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G = C4○D4.7Q8order 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○D44C4, 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×C8).589C23, (C2×C4).188C24, C2.3(D4○SD16), C23.433(C2×D4), M4(2)⋊C48C2, C4.Q8.184C22, C2.D8.210C22, C23.25D421C2, (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

C1C4 — C4○D4.7Q8
C1C2C22C2×C4C22×C4C2×C4○D4C2×C8○D4 — C4○D4.7Q8
C1C2C4 — C4○D4.7Q8
C1C22C2×C4○D4 — C4○D4.7Q8
C1C2C2C2×C4 — C4○D4.7Q8

Generators and relations for C4○D4.7Q8
 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 >

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

Smallest permutation representation of C4○D4.7Q8
On 64 points
Generators in S64
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 39 13 35)(10 40 14 36)(11 33 15 37)(12 34 16 38)(17 54 21 50)(18 55 22 51)(19 56 23 52)(20 49 24 53)(41 59 45 63)(42 60 46 64)(43 61 47 57)(44 62 48 58)
(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 43 45 47)(42 44 46 48)(49 55 53 51)(50 56 54 52)(57 59 61 63)(58 60 62 64)
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(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 52)(42 53)(43 54)(44 55)(45 56)(46 49)(47 50)(48 51)
(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 45)(2 64 32 48)(3 59 25 43)(4 62 26 46)(5 57 27 41)(6 60 28 44)(7 63 29 47)(8 58 30 42)(9 17 37 52)(10 20 38 55)(11 23 39 50)(12 18 40 53)(13 21 33 56)(14 24 34 51)(15 19 35 54)(16 22 36 49)

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

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

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

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

Matrix representation of C4○D4.7Q8 in 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] >;

C4○D4.7Q8 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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