = D4⋊(C4⋊C4)  order 128 = 27

5th semidirect product of D4 and C4⋊C4 acting via C4⋊C4/C2×C4=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: D45(C4⋊C4), C4.55(C4×D4), D4⋊C45C4, C4⋊C4.306D4, (C2×D4).28Q8, C2.8(C4×SD16), (C2×D4).274D4, (C2×C4).87SD16, C2.3(D4.Q8), C2.4(D4⋊D4), C2.6(D8⋊C4), C22.4Q169C2, C2.2(D42Q8), (C22×C4).685D4, C23.762(C2×D4), C22.146(C4×D4), C4.27(C22⋊Q8), C22.86C22≀C2, C2.3(C22⋊SD16), C22.51(C4○D8), C22.54(C2×SD16), C22.72(C8⋊C22), (C2×C42).270C22, (C22×C8).313C22, C4.8(C22.D4), C22.73(C22⋊Q8), C23.65C232C2, C22.7C4226C2, (C22×C4).1355C23, (C22×D4).459C22, C2.20(C23.8Q8), C4⋊C45(C2×C4), (C2×C8)⋊19(C2×C4), C4.14(C2×C4⋊C4), (C2×C4×D4).18C2, (C2×C4.Q8)⋊17C2, (C2×C4).990(C2×D4), (C2×C4).266(C2×Q8), (C2×D4).162(C2×C4), (C2×D4⋊C4).29C2, (C2×C4).751(C4○D4), (C2×C4⋊C4).759C22, (C2×C4).373(C22×C4), SmallGroup(128,596)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D4⋊(C4⋊C4)
C1C2C22C2×C4C22×C4C22×D4C2×C4×D4 — D4⋊(C4⋊C4)
C1C2C2×C4 — D4⋊(C4⋊C4)
C1C23C2×C42 — D4⋊(C4⋊C4)
C1C2C2C22×C4 — D4⋊(C4⋊C4)

Generators and relations for D4⋊(C4⋊C4)
 G = < a,b,c,d | a4=b2=c4=d4=1, bab=cac-1=a-1, ad=da, cbc-1=a-1b, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 396 in 181 conjugacy classes, 66 normal (44 characteristic)
C1, C2 [×7], C2 [×4], C4 [×4], C4 [×8], C22 [×7], C22 [×16], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×24], D4 [×4], D4 [×6], C23, C23 [×10], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×2], C2×C8 [×5], C22×C4 [×3], C22×C4 [×11], C2×D4 [×6], C2×D4 [×3], C24, C2.C42, D4⋊C4 [×4], D4⋊C4 [×2], C4.Q8 [×2], C2×C42, C2×C22⋊C4, C2×C4⋊C4 [×3], C2×C4⋊C4, C4×D4 [×4], C22×C8 [×2], C23×C4, C22×D4, C22.7C42, C22.4Q16, C23.65C23, C2×D4⋊C4 [×2], C2×C4.Q8, C2×C4×D4, D4⋊(C4⋊C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C4⋊C4 [×4], SD16 [×2], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C2×SD16, C4○D8, C8⋊C22 [×2], C23.8Q8, C4×SD16, D8⋊C4, D4⋊D4, C22⋊SD16, D42Q8, D4.Q8, D4⋊(C4⋊C4)

Smallest permutation representation of D4⋊(C4⋊C4)
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 10)(2 9)(3 12)(4 11)(5 64)(6 63)(7 62)(8 61)(13 18)(14 17)(15 20)(16 19)(21 26)(22 25)(23 28)(24 27)(29 36)(30 35)(31 34)(32 33)(37 41)(38 44)(39 43)(40 42)(45 52)(46 51)(47 50)(48 49)(53 60)(54 59)(55 58)(56 57)
(1 64 15 53)(2 63 16 56)(3 62 13 55)(4 61 14 54)(5 17 60 11)(6 20 57 10)(7 19 58 9)(8 18 59 12)(21 52 30 42)(22 51 31 41)(23 50 32 44)(24 49 29 43)(25 47 34 38)(26 46 35 37)(27 45 36 40)(28 48 33 39)
(1 23 11 27)(2 24 12 28)(3 21 9 25)(4 22 10 26)(5 40 64 44)(6 37 61 41)(7 38 62 42)(8 39 63 43)(13 30 19 34)(14 31 20 35)(15 32 17 36)(16 29 18 33)(45 53 50 60)(46 54 51 57)(47 55 52 58)(48 56 49 59)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4N4O4P4Q4R8A···8H
order12···222224···44···444448···8
size11···144442···24···488884···4

38 irreducible representations

dim1111111122222224
type++++++++++-+
imageC1C2C2C2C2C2C2C4D4D4D4Q8SD16C4○D4C4○D8C8⋊C22
kernelD4⋊(C4⋊C4)C22.7C42C22.4Q16C23.65C23C2×D4⋊C4C2×C4.Q8C2×C4×D4D4⋊C4C4⋊C4C22×C4C2×D4C2×D4C2×C4C2×C4C22C22
# reps1111211822224442

Matrix representation of D4⋊(C4⋊C4) in GL5(𝔽17)

10000
016000
001600
00001
000160
,
10000
016000
00100
00001
00010
,
10000
00100
016000
0001212
000125
,
40000
01000
001600
000013
00040

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,1,0],[1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,12,5],[4,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,13,0] >;

D4⋊(C4⋊C4) in GAP, Magma, Sage, TeX

D_4\rtimes (C_4\rtimes C_4)
% in TeX

G:=Group("D4:(C4:C4)");
// GroupNames label

G:=SmallGroup(128,596);
// by ID

G=gap.SmallGroup(128,596);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,680,422,2019,1018,521,248,2804,1411,718,172]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^4=d^4=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^-1*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations

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