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G = C4○D4.7Q8  order 128 = 27

5th non-split extension by C4○D4 of Q8 acting via Q8/C4=C2

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

Aliases: C8○D4⋊4C4, D4○(C4.Q8), Q8○(C4.Q8), C4○D4.7Q8, D4.9(C4⋊C4), Q8.9(C4⋊C4), (C2×D4).345D4, C4.6(C22×Q8), C8.14(C22×C4), C4.50(C23×C4), (C2×Q8).264D4, C4⋊C4.351C23, M4(2)⋊16(C2×C4), (C2×C4).188C24, (C2×C8).589C23, C2.3(D4○SD16), C23.433(C2×D4), M4(2)⋊C4⋊8C2, C2.D8.210C22, C4.Q8.184C22, C23.25D4⋊21C2, (C22×C4).907C23, (C22×C8).244C22, C22.135(C22×D4), C42⋊C2.78C22, (C2×M4(2)).258C22, C23.33C23.4C2, (C2×C8)⋊9(C2×C4), C4.21(C2×C4⋊C4), (C2×C4.Q8)⋊6C2, (C2×Q8)○(C4.Q8), (C2×C8○D4).6C2, C22.3(C2×C4⋊C4), (C2×C4).96(C2×Q8), C4○D4.32(C2×C4), C2.27(C22×C4⋊C4), (C2×C4).1081(C2×D4), (C2×C4⋊C4).572C22, (C2×C4).251(C22×C4), (C2×C4○D4).277C22, SmallGroup(128,1644)

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

Derived series
C1 — C4 — C4○D4.7Q8
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — C4○D4.7Q8
Lower central
C1 — C2 — C4 — C4○D4.7Q8
Upper central
C1 — C22 — C2×C4○D4 — C4○D4.7Q8
Jennings
C1 — C2 — C2 — C2×C4 — C4○D4.7Q8

Subgroups: 356 in 238 conjugacy classes, 172 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×6], C22 [×6], C8 [×8], C2×C4, C2×C4 [×15], C2×C4 [×14], D4 [×12], Q8 [×4], C23 [×3], C42 [×6], C22⋊C4 [×6], C4⋊C4 [×8], C4⋊C4 [×6], C2×C8, C2×C8 [×15], M4(2) [×12], C22×C4 [×3], C22×C4 [×6], C2×D4 [×3], C2×Q8, C4○D4 [×8], C4.Q8, C4.Q8 [×9], C2.D8 [×6], C2×C4⋊C4 [×6], C42⋊C2 [×6], C4×D4 [×6], C4×Q8 [×2], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×C4○D4, C2×C4.Q8 [×3], C23.25D4 [×3], M4(2)⋊C4 [×6], C23.33C23 [×2], C2×C8○D4, C4○D4.7Q8

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C2×C4⋊C4 [×12], C23×C4, C22×D4, C22×Q8, C22×C4⋊C4, D4○SD16 [×2], C4○D4.7Q8

Generators and relations
 G = < a,b,c,d,e | a4=c2=1, b2=d4=a2, e2=a-1d2, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc=a2b, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

100000
010000
000100
0016000
000001
0000160
,
100000
010000
0001600
001000
000001
0000160
,
1600000
0160000
000001
0000160
0001600
001000
,
1020000
970000
0012500
00121200
0000125
00001212
,
1430000
830000
0000143
000033
0014300
003300

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0],[10,9,0,0,0,0,2,7,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,5,12],[14,8,0,0,0,0,3,3,0,0,0,0,0,0,0,0,14,3,0,0,0,0,3,3,0,0,14,3,0,0,0,0,3,3,0,0] >;
 

44 conjugacy classes

class 1 2A2B2C2D···2I4A···4H4I···4X8A8B8C8D8E···8J
order12222···24···44···488888···8
size11112···22···24···422224···4

44 irreducible representations

dim11111112224
type++++++++-
imageC1C2C2C2C2C2C4D4D4Q8D4○SD16
kernelC4○D4.7Q8C2×C4.Q8C23.25D4M4(2)⋊C4C23.33C23C2×C8○D4C8○D4C2×D4C2×Q8C4○D4C2
# reps133621163144

In GAP, Magma, Sage, TeX 

C_4\circ D_4._7Q_8
 
% in TeX
 
G:=Group("C4oD4.7Q8");
 
// GroupNames label
 
G:=SmallGroup(128,1644);
 
// by ID
 
G=gap.SmallGroup(128,1644);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,521,2804,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=c^2=1,b^2=d^4=a^2,e^2=a^-1*d^2,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,c*b*c=a^2*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
 
// generators/relations
 

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