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G = C2×D4⋊D4  order 128 = 27

Direct product of C2 and D4⋊D4

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

Aliases: C2×D4⋊D4, C24.101D4, D4⋊7(C2×D4), Q8⋊7(C2×D4), (C2×D4)⋊48D4, (C2×Q8)⋊35D4, (C22×D8)⋊6C2, C4⋊C4.3C23, C4.37C22≀C2, (C2×D8)⋊36C22, C4.37(C22×D4), C4⋊D4⋊48C22, C22⋊C8⋊56C22, (C2×C8).127C23, (C2×C4).219C24, (C2×D4).25C23, C23.850(C2×D4), (C22×C4).712D4, D4⋊C4⋊70C22, Q8⋊C4⋊64C22, (C2×SD16)⋊67C22, (C22×SD16)⋊17C2, C22.86(C4○D8), (C2×Q8).352C23, C22.116C22≀C2, (C22×C8).134C22, (C23×C4).539C22, (C22×C4).957C23, C22.479(C22×D4), C22.111(C8⋊C22), (C22×D4).325C22, (C22×Q8).465C22, C2.7(C2×C4○D8), C2.9(C2×C8⋊C22), (C2×C4⋊D4)⋊44C2, (C2×C22⋊C8)⋊23C2, (C22×C4○D4)⋊7C2, (C2×D4⋊C4)⋊36C2, (C2×Q8⋊C4)⋊22C2, C2.37(C2×C22≀C2), (C2×C4).1093(C2×D4), (C2×C4○D4)⋊63C22, (C2×C4⋊C4).580C22, SmallGroup(128,1732)

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

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

Subgroups: 844 in 406 conjugacy classes, 116 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×6], C22 [×36], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×36], D4 [×4], D4 [×36], Q8 [×4], Q8 [×6], C23, C23 [×2], C23 [×24], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], C2×C8 [×4], D8 [×8], SD16 [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×18], C2×D4 [×8], C2×D4 [×28], C2×Q8 [×6], C2×Q8 [×3], C4○D4 [×32], C24, C24 [×2], C22⋊C8 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C2×C22⋊C4, C2×C4⋊C4, C4⋊D4 [×4], C4⋊D4 [×2], C22×C8 [×2], C2×D8 [×4], C2×D8 [×4], C2×SD16 [×4], C2×SD16 [×4], C23×C4, C23×C4, C22×D4 [×2], C22×D4 [×2], C22×Q8, C2×C4○D4 [×4], C2×C4○D4 [×10], C2×C22⋊C8, C2×D4⋊C4, C2×Q8⋊C4, D4⋊D4 [×8], C2×C4⋊D4, C22×D8, C22×SD16, C22×C4○D4, C2×D4⋊D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C4○D8 [×2], C8⋊C22 [×2], C22×D4 [×3], D4⋊D4 [×4], C2×C22≀C2, C2×C4○D8, C2×C8⋊C22, C2×D4⋊D4

Generators and relations
 G = < a,b,c,d,e | a2=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=d-1 >

Smallest permutation representation
►On 64 points
Generators in S64
(1 17)(2 18)(3 19)(4 20)(5 54)(6 55)(7 56)(8 53)(9 14)(10 15)(11 16)(12 13)(21 35)(22 36)(23 33)(24 34)(25 30)(26 31)(27 32)(28 29)(37 49)(38 50)(39 51)(40 52)(41 45)(42 46)(43 47)(44 48)(57 61)(58 62)(59 63)(60 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 51)(2 50)(3 49)(4 52)(5 23)(6 22)(7 21)(8 24)(9 47)(10 46)(11 45)(12 48)(13 44)(14 43)(15 42)(16 41)(17 39)(18 38)(19 37)(20 40)(25 63)(26 62)(27 61)(28 64)(29 60)(30 59)(31 58)(32 57)(33 54)(34 53)(35 56)(36 55)

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

100000
010000
0016000
0001600
0000160
0000016
,
1600000
0160000
0016000
0001600
000001
0000160
,
010000
100000
0016200
000100
0000314
00001414
,
0160000
100000
001000
0011600
0000013
0000130
,
100000
0160000
001000
0011600
000010
0000016

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,2,1,0,0,0,0,0,0,3,14,0,0,0,0,14,14],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,13,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;
 

38 conjugacy classes

class 1 2A···2G2H···2M2N2O4A···4H4I4J4K4L4M4N8A···8H
order12···22···2224···44444448···8
size11···14···4882···24444884···4

38 irreducible representations

dim111111111222224
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4D4D4C4○D8C8⋊C22
kernelC2×D4⋊D4C2×C22⋊C8C2×D4⋊C4C2×Q8⋊C4D4⋊D4C2×C4⋊D4C22×D8C22×SD16C22×C4○D4C22×C4C2×D4C2×Q8C24C22C22
# reps111181111344182

In GAP, Magma, Sage, TeX 

C_2\times D_4\rtimes D_4
 
% in TeX
 
G:=Group("C2xD4:D4");
 
// GroupNames label
 
G:=SmallGroup(128,1732);
 
// by ID
 
G=gap.SmallGroup(128,1732);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,2804,1411,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b^-1*c,e*c*e=b*c,e*d*e=d^-1>;
 
// generators/relations
 

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